s s Rewards& ISSUE 62 AUGUST 2013 Published by the Investment Section of the Society of Actuaries CHAIRPERSON’S CORNER 2013 SECTION LEADERSHIP 2012-2013 COUNCIL YEAR IN REVIEW Officers Thomas Anichini, Chairperson By Thomas Anichini Larry Zhao, Vice Chairperson Thomas Egan, Secretary Deep Patel, Treasurer

Council Members Martin Belanger Steven Chen Paul Donahue James Gannon Frank Grossman his year we initiated a few new efforts to keep you engaged and contribute to your Board Partner Evan Inglis continuing investment education syllabus. Here are a few highlights. Website Coordinator T Thomas Anichini EBSCO BUSINESS SOURCE CORPORATE PLUS (EBSCO) At the 2012 Annual Meeting, the Investment Section Council agreed to purchase access to Other Representatives Chad Hueffmeier, Vol. Work Group Coordinator EBSCO for section members. (Read my piece describing EBSCO elsewhere in this issue, Larry Zhao, Risks & Rewards Liaison “Your newest member benefit…”) If you do not otherwise have access to EBSCO outside Paul Donahue, Redington Prize Coordinator Ed Martin, 2013 Investment Symposium Rep the SOA Investment Section, I think you will find this one of the most valuable aspects Frank Grossman, 2013 Life & Annuity Symposium Rep of your section membership. Ryan Stowe, 2013 Life & Annuity Symposium Co-Rep Steven Chen, 2013 Annual Meeting Rep Response to this new benefit has been stellar. The week we rolled out EBSCO, I received an unsolicited email from a section member that began: “Tom: Wow – this is truly awe-

Joseph Koltisko, Newsletter Editor some. Thank you so much for getting it done. And BTW I’ve never written the SOA to (Chief Editor of this issue) complement them before ...” So if you have not yet explored EBSCO, take some time New York Life Insurance Co to explore it and discover what one of your fellow section members finds so “truly awe- 51 Madison Avenue rm 1113 New York, NY 10010 some.” ph: 212.576.5625 f: 212.447.4251

Nino A. Boezio, Newsletter Editor INVESTMENT SYMPOSIUM AUDIO RECORDINGS (Chief Editor of next issue) Next For the first time, the Investment Symposium sessions have been recorded, thanks to Segal Rogerscasey Canada Submission the generous sponsorship of the Investment Section, and these audio recordings are 65 Queen St. West, Suite 2020 deadline Toronto, ON M5H 2M5 Nov. 11 available for purchase ($20 for non-members, $0 for Investment Section members) via ph: 416.642.7790 f: 416.304.0570 the SOA web store. Visit the SOA website’s Professional Development / Presentation Archives / 2013 Presentations page, where you will find a link to the 2013 Investment SOA STAFF Symposium presentations. Sam Phillips, Jill Leprich, Staff Editor Section Specialist e: [email protected] e: [email protected] The SOA’s media team synchronized the audio recordings with the presentations, which I find makes the viewing experience virtually like that of a webcast. If you like what you Meg Weber, Julissa Sweeney, Staff Partner Graphic Designer hear make sure you attend next year’s Symposium in person. e: [email protected] e: [email protected] INVESTMENT CONTEST Published by the Investment Section of the Society of Actuaries Back in the fall, the council was contemplating some ideas to prompt social networking. This newsletter is free to section members. Current issues are available on the SOA website (www.soa.org). Some ideas we considered included prediction contests, e.g., guess the price of gold, To join the section, SOA members and non-members can the price of oil, the level of the Dow, the yield on the 10-year Treasury, etc. Then we locate a membership form on the Investment Section Web page at www.soa.org/investment eschewed the notion of merely guessing numbers in favor of actually building portfolios This publication is provided for informational and educational and decided to hold an asset allocation contest. purposes only. The Society of Actuaries makes no endorsement, representation or guarantee with regard to any content, and disclaims any liability in connection with the use or misuse of any information provided herein. This publication In April, we solicited entries from section members and received about 120 entries. should not be construed as professional or financial advice. Statements of fact and opinions expressed herein are those Entrants had to build portfolios using combinations of 10 exchange-traded products of the individual authors and are not necessarily those of the Society of Actuaries. (ETPs). We are offering prizes in three categories: cumulative return, lowest volatility, © 2013 Society of Actuaries. All rights reserved.

2 | RISKS AND REWARDS AUGUST 2013 and highest reward/risk ratio, measured over nearly a six-month period. We will announce winners at our hot breakfast at the 2013 Annual Meeting. Thomas M. Anichini, ASA, CFA, is a Senior Investment The original intent of the contest was as a catalyst to social activity—so visit the section Strategist at GuidedChoice. home page and look for the “2013 Investment Contest” link. You can see all the entrants’ He may be reached at tanich- names, results to date, and if you unhide all the sheets, their allocations and predictions. [email protected].

INTERACT WITH YOUR FELLOW SECTION MEMBERS AND YOUR COUNCIL Chances are, on the contest workbook or our LinkedIn sub-group page, you will see some names of friends and former colleagues—ping them via email or say hello via LinkedIn. You may find most council members listed in the SOA member directory, and several of us are members of our LinkedIn sub-group. Share your feedback and suggestions with the council. As a team of volunteers, we are here to facilitate your ideas.

AUGUST 2013 RISKS AND REWARDS | 3 2013 SYMPOSIUM PRESENTER’S DIARY: … | FROM PAGE 1

reforms. My presentation offered as models the regulatory One audience member during my session opined that the responses of Canada, the United States, and the European press had recently reported that U.S. default swap trading Union to three possible causes of the financial crisis: today so nearly resembles pre-crisis levels that Wall Street 1) concentration in over-the-counter (i.e., non-exchange employers are once again recruiting new graduates with an traded) derivatives, 2) questionable imprimaturs by the expertise in the field. Indeed, while the crisis succeeded major credit rating agencies, and 3) anonymous trading in in highlighting the almost unfathomable degree to which “dark pools.” This article both summarizes and supplements institutions trusted this business line, the practice continues that presentation. on a weighty world scale.

OTC DERIVATIVES CREDIT RATING AGENCIES A considerable amount of regulatory response since 2009 The Securities Exchange Commission was authorized by has been focused on credit default swaps, the oft-blamed the Dodd-Frank Act of 2010 to require greater disclo- but seldom understood hedges to many CDO trading sures from nationally registered credit rating agencies. strategies of the last decade. The nearly uniform regula- Subsequent Commission rulemaking mandated that the tory response of requiring “transparent” trading of these agencies, among other things, improve the quality of ratings instruments on regulated exchanges or trading facilities is and provide more transparency in attendant methodolo- designed to, among other things, both increase competition gies. But perhaps the more interesting reform contributed and provide for better pricing. by Dodd-Frank paves the way for potential class actions against the agencies by private plaintiffs. Specifically, by Nearly five years after the onset of the crisis, final rules eradicating its Rule 436(g) and concurrently clarifying the governing the new transparency remain debated. At first required proof of mental state for suits against agencies, the glance, the varied approaches to regulatory rulemaking SEC invited the private class action Bar to the table set by provide some insight into the delay: The European Union Congress in 1933 for plaintiffs against issuers, underwrit- employs an extra-territorial process that begins with the ers, and broker-dealers. European Commission and often ends with legislative action at the Member State level. Canadian regulation, Reforms in the European Union and Canada, while also while driven by the Canadian Securities Administrators guaranteeing agency registration and greater disclosure, (CSA), might vary from province to province. Even the do not similarly empower private plaintiffs. A pending EU United States has seen relevant turf wars between Congress, reform asks commenters for feedback on whether fines the SEC, and the CFTC. against certain entities would deter misleading ratings.

But the political promises of repeal of these reforms now A U.S. Department of Justice civil action from early 2013 appear quixotic, and mandatory measures aimed at greater did allege that a major rating agency overvalued CDOs disclosure are being rolled out in the United States and in 2007, perhaps succeeding foremost in raising issues of the European Union between 2013 and 2015. Meanwhile, timely government response than rating accuracy or ear- in Canada, the provincial responses such as the Quebec nestness. Overall, reforms to date have done little to alter Derivatives Act of 2009 have highlighted the need to pro- the “issuer pays” model of compensation, while also stop- vide exemptions for sophisticated entities serving as coun- ping short of subjecting the agencies to the degree of over- terparties to the subject swap trades. sight reserved for broker-dealers and issuing companies.

4 | RISKS AND REWARDS AUGUST 2013 One audience commenter openly asked whether stricter implemented with finality and publicized with force. As an regulation of the agencies is even possible, given the lack old adage posits, Wall Street can handle anything except of the direct customer relationship that fuels broker-dealer uncertainty itself. supervision. J. Scott Colesanti, LL.M., is an Associate Professor DARK POOLS at the Hofstra University Maurice A. Deane School Earlier this year, The New York Times reported that such of Law, where he has taught Securities Regulation “off Board” trading may have peaked near 40 percent since 2002. He can be reached at (516) 463-6413, of the activity of some exchange listed issues in 2013. [email protected] Nonetheless, there is no immediate plan for American regulators to fit that genie back into the bottle. Likewise, the operational requirements imposed by the EU on dark pools in recent years have actually been credited for their growth.

Canada alone has taken direct action to decrease the flow of trading away from established stock exchanges. CSA rules imposed in October 2012 obligate firms to demonstrate that customer trades filled internally were completed at a price commensurate with the market. Dark pool trading was said to have decreased in excess of 30 percent in the month immediately following, thus raising questions of whether greater regulation may succeed foremost in driving busi- ness to other markets.

CONCLUSION To be sure, reasonable minds can differ on the wisdom of greater scrutiny of credit rating agencies, slowing the wave of off-board stock trading, and publishing details of credit derivative transactions akin to publically available infor- mation about stock trades. Questions of agency capture, the dearth of criminal penalties, and the lingering moral hazard occasioned by both add to the debate. What might best restore confidence in world markets would be closure on the present slate of reforms. In a nutshell, approaches— even where crystallized—remain somewhat undetailed and futuristic at present, perhaps precluding meaningful evaluation of results. After my presentation, I was of firmer conviction that measures such as Dodd-Frank need to be

AUGUST 2013 RISKS AND REWARDS | 5 SOA 2013 LIFE & ANNUITY SYMPOSIUM UPDATE By Frank Grossman & Ryan Stowe

ALBERT MOORE (L) PLAYING CHESS WITH CALEB BOUSU (R).

he late May weather that greeted actuaries attending Eduard Nunes (all the way from Tokyo!) with a 4:1 score. the 2013 Life & Annuity Symposium in Toronto Here is their game from round four: T was particularly clement: sunny with temperatures in the mid-60s. Maple Leafs fans were gleefully walking Caleb Bousu v Eduard Nunes (Closed Sicilian) 1 e4 c5 on air, as their team returned to the Stanley Cup playoffs 2 Nc3 d6 3 f4 Nf6 4 Nf3 Nc6 5 Bb5 g6 6 O-O Bg7 7 d3 after a nine-year hiatus. Yet Mother Nature has a way of O-O 8 Bxc6 bxc6 9 Qe1 Rb8 10 Qh4 Ne8 11 e5 dxe5 12 asserting herself when presented with anomalous phe- fxe5 Rb4 13 Ne4 f5 14 exf6 exf6 15 c3 Rb5 16 a4 Rb7 nomena. In the seventh game of their preliminary round 17 Nxc5 Qb6 18 d4 Re7 19 Re1 Rxe1+ 20 Qxe1 Nd6 21 series with the Leafs, the Beantown Bruins scored twice Bf4 Re8 22 Qd1 Nc4 23 b4 a5? 24 Qb3! Be6 25 Nxe6 in the final 1:22 of regulation time to force overtime, and Rxe6 26 Qxc4 Kf7 27 Re1 c5 28 Rxe6 Qxe6 29 Qxe6+ duly sent the Toronto side and their supporters packing in Kxe6 30 dxc5 Kd7 31 b5 f5 32 Be5 Bf8 33 Bd4 h6 … 1:0 the extra frame—leaving the Ottawa Senators as the sole Canadian squad to advance to the quarter-finals. And the Plans for the second speed chess networking tournament, north wind delivered a frost warning to parts of southern to be held on the Tuesday evening of the upcoming 2013 Ontario during the week following the symposium, after Annual Meeting in San Diego, are already underway. [FG] most out-of-town attendees had safely returned home. INVESTMENT SECTION HOT BREAKFAST THOMAS C. BARHAM III SPEED CHESS (SESSION 04) TOURNAMENT (SESSION 01) A good mix of both larks and owls turned out for this early A dozen chess enthusiasts met on the Sunday afternoon Monday morning (7:00 am!) session. Section news and prior to the meeting for the inaugural Thomas C. Barham views were delivered by Ryan Stowe and Frank Grossman, III Speed Chess Tournament. This networking event is including a look back to the March Investment Symposium named for a prominent chess-playing member of the in New York, and a look forward to the Investment Section- SOA, and was jointly sponsored by the Technology and sponsored sessions at the 2013 Annual Meeting in San Investment Sections as an opportunity for members to make Diego. Nino Boezio arrived with perfect timing to share new acquaintances, and have fun playing 10-minute speed some observations about Risk & Rewards from his perspec- chess too. tive as our long-standing co-editor, inviting those in atten- dance to consider becoming contributors. And the section’s A local chess organizer, Ted Winick, and his colleagues new staff partner, David Schraub, was briefly introduced. at the Annex Chess Club were enlisted to run the event. Tournament directors Alex Ferreira, and Shabnam Abbarin, Geoff Hancock then gave a short talk titled, “Economic handled the pairings and myriad details with aplomb. Scenario Generators: All models are wrong—so now Interestingly, Ted’s day job as the president of the Chess what?” in which he offered a pungent commentary for Institute of Canada is devoted to establishing chess—as those actuaries able to smell the coffee at that early hour. a teaching tool for fundamental concepts such as pattern During his engaging presentation, Geoff surveyed things recognition, abstract thinking, and problem solving—on the that actuaries have done fairly well (e.g., increased facil- grade school mathematics curriculum. ity with stochastic modeling), some things that haven’t been done that well (e.g., over-reliance on point estimates Congratulations to Caleb Bousu who finished first after five when delivering results rather than ranges) and areas where rounds with a clean 5:0 record. And second place went to we’ve really fouled-up (e.g., making the use of ESGs and

6 | RISKS AND REWARDS AUGUST 2013 FROM LEFT TO RIGHT: TOURNAMENT CO-ORGANIZ- HANS AVERY (L) PLAYING CHESS WITH RYAN STOWE (R). ER ALBERT MOORE, AND TOURNAMENT DIRECTORS SHABNAM ABBARIN AND ALEX FERREIRA, AWARD FIRST PLACE CERTIFICATE AND PRIZE TO CALEB BOUSU. stochastic models “too academic” and opaque for senior fewer assets (on a per client basis; the middle market itself management to trust). The breakfast session concluded with is quite large) than the affluent and mass affluent markets, a book-draw, and Leonard Mangini won a copy of Dan and this is one of the challenges of serving the middle Areily’s Predictably Irrational. [FG] market. The panelists focused on the challenges advisors face in allocating assets of the middle market to maximize PERSPECTIVES ON LIFE INSURANCE AND the retirement income outcome dilemma through a concept ANNUITIES IN THE MIDDLE MARKET similar to the efficient frontier in the investment world. [RS] (SESSION PD-25) This session was a panel discussion delivered by Douglas DISCOUNT RATES FOR FINANCIAL Bennett, Robert Buckingham, and Walter Zultowski, REPORTING PURPOSES: ISSUES AND moderated by Ryan Stowe, and co-sponsored with the APPROACHES (SESSION PD-32) Marketing and Distribution Section. The life insurance This session, presented in conjunction with the Financial portion of this session focused on recent research from Reporting Section, dealt with the International Actuarial the second half of 2012. The life insurance middle mar- Association’s (IAA) recent work developing an educa- ket (defined as “Young Families” age 25–40 with annual tional monograph on discount rates for financial reporting household income between $35,000–$125,000, and at least purposes. David Congram lead off with some background one dependent in the household) was segmented into three about the project and its sponsors, as well as the IAA’s groups, each having different attitudes and beliefs about broader educational mandate. Andy Dalton then provided their need for life insurance, as well as different motiva- an overview of the monograph itself, followed by some tions for purchasing (or not purchasing) life insurance. comments contrasting risk free rates in theory and in prac- “Protectors” buy life insurance to meet a need rather than tice—accompanied by the session’s most memorable slide. based on a strong belief in the product. “Planners” most Next, Derek Wright spoke about setting discount rates for likely perceive and appreciate the long-term value of life pass-through products. David returned to the lectern to dis- insurance. “Opportunistic buyers” have less belief (e.g., cuss the evolving influence of sovereign and political risk, perceived value) in the product, and typically purchase less and Frank Grossman concluded the session with some brief coverage than other segments. They also tend to buy most comments about replicating portfolios. [FG] products at their place of employment. The presentation provided insight into the different methods that companies DO LONG-TERM GUARANTEES IN can use to segment their target markets through data mining INSURANCE PRODUCTS MAKE SENSE? and predictive modeling, all with a view to making their (SESSION D-69) marketing efforts more effective by targeting their custom- The format of this well attended session was unique; a ers’ preferred buying style. “facilitated debate” between four panelists, moderated by Emile Elefteriadis. Jeff Adams and David Harris offered a The annuity portion of the session offered a different per- Canadian perspective, while Michael Downing and Michael spective on the middle market. The majority of the people LeBoeuf delivered their comments from an American van- who purchase annuities are age 50 or older. In fact, 26 per- tage point. The panelists addressed various questions posed, cent of Americans are baby boomers, and each day 10,000 and attendees were also invited to share their views through of them turn 65. What does this mean? That the pre-retiree the use of handheld interactive voting devices. and retired population need the guaranteed benefits that annuities can deliver. By definition, the middle market has CONTINUED ON PAGE 8

AUGUST 2013 RISKS AND REWARDS | 7

PRESENT VALUES, INVESTMENT RETURNS AND DISCOUNT RATES – PART 1 By Dimitry Mindlin

his is the first of a two-part article. The first pro- returns are uncertain, a single discount rate cannot encom- vides the groundwork for exploring the different pass the entire spectrum of investment returns, hence the T formulations of the discount rate, to support various selection of a discount rate is a challenge. In general, the sorts of objectives. It provides some useful rule of thumb asset value required to fund an uncertain financial commit- for estimating quantiles in the distribution of discount rates ment via investing in risky assets—the present value of the and for relating geometric and arithmetic discount assump- commitment—is uncertain (stochastic).1 tions in a defined series of returns. The second part, to follow in early 2014, applies this approach to examples of While the analysis of present values is vital to the process of a stochastic distribution of returns. funding financial commitments, uncertain (stochastic) pres- ent values are outside of the scope of this paper. This paper The concept of present value lies at the heart of finance in assumes that a present value is certain (deterministic)—a general and actuarial science in particular. The importance present value is assumed to be a number, not a random vari- of the concept is universally recognized. Present values of able in this paper. The desire to have a deterministic present various cash flows are extensively utilized in the pricing of value requires a set of assumptions that “assume away” all financial instruments, funding of financial commitments, the uncertainties in the funding problem. financial reporting, and other areas. In particular, it is generally necessary to assume that all A typical funding problem involves a financial commitment future payments are perfectly known at the present. The (defined as a series of future payments) to be funded. A next step is to select a proper measurement of investment financial commitment is funded if all payments are made returns that serves as the discount rate for present value cal- when they are due. A present value of a financial commit- culations. This step—the selection of the discount rate—is ment is defined as the asset value required at the present to the main subject of this paper. fund the commitment. One of the main messages of this paper is the selection of Traditionally, the calculation of a present value utilizes the discount rate depends on the objective of the calcula- a discount rate—a deterministic return assumption that tion. Different objectives may necessitate different discount represents investment returns. If the investment return and rates. The paper defines investment returns and specifies the commitment are certain, then the discount rate is equal their relationships with present and future values. The key to the investment return and the present value is equal to measurements of investment returns are defined in the con- the sum of all payments discounted by the compounded text of return series and, after a concise discussion of capital discount rates. The asset value that is equal to this present market assumptions, in the context of return distributions. value and invested in the portfolio that generates the invest- The paper concludes with several examples of investment ment return will fund the commitment with certainty. objectives and the discount rates associated with these objectives. In practice, however, perfectly certain future financial com- mitments and investment returns rarely exist. While the 1. INVESTMENT RETURNS calculation of the present value is straightforward when This section discusses one of the most important concepts returns and commitments are certain, uncertainties in the in finance—investment returns. commitments and returns make the calculation of the pres- ent value anything but straightforward. When investment

10 | RISKS AND REWARDS AUGUST 2013

3. INFLATION IN A TRADITIONAL 5. ACCESSING EXPOSURE TO THE INVESTMENT PORTFOLIO INFLATION MARKET A traditional equities/bonds portfolio relies on stable cor- Inflation can be traded through a variety of instruments. relation assumptions to produce diversification. Stocks and These include: TIPS, Total Return Swaps, ETFs, Inflation- bonds should produce diversification in an economy domi- linked Notes, Inflation Swaps and Inflation Options. nated by growth. However, periods of increased inflation or inflation uncertainty can have a negative impact on both TIPS (TREASURY INFLATION PROTECTED equities and bonds. SECURITIES) TIPS are securities issued by the U.S. government that offer Equities: Dividend streams may be pressured if inflation investors inflation protection. The principal is accredited leads to slower economic growth while the discount rate daily based on the Headline CPI index and is repaid at is increased due to higher nominal rates and uncertainty. maturity subject to a minimum of par, thus providing defla- tion protection. Semi-annual coupons paid on TIPS are Bonds: A higher discount rate would similarly lead to based on the inflation-adjusted principal. The TIPS market lower valuations on fixed rate bonds. is the largest inflation-linked market in the world. Regular auctions are conducted in 5y, 10y and 30y TIPS. The stagflation period of the late 1970s and early 1980s in the United States was characterized by low single-digit The U.S. government issued approximately $150 billion portfolio returns coupled with high correlation (above 50 of TIPS in 2012 and is expected to issue the same or more 1 percent) between equities and bonds. in 2013. Total market value of outstanding TIPS exceeds $900 billion and average daily trading volume is $11 billion 4. RISK FACTOR APPROACH TO ASSET (Source: Federal Reserve Bank of New York). ALLOCATION In asset allocation circles, an increasingly favored approach In addition to investing in TIPS, funds can also obtain expo- is to focus on diversification among different sources of sure to TIPS as an overlay, by entering into a Total Return risk premiums rather than simply diversifying among asset Swap on a TIPS Index. In a total return swap, one party classes. pays the return of the index in exchange for a funding rate, which could be quoted either as a fixed rate or as a spread to Not only can CPI-linked instruments provide inflation pro- a floating rate. TIPS indices are typically market-weighted, tection, but it is also possible to use these tools to source and are available for both the aggregate TIPS market and risk premiums due to a number of features of the CPI-linked for specific maturity buckets. market. These features include: Some portfolio managers have employed total return swaps - Deflation risk premium in short-dated TIPS and swaps, to synthetically replicate inflation-linked credit portfolios. - Asset swap premium in TIPS, and The cash is utilized to invest in high-yield bonds to obtain a - Tail risk premium in inflation options. credit risk premium, and a total return swap on a TIPS index provides inflation exposure as well.

CONTINUED ON PAGE 20

AUGUST 2013 RISKS AND REWARDS | 19 THE U.S. INFLATION MARKET… | FROM PAGE 19

FOCUS ON DIFFERENT SOURCES OF “ RISK PREMIUMS. “

A third way to obtain exposure to TIPS is via ETFs that Banks and other parties may issue Inflation-linked Notes reference TIPS indices2 or ETNs that reference Inflation that include embedded inflation swaps and inflation options. Expectations as implied by the difference in yield between This provides access to these markets for clients who do not TIPS and Treasuries.3 typically participate in the underlying derivative markets.

5.2. INFLATION SWAPS AND OPTIONS Inflation swaps and options have grown significantly over Inflation swaps offer a mechanism to trade inflation over the past several years and are actively traded across a wide a given time horizon, with mechanics similar to nominal range of maturities and strikes. interest rate swaps. At maturity, one party pays the cumu- lative percentage increase in the reference inflation index 6. INFLATION RISK PREMIUM STRATEGIES over the life of the swap in exchange for an annually com- 6.1. SHORT-DATED INFLATION-LINKED pounded fixed rate, known as the breakeven inflation rate. BONDS AND SWAPS Short-dated inflation-linked bonds and swaps tend to have An example of the above is if the fixed rate quoted on a an embedded deflation risk premium. Hence, the implied five-year inflation swap was 2 percent, and the CPI index inflation rate in TIPS and inflation-linked swaps has tended rose from a level of 220 to 255 during the five-year period to systematically under-predict realized inflation over short- (3 percent per annum inflation), a net payment of 5.50 per- term horizons. cent of notional would be paid to the buyer of inflation. This is slightly more than 1 percent per annum due to the effect A structural reason for this effect is that most TIPS funds of compounding. are benchmarked to indices that only include TIPS with greater than one-year to maturity. Accordingly, most TIPS It is also possible to obtain exposure to a measure of core funds are forced to sell TIPS as soon as their maturities fall inflation via inflation swaps that reference the DB Core below one year in order to reduce tracking error. Money- U.S. CPI Index, thus mitigating volatility due to fluctua- market funds cannot buy these short-dated TIPS as they are tions in energy prices. limited to fixed-rate debt, and TIPS interest is floating (with inflation). Accordingly, the lack of natural buyers of short U.S. CPI can be used as a proxy hedge for other correlated dated TIPS results in implied inflation (TIPS breakeven markets such as Canadian Inflation. Similarly, DB Core rates) being underpriced at the front-end of the curve. U.S. CPI can be used as a proxy for more specific measures of inflation such as property inflation. Graph 2 (pg. 21, top) shows historical1y inflation swaps and one-year realized inflation over the corresponding periods. Inflation Options provide asymmetric returns relative to CPI. Caps provide payoffs when inflation exceeds a strike Inflation swaps, which typically imply higher inflation rates (e.g., 4 percent), and Floors pay out when inflation is below than TIPS, have under-predicted realized inflation by more an agreed strike (e.g., 0 percent in the case of a “deflation” than 0.50 percent. Note: past performance is no guarantee floor). There are two-types of inflation options that regu- of future results. larly trade. “Year-On-Year” Options pay annually based on each year’s inflation rate, whereas “Zero Coupon” Options Volatility of returns can be affected by energy prices. pay out on the final maturity date based on cumulative infla- Accordingly, returns are less volatile if energy moves are tion over the period. hedged-out. For example, this can be achieved by buying one-year DB Core U.S. CPI Inflation Swaps.

20 | RISKS AND REWARDS AUGUST 2013

THE U.S. INFLATION MARKET… | FROM PAGE 21

6.3 DEFLATION FLOOR RISK PREMIUM One way in which the deflation risk premium embedded in Inflation options are often bought as tail-risk hedges. Equity inflation floors can be earned is by systematically selling macro hedge funds have purchased deflation floors as an year-over-year deflation floors. alternative to equity put options due to their relatively low premiums and expected good performance in deflationary Graph 4 (below) reflects the performance of selling 5y year- markets. over-year 0 percent-strike deflation floors and rolling the position monthly. Note: past performance is no guarantee As a result of these purchases, inflation options appear rich of future results. under various metrics, such as comparing realized with implied volatility; or implied deflation probabilities relative The above strategy can be implemented by either sell- to the distribution of inflation rates predicted by surveys of ing the deflation floors directly or by entering into a total professional forecasters. Accordingly, inflation caps embed return swap on an index which is designed to replicate this an inflation risk premium, and inflation floors embed a strategy. deflation risk premium. Generally, floors are considered richer than caps, especially since the FOMC is averse to 7. CONCLUSION deflation and would likely take extreme steps to prevent Large and liquid markets for U.S. CPI-linked exposure can deflationary scenarios. be accessed to obtain inflation protection. Regularly traded CPI-linked instruments include TIPS, Inflation Swaps and GRAPH 4 Inflation Options. Products based on these instruments include Total Return Swaps and Asset Swaps, as well as ETFs, ETNs and Inflation-linked Notes.

150 Not only can these instruments be used to obtain inflation protection, but they can also be used to earn risk premiums 140 that arise due to structural imbalances in the CPI market. Examples include cheapness in the front-end of the inflation 130 curve, relative value between inflation swaps and TIPS, and richness of deflation floors. 120 Much of the content for this paper was sourced from the Deutsche 110 Bank presentation and webinar titled, “Inflation Risk Factor and Risk Premia Strategies.” For access to the presentation or webinar, or for 100 further information, please contact the author: email: allan.levin@ db.com.

90 Risk Magazine ranked Deutsche Bank No.1 for US Inflation Dec-09 Jun-10 Dec-10 Jun-11 Dec-11 Jun-12 Swaps and No.1 for US Inflation Options for 2012.

GRAPH 4 Greenwich Associates ranked Deutsche Bank No.1 for Global Fixed Income for 2012, 2011 & 2010 and No.1 Overall US Fixed Income for De ation Floor Strategy 2012, 2011 & 2010.

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LAYERING YOUR OWN VIEWS INTO A STOCHASTIC SIMULATION—WITHOUT A RECALIBRATION By Tony Dardis, Loic Grandchamp and David Antonio

Editor’s Note: This article summarizes the authors’ presentation at the May 2013 Life and Annuity Symposium.

here is a useful technique that is gaining some The maximum entropy for these scenarios will always popularity amongst practitioners that enables be achieved by equally weighting these scenarios, and T “own views” to be superimposed on a stochasti- is calculated as follows: cally generated scenario set without having to reca- FIVE SCENARIOS EQUALLY WEIGHTED librate the underlying model(s). The technique won’t One-year be appropriate for all applications, but it may be effec- projected tive for some, such as superimposing alternative esti- Scenario rate Weight Entropy mates or creating “what-if” scenarios and stress tests. 1 2.0% 0.20 0.3219 2 2.0% 0.20 0.3219 The approach involves the use of a statistic known as 3 3.0% 0.20 0.3219 entropy. The value of entropy is maximized when equal 4 4.0% 0.20 0.3219 weighting is given to each individual scenario in a given scenario set. Thus the objective of the “own views” exercise 5 4.0% 0.20 0.3219 is to re-weight scenarios so that they hit your specific target, Avg/total 3.0% 1.00 1.6094 while maximizing the value of entropy. Let us now say that we have an alternative view as to what will transpire in the future and instead would like to Mathematically, for a set of N scenarios each with weight re-weight the scenarios so that on average we hit a lower Wi the entropy S of a scenario set is defined as follows: rate—say, 2.5 percent. Our first inclination might be to give N weighting only to the lowest rates from our original set of wS i ln w i five scenarios, with entropy calculated as follows: i 1 TARGET = 2.5 PERCENT; There is potentially an infinite combination of weightings CHOOSE ONLY THE VERY LOW WEIGHTS that would hit any specified target that we may have. The One-year objective of the entropy technique is to find the optimal projected weights Wi so that we maximize S while hitting our specific Scenario rate Weight Entropy target. 1 2.0% 0.25 0.3466 2 2.0% 0.25 0.3466 A simple example will help reinforce the concept. 3 3.0% 0.50 0.3466

Let us assume that we have calibrated a one-year interest 4 4.0% 0.00 0.0000 rate model so that on average it hits 3 percent. We then use 5 4.0% 0.00 0.0000 the model to generate five scenarios as follows: Avg/total 2.5% 1.00 1.0397

SCENARIO ONE-YEAR However, weighting only the very low rates is missing a lot RATE PROJECTED of very important information about the overall distribution 1 2.0% of the rates and this becomes apparent from the entropy

2 2.0% value—a much lower number than what we started with for the original scenario set that as equally weighted. So let’s 3 3.0% now consider what happens if we give some weighting to 4 4.0% all the scenarios, while still hitting the “own views” target 5 4.0% of 2.5 percent: Average 3.0%

24 | RISKS AND REWARDS AUGUST 2013

LAYERING YOUR OWN VIEWS INTO A STOCHASTIC... | FROM PAGE 25

the newly emerging statutory stochastic requirement), there In Chart 2 (pg. 25, bottom) we show a revised distribution may be some applications, such as testing of a new product of the 20-year Treasury bond equivalent yield projected campaign where interest rate risk is the only market driver over a 10-year horizon that starts with the 12/31/2012 of the business, where this is appropriate. This in turn leads 10,000 Academy scenarios, but reweights using maximum to the natural question: if I download the Academy’s 10,000 entropy in order to maintain the current level of the 20-year scenarios, is there a way I can rebalance these so that they rate over the next five years. produce an average interest rate path that is different to what is assumed in the Academy calibration? There are some very interesting features of the new distribu- tion that should be highlighted: The entropy technique can be used extremely effectively in • The entropy technique has worked beautifully in hitting such an example, and has great flexibility. In Chart 1 (pg. our “own views” path on average. 25, top) we show the distribution of the 20-year Treasury • The overall characteristics of the probability distribution bond equivalent yield projected over a 10-year horizon in terms of dispersion and tails are similar under the origi- under 10,000 Academy scenarios. These scenarios were nal and the reweighted scenario sets. generated from the Academy generator, initialized to the • It will be noted that the lower band of the reweighted set at Treasury yield curve at 12/31/2012. the second percentile level is well below the original set. This doesn’t mean to say that we are weighting scenarios As will be immediately apparent, the average path of the that were outside the original scenario set, but rather that 20-year rate under the Academy calibration immediately we are now giving a lot more weighting to scenarios that sets off on an upward trend which persists throughout the were originally outside the second percentile level. This projection period. What if our “own view,” however, was highlights another important characteristic of the entropy that given the current economic climate, and the very high method—it will not work where our target falls outside expectation that the government will persist in a monetary any of the scenarios that were originally generated. In this policy that continues to keep interest rates at extremely low example, this would mean that we may not be able to use levels, a much more realistic expectation is that on average entropy to target ultra-low 10-year rates, e.g., at or close rates will remain at or very close to today’s levels for at to 1 percent for a prolonged period. least the next five years? • The entropy of our reweighted scenario set corresponds to a set of 8,353 equally weighted scenarios. This num- Our first port of call might be to download the Academy’s ber, called the effective number of scenarios is a useful interest rate generator from actuary.org and recalibrate that statistic of the entropy method which allows practitioners so that it hits our target. The path of the 20-year rate in the to gauge how far apart the original and reweighted sets generator can be controlled using two parameters: long- are. However, the technique should not be viewed as a term mean reversion level and a speed of mean reversion. scenario reduction technique. There isn’t, therefore, sufficient flexibility to target a more general path for interest rates and the user is also con- Note that while this article has focused on looking at inter- strained because he can only directly control the evolution est rate scenarios and how we can reweight according to of the 20-year rate. It may also not be obvious to the user “own views” around a target path, other variables and target what parameters should be input to achieve the desired path. metrics could be used equally effectively. For example, we Perhaps entropy can help? might be more interested in setting a target for returns rather than yield, and perhaps it is equity rather than interest rate scenarios that are of most interest. Indeed, theoretically it

26 | RISKS AND REWARDS AUGUST 2013

AUMANN-SHAPLEY VALUES: A TECHNIQUE FOR BETTER ATTRIBUTIONS By Joshua Boehme

“HAPPY IS THE ONE WHO KNOWS THE CAUSES OF THINGS.” –VIRGIL1

ctuaries encounter attribution problems on a regu- Although likely few of us have ever tried to formally list lar basis. Indeed, any situation where results the properties we want a “good” attribution to satisfy, intui- change, whether due to changes in assumptions, tively we have an idea of how a reasonable method should A th market conditions, or even just the passage of time, often behave. For example, if the i variable did not change (so

leads to the natural follow-up question: why did the results ui = vi), we would expect its contribution to the difference change? To answer this question, actuaries use various to equal zero. Similarly, if the ith variable has no impact on

techniques, each with its own strengths and weaknesses. the value of f (meaning that f(u) = f(v) whenever ui ≠ vi and

However, a technique not widely known among actuar- uj = vj for all j ≠ i), then we again expect its contribution to ies—the Aumann-Shapley value from game theory—in equal zero. many cases can produce attributions that better satisfy our intuitive expectations of a good attribution. ATTRIBUTION TECHNIQUES Aumann-Shapley Some authors have already applied the Aumann-Shapley The technique that this article focuses on, the Aumann- approach to various financial problems in non-actuarial set- Shapley value, requires f and its parameters to satisfy a few tings. Denault (1999), for example, used it to allocate mar- conditions. Specifically, f must have partial derivatives2 in gin requirements among portfolios of options. However,Attribution the Techniquesall of its parameters along the vector between u and v. We Aumann-Shapley approach has yet to receive Aumann-Shapleywidespread do not need a closed-form version of f, but we must have exposure within the actuarial community. The technique that this article focuses on, the Aumann-Shapley value, requires f and its a way to compute its partial derivatives at any given point 2 parameters to satisfyon the a path.few conditions. Specifically, f must have partial derivatives in all FORMALIZING THE PROBLEM of its parameters along the vector between u and v. We do not need a closed-form version of f, but we must have a way to compute its partial derivatives at any given point on the Suppose we have a multivariate function f and two vectors For attribution problems that satisfy these requirements, the path. of parameters u and Attributionv representing Techniques the previous and latest Aumann-Shapley approach produces some valuable results. parameters respectively.Aumann-Shapley In the most general formFor attribution of the problemsIt always that produces satisfy thesean attribution requirements, with the no Aumann-Shapley unexplained approach The technique that this article focuses on, the Aumann-Shapley3 value, requires f and its problem we place almost no restrictions on the functionproduces f orsome valuableamount. results.As we Itwill always see 2in produces later examples, an attribu itstion results with also no unexplained parameters to satisfy a few conditions. Specifically,3 f must have partial derivatives in all its parameters. In someof its applications parameters along f may the vectorcontainamount. between discon u As and- we v. showWe will do see anot certain needin later a desirableclosed-form examples, stability version its results with respectalso show to how a certain we set desirable tinuities or may lack ofa closed-formf, but we must havesolution. a way Theto computestability function its with partial f respect derivativesup the to problem. how at any we given set pointup the on problem.the could take non-continuouspath. parameters as well. For example, For each variable we calculate the attributed amount a as: a binary variable couldFor indicate attribution whether problems to that use satisfy one thesemethod requirements, For eachthe Aumann-Shapley variable we calculate approach the attributedi amount ai as: or another, such as curtateproduces versus some valuablecontinuous results. mortality. It always produces an attribution with no unexplained1 3 f amount. As we will see in later examples, its results also show a certain desirable ai  (vi  ui )  (1  z)u  zv dz stability with respect to how we set up the problem. 0 xi In an attribution problem we seek to explain the difference th The resulting integral does not always have a closed-form solution, but we can evaluate it f(v)-f(u) by assigningFor to each the variable i variable we calculate an amount the attributed a rep amount- The a as: resulting integral does not always have a closed-form numerically.i i resenting its contribution to the difference, where i ranges1 f solution, but we can evaluate it numerically.     from 1 to the number of inputs to the functionai f.(Alternatively,vi ui )  (1 wez) ucanzv viewdz the Aumann-Shapley approach as a three-step process: 0 xi The resulting integral does not always have a closed-formAlternatively, solution, but we we can can view evaluate the it Aumann-Shapley approach f 1. Find the partial deri vative of f with respect to its ith parameter, denoted here. Ideally the total of thenumerically. ai values would equal f(v)-f(u), but as a three-step process: xi in practice that doesAlternatively, not always we happen. can view Anythe Aumann-Shapley remaining approach as a three-step process: th 2. Integrate1. that Find partial the partialderivative derivative along th ofe linef with segment respect between to its ui and v. Here, the difference, which sometimes goes by the term untraced or th f 1. Find the partial derivative of f withdummy respect tovariable itsparameter, i parameter, z represents denoted denoted the linear here.here. interpolation between u (at z = 0) and v (at unexplained, represents some portion of the change that the xi z = 1). attribution method in question2. Integrate could that not partial allocate derivative to onealong of th e line segment between u and v. Here, the the input variables. dummy variable z represents 3.the linearMultiply interpolation the result between by theu (at change z = 0) and in v that (at parameter (vi – ui) z = 1). 3. Multiply the result by the change in that parameter (v – u ) Step-Through i i Many actuaries faced with an attribution problem will solve it by stepping through the Step-Through parameters one at a time, a technique with several important advantages. As long as we 28 | RISKS AND REWARDS AUGUST 2013Many actuaries faced with an attribution problem will solve it by stepping through the parameters one at a time, a techniquecan evaluatewith several the important function advantages. f at each As combination long as we along the step-through, f can have any can evaluate the function f at eachnumber combination of discontinuities along the step-through, and it fcan can lackhave anya closed-form solution. We can use a step- number of discontinuities and it throughcan lack a evenclosed-form when solution. f has non-continuous We can use a step- inputs. Furthermore, a step-through also through even when f has non-continuousproduces inputs. an attribution Furthermore, with a step-through no unexplained also amount. produces an attribution with no unexplained amount.

Step-throughs have one well-knownStep-throughs disadvantage, have though: one the well-known results depend disadvantage, on the though: the results depend on the arbitrary order we use to step througharbitrary the parameters. order we In use the toexamples step through in this article the parameters. we In the examples in this article we will use a modified technique to willovercome use athis modified issue: we technique will perform to the overcome step-through this issue: we will perform the step-through

2The existence of partial derivatives also implies the continuity of f. The partial derivatives can contain non-removable discontinuities, but f itself2The cannot existence contain ofany partial along the derivatives path. also implies the continuity of f. The partial derivatives can contain 3Except to the extent that whatever tool we use to calculate the results has finite precision, though this issue applies to any attribution technique. non-removable discontinuities, but f itself cannot contain any along the path. 3Except to the extent that whatever tool we use to calculate the results has finite precision, though this issue applies to any attribution technique. for every possible order, then average the attributions together.4 This removes the 4 dependency on an arbitrarily chosenfor order. every As possible we will order, see with then a averagelater example, the attributions though, together. This removes the even this modified step-through methoddependency still has on a significantan arbitrarily weakness. chosen order.Despite As that, we willin see with a later example, though, 2. Integrate that partial derivative along the line segmentsituations wherePartial we cannotderivatives satisfy have theeven requi asthis onerements modified major of step-through theadvantage Aumann- theirmethodShapley fre still approach,- has a significant a weakness. Despite that, in between u and v. Here, the dummy variable zstep-through repre- quentremains ease a viable of computation alternative.situations and where interpretation. we cannot satisfyFor example, the requi rements of the Aumann-Shapley approach, a sents the linear interpolation between u (at z = 0) and v from the duration ofstep-through a bond, a relatively remains a intuitiveviable alternative. concept, (at z = 1). Partial Derivativeswe can quickly estimate the change in its value due to a Partial Derivatives Actuaries alreadychange frequently in interest use derivativesrates. or approximations to the derivative to perform attributions.for Someevery possible partial order,derivativesActuaries then average alreadycome the up attributionsfrequently so often together. thatuse theyderivatives4 This have removes specific or approximationsthe to the derivative to 3. Multiply the result by the change in that parameternames, such as the dependency“Greeks” (delta,on an arbitrarilyperform gamma, chosen attributions.vega, order. rho, As theta, Somewe will etc.) partial see or with duration. derivatives a later example, In somecome though, up so often that they have specific even this modified step-through method still has a significant weakness. Despite that, in (vi – ui) cases, we useThe formulas main to difference directly calculatenames, between such the apartial as partial the derivatives;“Greeks” derivative (delta, other attribu gamma, times,- we vega, shock rho, theta, etc.) or duration. In some one of the parameterstion andsituations athe small Aumann-Shapleywhere amount we cases,cannot to numerically satisfywe use approach the formulas requi estimaterements comes to directly ofthe the from derivative. Aumann- calculate whereShapley the approach, partial derivatives;a other times, we shock Step-Through step-through remains a viable alternative. we evaluate the partialone derivative.of the parameters In the a Aumann-Shapley small amount to numerically estimate the derivative. Many actuaries faced with an attribution problem will Partial derivativesapproach, havePartial aswe Derivatives one evaluate major advantageit along the their entire frequent path betweenease of computation u and and solve it by stepping through the parameters one at ainterpretation. time, a For example,Actuaries already from frequentlythePartial duration derivativesuse derivatives of a bond, have or aapproximations reaslatively one major intui to advantagethetive derivative concept, their to frequent ease of computation and technique with several important advantages. As longwe as can we quickly v. estimateForperform the partialthe attributions. change derivative,interpretation. inSome its partialvalue we derivativesdueevaluate For to example,a changecome it at up a frominsosingle interestoften the thatpoint, duration rates.they have of specific a bond, a relatively intuitive concept, usuallynames, the suchbeginning as the “Greeks”we point. can (delta,quickly5 This gamma, difference,estimate vega, therho, though, changetheta, etc.) inleads or its duration. value dueIn some to a change in interest rates. can evaluate the function f at each combination along the cases, we use formulas to directly calculate the partial derivatives; other times, we shock step-through, f can have any number of discontinuiThe main- differenceto theone major between of the drawbackparameters a partial a smallofderivative a amountpartial attribution to derivative numerically and estimateapproach: the Aumann-Shapley the derivative. the approach comesattribution from where generally we evaluateThe has main thea nonzero partialdifference derivative. unexplained between In a the partial amount. Aumann-Shapley derivative attribution and the Aumann-Shapley ties and it can lack a closed-form solution. We can use Partial derivatives have as one major advantage their frequent ease of computation and approach, weThis evaluate may it suffice along the for entire approacha quick path estimate.comes between from uOther and where v. times, For we theevaluate though, partial the derivative, partial derivative. In the Aumann-Shapley a step-through even when f has non-continuous weinputs. evaluate it at a singleinterpretation. point, For usually example,approach, the from beginning we the evaluateduration point. of it a5 along bond,This a difference,the relatively entire intui path though,tive between concept, u and v. For the partial derivative, we maywe canwant quickly a complete estimate the attribution change in its ofvalue the due difference. to a change in interest rates. 5 Furthermore, a step-through also produces an attributionleads to the major drawback of a partialwe evaluate derivative it at approach: a single poi thent, attri usuallybution the generally beginning has point. This difference, though, with no unexplained amount. a nonzero unexplainedThe main amount. difference Thisleads between may to suffi athe partialce major for derivative a drawback quick attribution estimate. of a andpartial Other the Aumann-Shapley derivative times, approach: the attribution generally has though, we mayEXAMPLES wantapproach a complete comes from atatribution wherenonzero we of evaluateunexplained the difference. the partial amount. derivative. This In may the Aumann-Shapley suffice for a quick estimate. Other times, approach, we evaluatethough, it along we the mayentire wantpath between a complete u and v.at tributionFor the partial of thederivative, difference. Example 1: Zero-Coupon Bond 5 Step-throughs have one well-known disadvantage, though: we evaluate it at a single point, usually the beginning point. This difference, though, Examples Considerleads ato zero-couponthe major drawback 10-year of a partial bond derivative with aapproach: maturity the valueattribution generally has the results depend on the arbitrary order we use Exampleto step 1: Zero-Coupona nonzero Bond unexplained Examples amount. This may suffice for a quick estimate. Other times, through the parameters. In the examples in this articleConsider we a zero-couponof $1 though,million. 10-year we mayFor wantbonda givenExample a completewith yield a 1:maturity at Zero-Coupontotribution maturity value of the ofy,difference. Bond $1the million. following For a given will use a modified technique to overcome this issue:yield weto maturity formula y, the gives following its value foConsiderrmula at time gives a zero-coupont: its value at time10-year t: bond with a maturity value of $1 million. For a given Examples yield to maturity y, the following formula gives its value at time t: will perform the step-through for every possible order, then Example 1: Zero-Couponf (y,t) Bond e y(10 t ) 106 average the attributions together.4 This removes the depen- Consider a zero-coupon 10-year bond with a maturity value of $1 million. Fory(10 a givent ) 6 Suppose we have theyield following to maturity parameter y, the following sets: fo rmula gives its value at timef (y t:, t) e 10 dency on an arbitrarily chosen order. As we will see with Suppose we have theSuppose following we haveparameter the following sets: parameter sets: y t f(y, t) y(10 t ) 6 a later example, though, even this modified step-through f (y,t)  e  10 Suppose we have the following y parameter t f(y, sets: t) method still has a significant weakness. Despite uthat, 5.00% in 1 637,628 y t f(y,u t) 5.00% 1 637,628 situations where we cannot satisfy the requirementsv of8.00% the 2 527,292 u 5.00% 1 637,628 Aumann-Shapley approach, a step-through remains a viable v 8.00% 2 527,292 Difference -110,336v 8.00% 2 527,292 alternative. Difference -110,336 Difference -110,336 We then get the following attributions from the three

Partial Derivatives We then get themethods following discussed attributions above: from the three methods discussed above: We then get the followingWe then attributions get the from following the three attributions methods discussed from above: the three methods discussed above: Actuaries already frequently use derivatives or approxima - y t Total Attributed Unexplained y t Total Attributed Unexplained tions to the derivative to perform attributions. Some partial y t Total Attributed Unexplained derivatives come up so often that they have specific names, 4 Aumann-Shapley -147,619 37,284 -110,336 0 This corresponds to the4This Shapley corresponds value to fromthe Shapley game value theory. from game theory. such as the “Greeks” (delta, gamma, vega, rho, theta,5 etc.) 5 4 Evaluating it at the end Step-throughEvaluating point instead it at the or end at -146,952 pointbothThis instead pointcorresponds 36,616 sor does at both not to point theeliminate -110,336s Shapleydoes not its eliminate value drawbacks. from its 0drawbacks. game For thetheory. For examples the examples or duration. In some cases, we use formulas to indirectly this article we will usein this the article partial we derivativewill use5 Evaluatingthe partialat the derivative beginning it at the at end thepoint. beginning point instead point. or at both points does not eliminate its drawbacks. For the examples Partial derivative -172,160 31,881 -140,278 29,942 calculate the partial derivatives; other times, we shock one in this article we will use the partial derivative at the beginning point. of the parameters a small amount to numerically estimate As expected, both the step-through and the Aumann-Shapley approach fully attribute the the derivative. change. They also produce comparable results. The partial derivative results, though easily calculable,6 do not accuratelyCONTINUED capture the total ONchange PAGE in value. 30

Example 2: Zero-Coupon Bond, Revisited Many times, we can formulate a problem in multiple ways. Suppose that instead of expressing the yield for the zero-coupon bond in terms of a single variable, as in the previous example, we express it in termsAUGUST of two components: 2013 RISKS a preva ANDiling interest REWARDS rate r, | 29 and a credit spread c. As a formula: f (r,c,t)  e(rc)(10t) 106 Given the modified parameter sets:

r c t f(r, c, t) u 4.00% 1.00% 1 637,628 v 5.00% 3.00% 2 527,292 Difference -110,336 The initial and final values have not changed from the previous example—we have merely separated the yields to maturity into two components. Now that we have shifted perspective, what happens to the attributions? r c t Total Attributed Unexplained Aumann-Shapley -49,206 -98,413 37,284 -110,336 0 Step-through -49,162 -98,027 36,853 -110,336 0 Partial derivative -57,387 -114,773 31,881 -140,278 29,942 Note in particular the step-through values. By formulating the problem in a slightly different way, the value attributed to the time variable t has changed! In contrast, the amounts attributed to t by both the Aumann-Shapley approach and the partial derivative have not changed from before.

Now, some readers may object that this change in the attributed value for t comes from the fact that we stepped through every possible order of variables. In practice, most actuaries would use only a single order, and most likely we would step through the two interest rate components consecutively. Under those circumstances, the value attributed to t would come out equal under both formulations. For example, if we use the order y, t

6The attribution for t, for example, equals 5 percent of the initial value.

AUMANN-SHAPLEY VALUES:… | FROM PAGE 29

As expected, both the step-through and the Aumann- actuaries would use only a single order, and most likely we Aumann-Shapley -147,619 37,284 -110,336 0 Shapley approach fully attribute the change. They also would step through the two interest rate components con- Aumann-Shapleyproduce comparable-147,619 Step-through37,284 results. -110,336 The -146,952 partial derivative 036,616 results, -110,336 secutively. 0 Under those circumstances, the value attributed Step-throughthough easily -146,952 calculable, Partial36,616 derivative6 do -110,336 not -172,160accurately 0 31,881 capture the -140,278 total to t 29,942would come out equal under both formulations. For Partial derivative -172,160 31,881 -140,278 29,942 change in value. example, if we use the order y, t in the first example and c,

As expected, both the step-through and the Aumann-Shapleyr, t in approach the second, fully attribute then in the both cases we attribute $40,540 As expected,Example both the 2: step-through Zero-Couponchange. and They the AuBond, alsomann-Shapley produce Revisited comparable approach fully resu attributelts. The the partialto derit. However,vative results, any though particular order comes from an arbitrary change. They also produce comparable results. The6 partial derivative results, though easily calculable,ManyAumann-Shapley times,6 do not weaccurately -147,619easily can formulatecalculable,capture 37,284 the total ado -110,336 problem change not accurately in value. in multiple 0 capture ways.the total changechoice in value. on our part. Nothing intrinsic in the order itself SupposeStep-through that instead -146,952 of36,616 expressing -110,336 the yield 0for the zero- would lead us to conclude that we should choose one order Examplecoupon Partial2: Zero-Coupon derivative bond inBond, -172,160Exampleterms Revisited of 31,8812: a Zero-Couponsingle -140,278 variable, Bond, as Revisited 29,942in the previous over another. The equally natural order t, y (or t, c, r) would Many times, we can formulateMany a problem times, in wemultiple can formulateways. Suppose a problem that instead in multipleof ways. Suppose that instead of expressingexample, the yield forwe the express zero-coupon it in bond terms in terms of oftwo a single components: variable, as in athe pre - lead us to attribute $32,692 to t. As expected, both theexpressing step-through the and yield the Au formann-Shapley the zero-coupon approach bondfully attribute in terms the of a single variable, as in the previousvailing example,change. Theyinterest we expressalso produce rate it in r, termscomparable and of atwo credit resu components:lts. Thespread partial a prevac. deri Asilingvative a interestformula: results, rate though r, and a credit spread c. As a6 formula:previous example, we express it in terms ofin two the components: first example a preva andiling c, r, interest t in the rate sec r,ond, then in both cases we attribute $40,540 to t. easily calculable, do not accurately capture the total change in value. In the end, when using a step-through, we must either and a credit spread(rc)(10 c. tAs) a 6formula: However,in the first any example particular and c,order r, t in comes the sec fromond, anthen arbitrary in both caseschoice we on attribute our part. $40,540 Nothing to t. f (r,c,t)  e  10 intrinsicHowever,(rc)(10 int) anytheaccept order6particular that itself we order would have comes leadchosen from us to anan conclude arbitrary thatchoice order, we on sorhould our we part. choosemust Nothing one order Given theExample modified 2: parameter Zero-Coupon sets: Bond, Revisited f (r,c,t)  e  10 GivenMany times, the modifiedwe can formulate parameter a problem insets: multiple ways. Suppose that insteadoverintrinsic another.of in acceptthe The order equallythat itself the naturalwould results lead order couldus t,to y conclude (orvary t, c,if r)thatwe would were-formulate s houldlead uschoose to attributethe one order $32,692 r c t f(r, Givenc, t) the modified parameter sets: expressing the yield for the zero-coupon bond in terms of a single variable,to asover t. in the another. problem The equally in an natural equivalent order t, way.y (or t,Either c, r) would way, lead step-throughs us to attribute $32,692 u 4.00%previous 1.00% example, 1 637,628 we express r it in terms c of two t components: f(r, c, t) a prevailing interestto t. rate r, and a credit spread c. As a formula: produce non-unique results. v 5.00% 3.00% 2 527,292 In the end, when using a step-through, we must either accept that we have chosen an u 4.00% 1.00% (r1 c)(10 637,628t) 6 Difference -110,336 f (r,c,t)  e  10 arbitraryIn the end, order, when or using we must a step-through, accept that we the must results either could accept vary that if wewe re-formulatehave chosen an the Given the modified parameterv 5.00% sets: 3.00% 2 527,292 arbitrary order,Example or we 3:must Binary accept Call that the results could vary if we re-formulate the The initial and final values have not changed from the previous example—we haveproblem in an equivalent way. Either way, step-throughs produce non-unique results. merely separated r the yields c to t maturity f(r, c, t)into Difference two components. -110,336 Now that we have shiftedproblem inSuppose an equivalent we ownway. aEither binary wa cally, step-throughs on a particular produce security, non-unique with results. perspective, what happens to the attributions? Theu 4.00%initial 1.00% and The 1 final 637,628 initial values and final have values not have changed not changed fromExample from the the 3: previous aBinary strike example—we Call K at 100. haveFor illustrative purposes we will hold Example 3: Binary Call v 5.00% 3.00%r merely2 527,292 c separated t the Total yields Attributed to maturity Unexplained intoSuppose two components. we own Now a binary that we call have on shifteda particular security, with a strike K at 100. For previous example—we have merely separated the yieldsSuppose to wethe own interest a binary rate call r onconstant a particular at 2 security,percent withand athe strike volatility K at 100. σ For Aumann-Shapley -49,206Difference perspective, -98,413-110,336 37,284 what happens -110,336 to the attributions? 0 illustrative purposes we will hold the interest rate r constant at 2 percent and the volatility maturity into two components. Now that we have shiftedillustrative constantpurposes weat 25will percent, hold the andinterest we rawillte r constantassume atthe 2 percentunderlying and the volatility The initial and final values have not changed from the previous example—weσ constant have at 25 percent, and we will assume the underlying security pays no dividends. Step-through -49,162 -98,027 36,853 -110,336r c 0 t σ constant Total Attributedsecurityat 25 percent, pays Unexplained and no wedividends. will assume For the the underlying current assetsecurity spot pays price no dividends. perspective,merely separated what the yields happens to maturity to intothe two attributions? components. Now that weFor have the shifted current asset spot price S and time to option maturity t, the value of our call Partial derivativeperspective, -57,387 what happens -114,773 to the attributions? 31,881 -140,278 29,942 For the current asset spot price S and time to option maturity t, the value of our call Aumann-Shapley -49,206 -98,413 37,284equals: -110,336S and time to option 0 maturity t, the value of our call equals: Note in particular the step-throughr values. c By formulating t Total the problemAttributed in a Unexplained slightlyequals: different way, the value attributStep-throughed to the time variable -49,162 t has changed! -98,027 In contrast, 36,853 the -110,336 0 rt  rt  amounts attributedAumann-Shapley to t by both-49,206 the Aumann-Sh -98,413 apley 37,284 approach -110,336 and the partial derivative 0 ff((SS,t,)t) ee (d(d)2) Partial derivative -57,387 -114,773 31,881 -140,278 29,942 2 have not changedStep-through from before. -49,162 -98,027 36,853 -110,336 0 where x x 1 2 Partial derivative -57,387Note in -114,773particular 31,881 the step-through -140,278 values. 29,942 By formulating the problem in a slightly 11  1u2 u Now, some readers may object that this change in the attributed value for t comes from ((xx)) e e2 2 dudu Note in particular thedifferent step-through way, values. the Byvalue formulating attribut theed problemto the time in where a slightlywhere variable t has changed! In contrast, the  the fact that we stepped through every possible order of variables. In practice, most 22 different way, the valueamounts attribut edattributed to the time to variable t by both t has the changed! Aumann-Sh In contrast,apley the approach and the partial derivative actuaries would use only a single order, and most likely we would step through the two interestNote rateamounts components in attributedparticular consecutively. tohave t by the bothnot step-through the changedUnde Aumann-Shr those from circumstances,apley before.values. approach By the and valueformulating the partial attributed derivative to t wouldhave come not out changed equal fromunder before. both formulations. For example, if we use the order y, t the problem in aNow, slightly some different readers may way, object the th valueat this attributedchange in the attributed value for t comes from Now, some readers may object that this change in the attributed value for t comes from 2 Sand   2 6The attributionto the for time t, for example, variable theequals fact t5 percenthas that changed! ofwe the stepped initial value. In through contrast, every the possibl amountse order of variables. In practice, most ln Sand  (r  )t the fact that we stepped through every possible order of variables. In practice, most lnKK  (r 2 2 )t d2  attributedactuaries would to uset by onlyactuaries both a single thewould order, Aumann-Shapley anduse most only likely a single we would order, approach step and through most and the likely two we would step through thed two2  t interest rate components consecutively. Under those circumstances, the value attributed  t the partial derivativeinterest have rate componentsnot changed consecutively. from before. Under those circumstances, the value attributed to t would come out toequal t would under bothcome formulations. out equal For under example, both if formulations. we use the order Fory, t example, if we use the order y, t At the boundary where t = 0, its value equals 6 Now,The attribution some for t,readers for example, equalsmay 5 percentobject of the that initial value.this change in the 6The attribution for t, for example, equals 5 percent ofAt the boundary where t = 0, its value equals the initial value.   At the boundary where t = 0, its value equals 1 if S K attributed value for t comes from the fact that we stepped f (S,0)   1 if S  K f (S,0) 0 if S  K through every possible order of variables. In practice, most 0 if S  K

Given the parameters: Given the parameters: S t f(S, t) S t f(S, t) u 90 1 0.314 u 90 1 0.314 v 110 0 1.000 v 110 0 1.000 30 | RISKS AND REWARDS AUGUST 2013 Difference 0.686 Difference 0.686 The three methods disagree significantly about the nature and magnitude of the time Thecomponent’s three methods contribution: disagree significantly about the nature and magnitude of the time component’s contribution:S t Total Attributed Unexplained S t Total Attributed Unexplained in the first example and c, r, t in the second, then in both cases we attribute $40,540 to t. in the first example andHowever, c, r, t in the any sec particularond, then orderin both comes cases fromwe attribute an arbitrary $40,540 choice to t. on our part. Nothing However, any particularintrinsic order comes in the from order an itself arbitrary would choice lead onus ourto conclude part. Nothing that we should choose one order intrinsic in the order itselfover would another. lead The us to equally conclude natural that we order should t, y choose(or t, c, one r) would order lead us to attribute $32,692 over another. The equallyto t. natural order t, y (or t, c, r) would lead us to attribute $32,692 to t. In the end, when using a step-through, we must either accept that we have chosen an In the end, when usingarbitrary a step-through, order, weor wemust must either accept accept that that the we results have chosen could varyan if we re-formulate the arbitrary order, or we mustproblem accept in thatan equivalent the results way.could Eithervary if wawey, re-formulate step-throughs the produce non-unique results. problem in an equivalent way. Either way, step-throughs produce non-unique results. Example 3: Binary Call Example 3: Binary CallSuppose we own a binary call on a particular security, with a strike K at 100. For Suppose we own a binaryillustrative call on a purposes particular we security, will Aumann-Shapleyhold withAumann-Shapley the a strike interest K at ra 100.te r0.435 constantFor0.435 0.251 at 0.2512 percent and 0.686 the 0.686 volatility 0 0 illustrative purposes weσ willconstant hold theat 25 interest percent, rate and r constant we will at assume 2 percent the and underlying the volatility security pays no dividends. σ constant at 25 percent, and we will assume the underlyingStep-throughStep-through security pays no 0.653 dividends. 0.653 0.033 0.033 0.686 0.686 0 0 For the current asset spot price S andAumann-Shapley time to option maturity 0.435 t, the 0.251 value of our 0.686 call 0 For the current asset spotequals: price S and time to option maturity t, the value of our call equals: PartialPartial derivative derivative 0.312 0.312 -0.060 -0.060 0.252 0.252 0.434 0.434 Step-throughrt 0.653 0.033 0.686 0 rt f (S,t)  e (d ) f (S,t)  e (d ) 2 2 Partial derivative 0.312 -0.060 0.252 0.434 The price crossesx the strike during the attribution period, but the call does not reach full x The price crosses1 the 1 ustrike2 during the attribution period, but the call does not reach full 1  1 u2 2 2 (x)  e du where where (x)   e valueduvalue immediately immediately at thatat that time. time. Its Itsvalue value still s tillincludes includes a discount a discount for fother theprobability probability of a of a  2  2  subsequentThe price decrease.crosses the The strike passage during of the time attribution eventually period, drives but that the probability call does not to reachzero, full subsequent decrease. The passage of time eventually drives that probability to zero, bringingbringing value the immediately the option option to itsto at fullits that full value. time. value. ItsAt valuetheAt thebeginning s tillbeginning includes of theof a theattributiondiscount attribution fo period,r the period, probability though, though, the of athe oppositeoppositesubsequent pattern pattern decrease.holds: holds: the Thethe possibility possibilitypassage that of thattime volatility volatility eventually will will cause drives cause the that theasset probability asset price price to toexceed to zero, exceed 2 2 ln Sand  (r   )thet the strikebringing strike recedesln recedesSand the ( roption as  weas)t weapprto itsapproach fulloach value.maturity, maturity, At themeaning meaningbeginning that that theof the thepassage attributionpassage of time of period, time reduces reduces though, the the the  K 2 K 2 d2 option’sopposited2  value. pattern The holds: partial the derivative possibility results that volatility reflect the will latter cause effect. the asset price to exceed  t option’s value. Thet partial derivative results reflect the latter effect. the strike recedes as we approach maturity, meaning that the passage of time reduces the

Thus,Thus,option’s f’s f’ssensitivity sensitivity value. Theto timeto partial time varies variesderivative considerably considerably results over reflect over the the theattribution latterattribution effect. region. region. By Byonly only consideringconsidering the the edges edges of theof the region, region, the thestep-through step-through does does not notaccurately accurately capture capture the thefull full At the boundary where t = 0, its value equals At the boundary where t = 0, its value equals 1 if S  KsensitivitysensitivityThus, f’sand andsensitivity ends ends up upattributing to attributing time varies little little considerably of theof thechange change over to the tothe thetime attribution time component. component. region. By only f (S,0)   1 if S  K  consideringf (S,0)   the edges of the region, the step-through does not accurately capture the full 0 if S K 0 if S  K An An Intuitivesensitivity Intuitive Argument and Argument ends upfor attributingfor Why Why Aumann-Shapley Aumann-Shapley little of the change Produces Producesto the timea Complete a component. Complete Attribution Attribution

Given the parameters: AlthoughAlthough the the examples examples have have shown shown that that the theAumann-Shapley Aumann-Shapley approach approach produces produces a a Given the parameters:Given the parameters: We have defined f as a multivariate function of the x completeAn Intuitive attribution Argument in those for cases, Why theyAumann-Shapley do not explain Produces why it works a Complete in general.i Attribution A quick S t f(S, t) complete attributionvariables. in those cases,However, they overdo not the explain attribution why it worksregion in we general. can A quick S t f(S, t) (though(thoughAlthough non-rigorous) non-rigorous) the examples argument argument have will shown will help helpthat illustrate theillustrate Aumann-Shapley the thelogic logic underpinning underpinning approach the produces thetechnique. technique. a u 90 1 0.314 also view f as a function of z alone, denoted f(z) for clarity. u 90 1 0.314 complete attribution in those cases, they do not explain why it works in general. A quick v 110 0 1.000 The linear interpolation between u and v connects the two: WeWe have(though have defined defined non-rigorous) f as f aas multivariate a multivariate argument function will function help of illustrate theof thexi variables. x thei variables. logic However, underpinning However, over over the the technique. the v 110 0 1.000 (z) (z) Difference 0.686 attributionattribution region region wef (z) wecan =can alsof((1-z)u+zv). also view view f as f aThusas function a function f (0), of for zof alone, example,z alone, denoted denoted would f(z) meanforf(z) forclarity. clarity. The The Difference 0.686 We have definedto calculate f as a multivariate the values function of each ofx thefor (z)xz=0,i variables.(z) then evaluate However, f at over (z)the (z) linearlinear interpolation interpolation between between u and u and v connects v connects the thetwo:i two: f (z) f (z)= f((1-z)u+zv). = f((1-z)u+zv).(z) Thus Thus f (0), f (0), attribution region we can also view f as a function(z) of z alone, denoted f for clarity. The The three methods disagree significantly aboutfor for theexample, example,nature would wouldthose mean mean values—in to calto culatecalculate other the thewords,values values off each(0)of eachequals xi for x forourz=0, z=0,initial then then evaluate value evaluate f at f at The three methods disagree significantly about the nature and magnitude of the time (z)i (z) thoselinear values—in interpolation other words, between f(z) u(0) (z)and equals v connects our initial the two:value f f(u).(z) =We f((1-z)u+zv). can now write Thus out f (0), andcomponent’s magnitude contribution: of theThe time three component’s methods disagree contribution: significantlythose values—in about thef(u). othernature We words, and can magnitude now f (0) write equals of theout time ouran initial equation value for f(u). the differenceWe can now write out an anequationfor equation example, for for the would the difference difference mean weto cal weseekculate seek to attribute: tothe attribute: values of each xi for z=0, then evaluate f at Scomponent’s t Totalcontribution: Attributed Unexplainedthose values—inwe seekother towords, attribute: f(z)(0) equals our initial value f(u). We can now write out (z) (z) (z) (z) S t an Totalequation Attributed for the Unexplaineddifferencef ( vwef)(v fseek)(uf)( uto) fattribute: (1)f (1)f (0)f (0) Aumann-Shapley 0.435 0.251 0.686 0 AssumingAssuming that that we weAssuminghave have a sufficiently a sufficientlythat wef (havesmoothv) smoothf (au sufficiently)function, function,f (z)(1) wef smooth ( wezcan)(0) can express function,express the thedifference we difference as as Step-through 0.653 0.033 0.686 an anintegral: integral: 0 can express the difference as an integral: Assuming that we have a sufficiently smooth function, we can express the difference as Partial derivative 0.312 -0.060 0.252 0.434 1 (z) an integral: df1 df (z)    (z)dz(z) dz dz1 (z) The price crosses the strike during the attribution period, 0 0 dfdz The price crosses the strike during the attribution period, but the call does not reach full   (z)dz but the call does not reach full value immediatelyWe still at need that to relate this back to the original xdzi variables, and we do this by applying value immediately at that time. Its value still includes a Wediscount still foneedr the toprobabilityWe relate still this needof aback to relateto the thisoriginal0 back x ito variables, the original and xwei variables, do this by applying time.subsequent Its value decrease. still Theincludes passage a ofdiscount time eventually for thethe drives probabilitythe chain chainthat rule. probability rule. Note Noteand thatto wezero, that we do wehave this have byswitched applyingswitched back theback to chain the to theoriginal rule. original Note multivariate thatmultivariate we have function function f: f: ofbringing a subsequent the option decrease. to its full value.The passageAt the beginning of time of eventuallytheWe attribution still need period, to relate though, this the back to the original xi variables, and we do this by applying switched back1 to1 the original multivariate function f: opposite pattern holds: the possibility that volatility will causethe chain the asset rule. price Note to exceed that we have switchedf fdxi dx back to the original multivariate function f: drives that probability to zero, bringing the option to its full      ((1i ((1z)uz)uzv)zvdz) dz the strike recedes as we approach maturity, meaning that the passage of time reduces the  1 x dz   value. At the beginning of the attribution period, though, 0 0i  i i xif dzdx  option’s value. The partial derivative results reflect the latter effect.     i ((1  z)u  zv)dz the opposite pattern holds: the possibility that volatility  SinceSince z represents z represents our our linear linear interpolation interpolation0  i xvariable,i variable,dz the thederivative derivative of x ofi equals x equals the the Thus, f’s sensitivity to time varies considerably over the attribution region. By only i will cause the asset price to exceed the strikedifference recedesdifference betweenas between the the final final and and initial initial values values. With. With that that substituti substitution, on,and and separating separating out out considering the edges of the region, the step-through does Sincenot accurately z represents captureSince our thez linear representsfull interpolation our linear variable, interpolation the derivative variable, of xi equalsthe the we approach maturity, meaning that the passagethethe individual of individual time terms terms inside inside the the integral, integral, we wefinally finally obtain: obtain: sensitivity and ends up attributing little of the change to thedifference time component. betweenderivative the final of x and equals initial the values difference. With thatbetween substituti the finalon, and and separating out reduces the option’s value. The partial derivativethe results individual terms inside thei integral, we finally obtain: reflectAn Intuitive the latter Argument effect. for Why Aumann-Shapley Produces a Completeinitial Attribution values. With that substitution, and separating out the Although the examples have shown that the Aumann-Shapley approach producesindividual a terms inside the integral, we finally obtain: complete attribution in those cases, they do not explain why it works in general. A quick Thus,(though f’s non-rigorous) sensitivity argumentto time willvaries help considerablyillustrate the logic over underpinning the the technique. 1 f attribution region. By only considering the edges of the f (v)  f (u)  (vi  ui )  ((1  z)u  zv)dz i 0 xi region,We have the defined step-through f as a multivariate does not function accurately of the capture xi variables. the fullHowever, over the attribution region we can also view f as a function of z alone, denoted f(z) for clarity. The where the term onwhere the rightthe term hand on side the rightcorresponding hand side tocorresponding each xi gives to that each variable’s sensitivitylinear interpolation and ends between up attributing u and v connects little of the the two: change f(z)(z) =to f((1-z)u+zv). the Thus f(z)(0), attribution. x gives that variable’s attribution. timefor example,component. would mean to calculate the values of each xi for z=0, then evaluatei f at those values—in other words, f(z)(0) equals our initial value f(u). We can now write out ANan equation INTUITIVE for the difference ARGUMENT we seek to attribute:FOR W HYFrom this argument,From we this can argument, also see wewhy can jump also discontinuities see why jump causediscontinui this approach- to fail, f(v) f (u)  f (z)(1) f (zsince)(0) they causeties changes cause inthis the approach value of tof that fail, do since not theyget captured cause changes by the inderivative. AUMANN-SHAPLEY PRODUCES A However, as long as we have a sufficiently smooth function (and a sufficiently accurate the value of f that do not get captured by the derivative. COMPLETEAssuming that we A haveTTRIBUTION a sufficiently smooth function,calculator we can express for the difference integral) as we will always get a complete attribution from the Aumann- an integral: However, as long as we have a sufficiently smooth function Although the examples have shown that theShapley Aumann- approach. 1 df (z) (and a sufficiently accurate calculator for the integral) we Shapley approach produces a complete  attribution(z)dz in those dz will always get a complete attribution from the Aumann- cases, they do not explain why it works0 in general.Accuracy A quick of Numeric Integration Shapley approach. (thoughWe still non-rigorous)need to relate this argument back to the will original help xillustratei variables,Since the and thelogic we Aumann-Shapley do this by applying results come from a numeric integration, a natural question the chain rule. Note that we have switched back to the arises:original howmultivariate much functionconfidence f: should we have in their accuracy? In practice we can obtain underpinning the technique. 7 1  f dx very rapid convergence using Simpson’s rule. CONTINUED Example 1’s resultsON PAGE earlier 32 used 1,000     i ((1  z)u pointszv)dz to ensure a highly accurate result, but even if we evaluate the integrals at just three  x dz  0 i i points and use Simpson’s rule we get almost identical results: Since z represents our linear interpolation variable, the derivative of xi equals the difference between the final and initial values. With that substituti on, and separating out Contributions to integrals the individual terms inside the integral, we finally obtain: z y t f f AUGUSTf 2013 RISKSf AND REWARDSf | 31  (10  t) f  yf   y t y t 0.00 5.00% 1.00 637,628 -5,738,653 31,881 -956,442 5,314 0.50 6.50% 1.50 575,509 -4,891,829 37,408 -3,261,219 24,939 1.00 8.00% 2.00 527,292 -4,218,339 42,183 -703,057 7,031 Total -4,920,718 37,283

(vi-ui) 3.00% 1.00

(vi-ui) Total -147,622 37,283 Previous Results for Comparison -147,619 37,284

Thus, even a quick calculation can produce reasonable results. Furthermore, by using a spreadsheet or programming language we can easily evaluate more points to improve the accuracy.

Additional Considerations In this article we have looked at some artificially simple examples. In real life, though, we may face more complex attribution problems. For example, we may wish to reflect the fact that interest rates vary based on the time to maturity. Even if the yield curve does not shift, the yield on an asset could change simply due to the passage of time. In that

x 7 f (x ) 4 f (x)  f (x  )  f (z)dz  2 x 6 AUMANN-SHAPLEY VALUES:… | FROM PAGE 31

1 f f (v)  f (u)  (vi  ui )  ((1  z)u  zv)dz i 0 xi The Aumann-Shapley approach applies in those more com- ACCURACYwhere the term on the rightOF hand NU side MERICcorresponding ItoNTEGRATION each xi gives that variable’s Sinceattribution. the Aumann-Shapley results come from a numeric plex cases as well. We do need a way to interpolate smooth-

integration,From this argument, a natural we can also question see why jump arises: discontinuities how much cause this confidence approach to fail, ly between the observed points. However, to perform any since they cause changes in the value of f that do not get captured by the derivative. attribution we generally need some form of interpolation shouldHowever, we as long have as we in have their a sufficientl accuracy?y smooth In function practice (and a wesufficiently can obtainaccurate verycalculator rapid for the convergence integral) we will always using get a Simpson’scomplete attribution rule. from7 theExample Aumann- anyway since our valuation points will rarely correspond Shapley approach. 1’s results earlier used 1,000 points to ensure a highly exactly to the market-observable points. Some common Accuracy of Numeric Integration methods include linear interpolation and cubic splines, both accurateSince the Aumann-Shapley result, but results even come if from we a numericevaluate integration, the a integralsnatural question at justarises: three how much points confidence and should use we Simpson’shave in their accuracy? rule Inwe practice get we almost can obtain of which provide differentiable interpolations. very rapid convergence using Simpson’s rule.7 Example 1’s results earlier used 1,000 identicalpoints to ensure results: a highly accurate result, but even if we evaluate the integrals at just three points and use Simpson’s rule we get almost identical results: Although this article has focused on asset valuation exam- Contributions to integrals ples, we can use the Aumann-Shapley approach for other z y t f f f f f  (10  t) f  yf   y t y t applications. However, we do need to verify that our 0.00 5.00% 1.00 637,628 -5,738,653 31,881 -956,442 5,314 function f meets the requirements. Liabilities in particular, 0.50 6.50% 1.50 575,509 -4,891,829 37,408 -3,261,219 24,939 though, often contain features that can potentially create 1.00 8.00% 2.00 527,292 -4,218,339 42,183 -703,057 7,031 discontinuities, including: Total -4,920,718 37,283 1. Charges, guaranteed rates of return, or other features (vi-ui) 3.00% 1.00 based on market values rounded to the nearest percent, (vi-ui) Total -147,622 37,283 Previous Results for Comparison -147,619 37,284 nearest 25 basis points, or some other multiple. 2. Any feature involving rebalancing something back to Thus, even a quick calculation can produce reasonable results. Furthermore, by using a a target, but only if outside some tolerance band. For Thus,spreadsheet even or programming a quick language calculation we can easily can evaluate produce more points reasonable to improve the accuracy. example, due to an investment strategy we have adopted results. Furthermore, by using a spreadsheet or program- Additional Considerations or due to a contractual agreement we have entered into, ming language we can easily evaluate more points to In this article we have looked at some artificially simple examples. In real life, though, we might rebalance a particular asset allocation back improvewe may face the more accuracy. complex attribution problems. For example, we may wish to reflect the fact that interest rates vary based on the time to maturity. Even if the yield curve does to 80 percent equity at end of each quarter if the cur- not shift, the yield on an asset could change simply due to the passage of time. In that rent asset allocation deviates from that by more than 5 ADDITIONALx CONSIDERATIONS 7 f (x ) 4 f (x)  f (x  )  f (z)dz  2 percent. In xthis article we have6 looked at some artificially simple examples. In real life, though, we may face more complex 3. Franchise deductibles. attribution problems. For example, we may wish to reflect 4. Digital payoffs. the fact that interest rates vary based on the time to matu- rity. Even if the yield curve does not shift, the yield on an 5. Features activated or deactivated at certain thresholds, asset could change simply due to the passage of time. In including knock-in and knock-out features. that case, it may make more sense to assign the change in Keep in mind, though features such as these do not automat- value solely to time, not due to a nonexistent movement in ically imply a problem. In example 3 our function contained the yield curve. a discontinuity at the point (S, t) = (100, 0), but the path To further complicate matters, when valuing options we between u and v did not pass through that point. In gen- could view volatility as depending on both the time to eral, the specific circumstances of the problem will dictate expiration and the moneyness of the option, leading to a whether we can use the Aumann-Shapley approach or not. two-dimensional volatility surface.

32 | RISKS AND REWARDS AUGUST 2013