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1. Duration Definition A measure of the sensitivity of an asset's value to movements (% change in price corresponding to a parallel shift in yields or YTM) ex - a 5yr duration means the will decrease in val by 5% if int rates rise 1% and increase in val by 5% if interest rates fall 1%

2. Duration Characteristics * tends to increase w/ but at a decreasing rate * if an investor expects IR to fall, a bond with long duration would be appealing if expected to fall, chose a shorter dur * all else equal, higher rates=lower dur

3. to Maturity * the one discount rate that equilibrates PV of cash flows to the market price * the IRR that a buyer would receive if they purchased the bond at market price

4. What are the key 1) assumes that the bond is held to maturity assumptions of YTM and that 2) the interim cash flows are reinvested at the same YTM

5. Macaulay Duration *Defined as the present-value wgted Definition avg time to receipt of cash flows, units=years

6. Problems with 1) assume that prices are a function of one-factor: YTM Macaulay/Modified 2) do not work for large IR changes (only the Duration linear component of the price/yield relationship) 3) not appropriate for sec with embedded options or where CF will be affected by IR changes

7. Analytic vs. numerical We say we have an analytic solution if we can actually solve the equation explicitly for the unknown solutions variable. In this case, it is easy to see that the explicit analytic solution is x = 5, and that this is the exact solution. If we were not so smart, we might develop an "algorithm" on a computer to solve this equation numerically.

8. Macaulay Duration Formula D = -(∂P/∂Y) 1/P (1+Y) (using partials) where P = Price Y=YTM

9. Macaulay Duration Analytic for fixed cash flow securities: Formula D = SIGMA (Pi*ti / V ) where Pi=PV of Cashflow i ti= time i V = PV of security ()

10. Modified Duration % change in price wrt a 1% change in YTM Definition

11. Modified Duration Formula MD = -∂P/∂Y * 1/P where P = dirty price, Y = YTM

12. Modified Duration - Modified Dur = Macaulay Duration / (1+YTM/n) Macaulay Duration Relationship

13. How can Modified Duration delta P = -modified duration * yield change be used to estimate the ex: MD=7, YTM change =+1%, P* = -7% change in price given a change in Yield?

14. Define Dollar Duration $ duration shows the effect of a change in yield (YTM) on the dollar value of a bond $D = -∂V/∂Y or Modified duration * price 15. Define PV01 and DV01 DV01 = $ val of a basis point = $D / 10,000 if currency is USD, DV01 = PV01 DV01 = DVBP, PV01 = PVBP

16. Define the Effective Duration ED = Option Adjusted Duration OAD = - (1/p) * (∂p/∂r) where r = parallel spot curve shock (one factor model) allows for CF to change as Yield changes as will happen with securities with embedded options

17. How is Effective Duration One method using a binomial tree: Calculated 1) Move the current YC up and down by a spec amt 2) recalculate the IR tree for each path 3) Re-estimate the CF along each path, holding OAS constant at current value 4) recalculate the avg price for the up and down trees Calc ED = {(Value IR down) - (Value IR up) } / (2V0deltaY)

18. Define Convexity Convexity measures the sensitivity of to changes in interest rates. It is the second order taylor expansion and accounts for the nonlinearity in the price/ytm relationship

19. Describe the effects of Assuming duration is negative: positive/negative convexity If pos convex = yields decr -> Slope (dp/dr) decreases -> Dur (-1/p*dp/dr) increases

yields increase -> slope decreases -> Dur decreases

higher convex = less affected by rise in IR, more affected by fall & higher price regardless of whether rates rise or fall

20. Explain the parametric approach assume Price is a function of N factors: P=P(F1,...FN) to risk mgmt *Param app investigates the price sensitivity of securities and ports to each risk factor in isolation, others fixed *Assume price is a function of a single factor, x: P = P (x) *Use a TS to estimate the behavior of percent changes around an arbitrary point x0 The first order expressions: -1/P (∂P/∂x) are partial dur (OAD, Key rate dur, spread dur, vol dur, prepay dur, etc.) * second order=convexities

21. Spread Duration The Approximate percentage change in a bond's price for a 100 basis point change in the credit spread assuming that the Treasury rate is unch Also can measures a bond's market price to a change in Option Adjusted Spread Does not account for prepayments

22. OAD/OAC drift drift = -∂OAD/∂r, OAC drift = -∂OAC/∂r

23. MBS Duration measures • Volatility duration - measures the % change in the price of an MBS that results from a 1% change in vol • Prepayment duration - % change in price resulting from a 1% change in the prepayment model • Current Spread Duration - % change in the price resulting from a 10bps change in cc spread • Refi Duration / Turnover Duration / Elbow duration - changes in refi dial, turnover dial, and elbow effect (lower costs=more refi) • Coupon Curve Duration measures the price sensitivity of MBS to changes in IR using the prices of MBS with the same agency and term.

24. Define a Bullet Bond Non-callable, coupon paying bond that pays principal at maturity (no amortization payments) amortizing bond - bond that repays part of principal along w/ coupon (mbs, abs)

25. Define Swap Spread Swap Spread = Swap Yield - Tsy Yield w/ same maturity regarded as a systematic credit premium spread over swap = specific credit risk 26. Define Nominal Nominal : diff in YTM of a bond and YTM of benchmark Yield Spread

27. Define Zero Zero Volatility Spread (zspread) - The constant spread that, when added to the yield at each point on the spot Volatility Spread rate Treasury curve where a bond's cash flow is received, will make the price of a security equal to the present value of its cash flows. for option-free bonds, zspread=oas

28. Define OAS the flat spread which has to be added to the treasury in a pricing model (that accounts for embedded options) to discount a security payment to match its market price. OAS is hence model dependent.

to calculate the OAS, the spot rate curve is given multiple interest rate paths. In other words, many different spot rate curves are calculated and the different interest rate paths are averaged An OAS accounts for interest rate vol and the prob of the prepayment of principal of the bond.

29. Define Discount Discount Margin (DM) - Bonds with var int rates are usually priced close to their . because the interest Margin rate (coupon) on a var rate bond adjusts to current IR based on changes in the bond's ref rate. The DM is the spread that, when added to the bond's current reference rate, will equate the bond's CF to its current P.

30. How is OAS 1) Estimate a series of IR paths representative of possible future rates so that no arbitrage exists calculated? 2) Project cash flows of the security along each of the paths 3) Estimate the OAS as the spread that when added to the yields used to discount cash flows, makes the avg of discounted cash flows equal to the price

31. How is Price • Using only duration: ∂ Price = (∂P * Duration) Approximation • Using duration + convexity: ∂ price = (∂yduration) + (0.5 Conv * (∂y)^2) done using duration/convexity?

32. Describe the YB 1) Determine which sector each position is in method for 2) For each position, assemble Risk Factors: Level of IR, vol of rates, credit spread, liquidity spread, hedging estimating TE risk, exchange rate risk, etc. 3) Decompose the covariance matrix for each sector risk factor set into principal components 4) Based on a multivariate normal distribution, the principal components cov matrix, and the initial values, sample a draw from the set of risk factors 5) Estimate the return of each position in the portfolio and benchmark (do this by estimating the change in return do to the change in each risk factor from its initial state and summing the differences) 6) repeat 4 and 5 10,000 times

Rather than estimate the entire covariance matrix that would incorporate all relationships of risk factors and positions, YB uses principal components which can explain 90% of the total variance of each subsector

33. Define OAC/OAD describes how OAC and OAD change as rates change: drift OAD Drift = - dOAD/dr OAC drift = -dOAC/Dr

34. Define Constant measures empirical duration for securities in particular price ranges Dollar Duration

35. List some MBS All measure price sensitivity Duration Measures Volatility Duration - to volatility Prepayment Duration - to prepayments Current Coupon Spread Duration - to cc spread Refi Duration - to refinancing dial Coupon Curve Duration - to changes in IR using prices of mbs w/ same agency and term

36. Describe the Monte Delta normal = dur and conv are calculated at the current price and scenarios with IR changes use that duration Carlo Method of est and convexity estimate which will lead to false results for path-dependent securities where the cash flow is TE dependent upon the IR. Return dist of each security is assumed to be a linear function of the underlying risk factors and risk factors are assumed to be normally distributed. Cov matrix comes from historical data Full Valuation = each bond is evaluated fully in each trial according to risk factors (yield, spread, vol, etc) 37. Describe the general 1) ID Relevant systematic risk factors (GDP, mort orig rates, Tsy yields, FX rates, swap spreads, etc) stages of market 2) Measurement of market exposure to risk factors risk management 3) Estimation of joint probability distribution of risk factors (Covariance matrix) 4) Computation of risk measures and explicit risk mitigation and management

38. List the primary Nominal Yield spread types of yield spread Zero Volatility Spread (Z-spread) calculations OAS Discount Margin

39. Define Zero Zero Volatility Spread (zspread) - The constant spread that, when added to the yield at each point on the spot Volatility Spread rate Treasury curve where a bond's cash flow is received, will make the price of a security equal to the present value of its cash flows.

40. Define Bullet Bond Non-callable, coupon paying bond that pays and Amortizing principal at maturity (no amortization payments) Bond amortizing bond - bond that repays part of principal along w/ coupon (mbs, abs)

41. How do you Absolute ~= $dur * delta y calculate Absolute relative = - MD * delta Y and Relative P&L basis point value = -$dur/10,000 using duration?

42. Define Marginal vs. Marginal TE = 1st derivative (∂TE/∂Weight of Position), an est of an infinitesimally small change Incremental Incremental TE = Original TE - TE (holdings in a sector eliminated) (independent)

Component TE = [w(sector) ∂TE/∂w(sector) - benchmark w(sector) ∂TE/∂benchmark w (sector) ] * 1/TE = Additive !

43. Define Carry Carry is the total expected yield of holding a position, including net coupon, impact of time, cost of financing, cost of hedging

44. How are weights Weights for: Swaps = (1+P) notional / Mkt val of portfolio; Future/Fwd = Pnotional val / mkt val of calculated for portfolio Swaps in YB (and most risk systems)

45. Define Key Rate "Key Rate Duration (KRD): partial durs that measure the 1st order price sensitivity to the isolated movements of Duration different segments of the price curve krdi= -(1/P) * (∂P/∂ri) The sum of the key rate durations can be used to approximate the portfolio duration: ∆P/P ≈ -∑ krdi * ∆ri However, the sum of KRDs may not equal the OAD because of CF uncertainties due to the nonlinearity fo the value surface (KRDs may be scaled to equal OAD) Can be calculated using a valuation model as krdi= (-1/P) (P i, up - P i, down)/ 2 ∆ri KEEPING OAS CONSTANT!!! Spot rates do not move in isolation: if a single spot rate is shocked, an entire region around it should be shocked as well KRD shocks should add up to the parallel shock"

46. Define Key Treasury Key Tsy Rate Duration (KTRD): partial durations that measure the 1st order price sensitivity to the isolated Rate Duration movements of different segments of the TSY curve ktrdi= -(1/P) * (∂P/∂ri)

47. Typical Spread High-Yield Bonds: High-yield bonds are usually priced at a nominal yield spread to a specific on-the-run U.S. Calculation: Treasury bond. However, sometimes when the credit rating and outlook of a high-yield bond deteriorates, the HY Bonds bond will start to trade at an actual dollar price. For example, such a bond trades at $75.875 as opposed to 500 basis points over the 10-year Treasury

48. Typical Spread • Corporate Bonds: A is usually priced at a nominal yield spread to a specific on-the-run U.S. Calculation: Treasury bond that matches its maturity. For example, 10-year corporate bonds are priced to the 10-year Corporate Bonds Treasury. 49. Typical • Mortgage-Backed Securities: There are many different types of MBS. Many of them trade at a nominal yield spread at Spread their weighted average life to the U.S. Treasury I-curve. Some adjustable-rate MBS trade at a DM, others trade at a Z- Calculation: spread. Some CMOs trade at a nominal yield spread to a specific Treasury. For example, a 10-year planned amortization MBS class bond might trade at a nominal yield spread to the on-the-run 10-year Treasury, or Z-bond might trade at a nominal yield spread to the on-the-run 30-year Treasury. Because MBS have embedded call options (borrowers have the free option of prepaying their mortgages), they are frequently evaluated using an OAS

50. Typical • Asset-Backed Securities: ABS frequently trade at a nominal yield spread at their weighted average life to the swap curve. Benchmark and Spread Calculation: ABS

51. Typical • Agencies: Agencies frequently trade at a nominal yield spread to a specific Treasury, such as the on-the-run 10-year Benchmark Treasury. Callable agencies are sometime evaluated based on an OAS where the spot rate curve(s) are derived from the and Spread yields on non-callable agencies. Calculation: Agencies

52. Typical • Municipal Bonds: Because of the tax advantages of municipal bonds (usually not taxable), their yields are not as highly Benchmark correlated with U.S. Treasury yields as other bonds. Therefore, munis frequently trade on an outright or and Spread even a dollar price. However, a muni's yield as a ratio to a benchmark Treasury yield is sometimes used as a relative value Calculation: measure. Munis

53. Define • General Autoregressive Conditional Heteroskedasticity GARCH and • Conditional = volatility today is dependent on the most recent value describe • Heteroskedastic = variances are not constant, they move over time Sigma^2n = gammaVL + au^2n-1 + B*sigma^2n-1 where sigma=vol, VL = lt vol, u is return

54. Describe • Is a special case of GARCH EWMA • Lacks mean reversion • Where lambda is the decay factor Sigma n = gammasigma^2n-1 + (1-gamma)r^2n-1

55. What is the The Tenor maturity of a swap called?

56. Which side The fixed-rate payer is the buyer, the floating-rate payer is the seller is the "buyer" of a swap?

57. How do you 1) Calculate the numerator as the float rate PV where the payment in each period = the fwd rate for that period * notional, calculate discounted using the derived spot rate from LIBOR the swap rate for a 2) Calculate the PV of the notional (1/2 paid semiannually), discounting the cfs over the term plain vanilla 3) Swap rate is 1) divided by 2) Swap? Swap spread = Swap rate - Tsy rate for the same maturity

58. How is a quoted as a fixed interest rate and an index on which the floating rate is based swap price • Example: "72-76 flat" on a 3 year swap means that the dealer will buy the swap (pay fixed) at 72bps over 3-yr Tsy, or sell quoted? the swap (pay float) for 76bps over 3-year Tsy

59. What Is London Inter Bank Offered Rate - An interest rate at which banks can borrow funds from other banks in the London LIBOR? interbank market. Although reference is often made to the LIBOR interest rate, there are actually 150 different LIBOR interest rates. LIBOR is calculated for 15 different maturities and for 10 different currencies. 60. What Variable rate demand obligations. Debt security which bears interest at a floating (variable) rate adjusted at specified intervals is a (such as daily, weekly, or monthly) and can be redeemed at its holder's option at par when the rate changes. Also called low VRDO? floater, variable rate demand note, or variable rate demand bond. Terms may be as much as 40 years, tax-exempt. VRDO's are municipal bonds. Always purchased at par. Generally low risk b/c they usually have credit enhancement (bank letter of credit or bond insurance), high liquidity.

61. What a tax-exempt, weekly reset index composed of 650 different high-grade, tax-exempt, VRDOs. It is a widely used benchmark for is the borrowers and dealer firms of variable-rate tax-exempt obligations SIFMA Index?

62. What approximation of the avg muni VRDO yields over the long run. In theory, future VRDO rates should equal the after-tax is the equivalent of LIBOR: ((1-marginal tax rate) LIBOR) plus a spread to reflect liquidity and other default risks. SIFMA Estimated as 0.67 1-month LIBOR as a benchmark %?

63. What = (treasury rate of comparable maturaty + LIBOR spread ) * SIFMA Percentage. is the Example: (Current Mkt YTM on a 3-yr US Tsy note + Current 3-yr LIBOR Swap spread over 3-year US Tsy Note)* 3 year SIFMA SIFMA percentage Swap Rate?