Modeling Portfolios That Contain Risky Assets II: Efﬁcient Frontiers for Various Models

Modeling Portfolios that Contain Risky Assets II: Efﬁcient Frontiers for Various Models

C. David Levermore University of Maryland, College Park

ICERM Lecture, 16 November 2011 Extracted from Math 420: Mathematical Modeling c 2011 Charles David Levermore Lecture II: Efﬁcient Frontiers for Various Models

Outline

1. Efﬁcient-Market Hypothesis 2. Modeling Portfolios with Risk-Free Assets 3. Modeling Long Portfolios 4. Modeling Long Portfolios with a Safe Investment 1. Efﬁcient-Market Hypothesis

The efﬁcient-market hypothesis (EMH) was framed by Eugene Fama in the early 1960’s in his University of Chicago doctoral dissertation, which was published in 1965. It has several versions, the most basic is the following.

Given the information available when an investment is made, no investor will consistently beat market returns on a risk-adjusted basis over long periods except by chance.

This version of the EMH is called the weak EMH. The semi-strong and the strong versions of the EMH make bolder claims that markets reﬂect information instantly, even information that is not publicly available in the case of the strong EMH. While it is true that some investors react quickly, most investors do not act instantly to every piece of news. Consequently, there is little evidence supporting these stronger versions of the EMH. The EMH is an assertion about markets, not about investors. If the weak EMH is true then the only way for an actively-managed mutual fund to beat the market is by chance. Of course, there is some debate regarding the truth of the weak EMH. It can be recast in the language of MPT as follows.

Markets for large classes of assets will lie on the efﬁcient frontier.

You can therefore test the weak EMH with MPT. If we understand “market” to mean a capitalization weighted collection of assets (i.e. an index fund) then the EMH can be tested by checking whether index funds lie on or near the efﬁcient frontier. You will see that this is often the case, but not always.

Remark. It is a common misconception that MPT assumes the weak EMH. It does not, which is why the EMH can be checked with MPT! Remark. It is often asserted that the EMH holds in rational markets. Such a market is one for which information regarding its assets is freely available to all investors. This does not mean that investors will act rationally based on this information! Nor does it mean that markets price assets correctly. Even rational markets are subject to the greed and fear of its investors. That is why we have bubbles and crashes. Rational markets can behave irrationally because information is not knowledge!

It could be asserted that the EMH is likely to hold in free markets. Such markets have many agents, are rational and subject to regulatory and legal oversight. These are all elements in Adam Smith’s notion of free market, which refers to the freedom of its agents to act, not to the freedom from any government role. Indeed, his radical idea was that government should nurture free markets by playing the role of empowering individual agents. He had to write his book because free markets do not arise spontaneously, even though his “invisible hand” insures that markets do. 2. Modeling Portfolios with Risk-Free Assets

Until now we have considered portfolios that contain only risky assets. We now consider two kinds of risk-free assets (assets that have no volatility associated with them) that can play a major role in portfolio management.

The ﬁrst is a safe investment that pays dividends at a prescribed interest rate µsi. This can be an FDIC insured bank account, or safe securities such as US Treasury Bills, Notes, or Bonds. (US Treasury Bills are most commonly used.) You can only hold a long position in such an asset.

The second is a credit line from which you can borrow at a prescribed interest rate µcl up to your credit limit. Such a credit line should require you to put up assets like real estate or part of your portfolio (a margin) as collateral from which the borrowed money can be recovered if need be. You can only hold a short position in such an asset. We will assume that µ µ , because otherwise investors would make cl ≥ si money by borrowing at rate µcl in order to invest at the greater rate µsi. (Here we are again neglecting transaction costs.) Because free money does not sit around for long, market forces would quickly adjust the rates so that µ µ . In practice, µ is about three points higher than µ . cl ≥ si cl si

We will also assume that a portfolio will not hold a position in both the safe investment and the credit line when µcl > µsi. To do so would effectively be borrowing at rate µcl in order to invest at the lesser rate µsi. While there can be cash-ﬂow management reasons for holding such a position for a short time, it is not a smart long-term position.

These assumptions imply that every portfolio can be viewed as holding a position in at most one risk-free asset: it can hold either a long position at rate µsi, a short position at rate µcl, or a neutral risk-free position. Markowitz Portfolios. We now extend the notion of Markowitz portfolios to portfolios that might include a single risk-free asset with return rate µrf. Let brf(d) denote the balance in the risk-free asset at the start of day d. For a long position µrf = µsi and brf(d) > 0, while for a short position µrf = µcl and brf(d) < 0.

A Markowitz portfolio drawn from one risk-free asset and N risky assets is uniquely determined by a set of real numbers f and f N that satisﬁes rf { i}i=1 N f + f = 1 , f < 1 if any f = 0 . rf i rf j 6 iX=1 The portfolio is rebalanced at the start of each day so that b (d) n (d) s (d 1) rf = f , i i − = f for i = 1, , N. Π(d 1) rf Π(d 1) i ··· − − The condition frf < 1 if any fj = 0 states that the safe investment must contain less than the net portfolio6 value unless it is the entire portfolio. Its value at the start of day d is N Π(d 1) = b (d)+ n (d) s (d 1) , − rf i i − iX=1 while its value at the end of day d is N 1 Π(d)= brf(d) 1+ D µrf + ni(d) si(d) . iX=1 Its return rate for day d is therefore Π(d) Π(d 1) r(d)= D − − Π(d 1) − N b (d) µ ni(d) si(d) si(d 1) = rf rf + D − − Π(d 1) Π(d 1) − iX=1 − N T = frf µrf + fi ri(d)= frf µrf + f r(d) . iX=1 The portfolio return rate mean µ and variance v are then given by

D D 1 1 fTr µ = D r(d)= D frf µrf + (d) dX=1 dX=1 D = f µ + fT 1 r(d) = f µ + fTm , rf rf D  rf rf dX=1 D   D 1 2 1 fTr fTm 2 v = D(D 1) r(d) µ = D(D 1) (d) − − − − dX=1 dX=1 D T = fT 1 r(d) m r(d) m f = fTVf . D(D 1)  − − − dX=1 We thereby obtain the formulas 

µ = µ 1 1Tf + mTf , v = fTVf . rf − Capital Allocation Lines. These formulas can be viewed as describing a point that lies on a certain half-line in the σµ-plane. Let (σ, µ) be the point in the σµ-plane associated with the Markowitz portfolio characterized by the distribution f = 0. Notice that 1Tf = 1 f > 0 because f = 0. 6 − rf 6 Deﬁne f ˜f = . 1Tf

Notice that 1T˜f = 1. Let µ˜ = mT˜f and σ˜ = ˜fTV˜f. Then (˜σ, µ˜) is the point in the σµ-plane associated with the Markowitzq portfolio without risk-free assets that is characterized by the distribution ˜f. Because

µ = 1 1Tf µ + 1Tf µ˜ , σ = 1Tf σ,˜ − rf we see that the point (σ, µ) in the σµ-plane lies on the half-line that starts at the point (0,µrf) and passes through the point (˜σ, µ˜) that corresponds to a portfolio that does not contain the risk-free asset. Conversely, given any point (˜σ, µ˜) corresponding to a Markowitz portfolio that contains no risk-free assets, consider the half-line (σ, µ)= φ σ,˜ (1 φ)µ + φ µ˜ where φ> 0 . − rf If a portfolio corresponding to (˜σ, µ˜) has distribution ˜f then the point on the half-line given by φ corresponds to the portfolio with distribution f = φ˜f. This portfolio allocates 1 1Tf = 1 φ of its value to the risk-free asset. The risk-free asset is held− long if φ− (0, 1) and held short if φ > 1 while φ = 1 corresponds to a neutral∈ position. We must restrict φ to either (0, 1] or [1, ) depending on whether the risk-free asset is the safe investment or the∞ credit line. This segment of the half-line is called the capital allocation line through (˜σ, µ˜) associated with the risk-free asset.

We can therefore use the appropriate capital allocation lines to construct the set of all points in the σµ-plane associated with Markowitz portfolios that contain a risk-free asset from the set of all points in the σµ-plane associated with Markowitz portfolios that contain no risk-free assets. Efficient Frontier. We now use the capital allocation line construction to see how the efficient frontier is modified by including risk-free assets. Recall that the efficient frontier for portfolios that contain no risk-free assets is given by

2 2 µ = µmv + νas σ σmv for σ σmv . q − ≥ Every point (˜σ, µ˜) on this curve has a unique frontier portfolio associated with it. Because µrf < µmv there is a unique half-line that starts at the point (0,µrf) and is tangent to this curve. Denote this half-line by µ = µ + ν σ for σ 0 . rf tg ≥ Let (σtg,µtg) be the point at which this tangency occurs. The unique frontier portfolio associated with this point is called the tangency portfolio f f associated with the risk-free asset; it has distribution tg = f(µtg). Then the appropriate capital allocation line will be part of the efﬁcient frontier. The so-called tangency parameters, σtg, µtg, and νtg, can be determined from the equations

µ = µ + ν σ , µ = µ + ν σ 2 σ 2 , tg rf tg tg tg mv as tg − mv q νas σtg νtg = . σ 2 σ 2 tg − mv q The first equation states that (σtg,µtg) lies on the capital allocation line. The second states that it also lies on the efficient frontier curve for portfolios that contain no risk-free assets. The third equates the slope of the capital allocation line to that of the efficient frontier curve at the point (σtg,µtg). By using the last equation to eliminate νtg from the first, and then using the resulting equation to eliminate µtg from the second, we find that σ 2 2 µmv µrf tg 2 2 σmv − = σtg σmv = . νas σ 2 σ 2 − − σ 2 σ 2 tg − mv q tg − mv q q We thereby obtain

2 2 2 νas σmv νas σmv σtg = σmvv1+ , µtg = µmv + , u µmv µ ! µmv µ u − rf − rf t 2 µmv µrf νtg = νasv1+ − . u νas σmv ! u f t The distribution tg of the tangency portfolio is then given by

µtg µmv 1 f = f (µ )= f + − V− m µ 1 tg f tg mv ν 2 − mv as σ 2 = σ 2 V 11 + mv V 1 m µ 1 mv − µ µ − − mv mv rf 2 − σmv 1 = V− m µ 1 . µ µ − rf mv − rf Remark. The above formulas can be applied to either the safe investment or the credit line by simply choosing the appropriate value of µrf.

Example: One Risk-Free Rate Model. First consider the case when

µsi = µcl < µmv. Set µrf = µsi = µcl and let νtg and σtg be the slope and volatility of the tangency portfolio that is common to both the safe investment and the credit line. The efﬁcient frontier is then given by

µ (σ)= µ + ν σ for σ [0, ) . ef rf tg ∈ ∞ f Let tg be the distribution of the tangency portfolio that is common to both the safe investment and the credit line. The distribution of the associated portfolio is then given by f σ f ef(σ)= tg for σ [0, ) , σtg ∈ ∞ These portfolios are constructed as follows: 1. if σ = 0 then the investor holds only the safe investment;

2. if σ (0, σ ) then the investor places ∈ tg σ σ tg − of the portfolio value in the safe investment, • σtg σ f of the portfolio value in the tangency portfolio tg; • σtg

f 3. if σ = σtg then the investor holds only the tangency portfolio tg; 4. if σ (σ , ) then the investor places ∈ tg ∞ σ f of the portfolio value in the tangency portfolio tg, • σtg σ σ by borrowing − tg of this value from the credit line. • σtg Example: Two Risk-Free Rates Model. Next consider the case when µsi < µcl < µmv. Let νst and σst be the slope and volatility of the so- called safe tangency portfolio associated with the safe investment. Let νct and σct be the slope and volatility of the so-called credit tangency portfolio associated with the credit line. The efﬁcient frontier is then given by

µ + ν σ for σ [0, σ ] , si st ∈ st  2 2 µef(σ)= µmv + νas σ σmv for σ [σ , σ ] ,  − ∈ st ct µ + ν σq for σ [σ , ) , cl ct ∈ ct ∞  where  2 2 µmv µsi νas σmv νst = νasv1+ − , σst = σmvv1+ , u νas σmv ! u µmv µ ! u u − si t 2 t 2 µmv µcl νas σmv νct = νasv1+ − , σct = σmvv1+ . u νas σmv ! u µmv µ ! u u − cl t t The return rate means for the tangency portfolios are ν 2 σ 2 ν 2 σ 2 µ = µ + as mv , µ = µ + as mv , st mv µ µ ct mv µ µ mv − si mv − cl while the distributions of risky assets for these portfolios are 2 2 σmv 1 σmv 1 f = V− m µ 1 , f = V− m µ 1 . st µ µ − si ct µ µ − cl mv − si mv − cl By the two mutual fund property, the distribution of risky assets for any efﬁcient frontier portfolio is then given by σ f st for σ [0, σst] , σst ∈ µ µ (σ) µ (σ) µ f  ct ef f ef st f ef(σ)=  − st + − ct for σ (σst, σct) ,  µ µ µ µ ∈  σ ct − st ct − st f for σ [σ , ) . σ ct ∈ ct ∞  ct    These portfolios are constructed as follows:

1. if σ = 0 then the investor holds only the safe investment;

2. if σ (0, σ ) then the investor places ∈ st σ σ st − of the portfolio value in the safe investment, • σst σ f of the portfolio value in the safe tangency portfolio st; • σst

f 3. if σ = σst then the investor holds only the safe tangency portfolio st; 4. if σ (σ , σ ) then the investor holds ∈ st ct µ µ (σ) ct − ef of the portfolio value in the safe tangency portfolio f , • µ µ st ct − st µ (σ) µ ef − st of the portfolio value in the credit tangency portfolio f ; • µ µ ct ct − st

f 5. if σ = σct then the investor holds only the credit tangency portfolio ct;

6. if σ (σ , ) then the investor places ∈ ct ∞ σ f of the portfolio value in the credit tangency portfolio ct, • σct σ σ by borrowing − ct of this value from the credit line. • σct Remark. Some brokers will simply ask investors to select a σ that reﬂects the risk they are willing to take and then will build a portfolio for them that f is near the efﬁcient frontier portfolio ef(σ). To guide this selection of σ, the broker will describe certain choices of σ as being “very conservative”, “conservative”, “aggressive”, or “very aggressive” without quantifying what these terms mean. The performance of the resulting portfolio will often be disappointing to those who selected a “very conservative” σ and painful to those who selected a “very aggressive” σ. We will see reasons for this when we develop model-based methods for selecting a σ.

Credit Limits. The One Risk-Free Rate and Two Risk-Free Rates models breakdown when µ µ because in that case the capital allocation cl ≥ mv line construction fails. Moreover, both models become unrealistic for large values of σ, especially when µcl is close to µmv, because they require an investor to borrow or to take short positions without bound. In practice such positions are restricted by credit limits. If we assume that in each case the lender is the broker and the collateral is part of your portfolio then a simple model for credit limits is to restrict the total short position of the portfolio to be a fraction ℓ 0 of the net portfolio ≥ value. The value of ℓ will depend upon market conditions, but brokers will often allow ℓ> 1 and seldom allow ℓ> 5. This model becomes 1 1Tf (1 1Tf) f 1Tf N | − |− − + | |− ℓ, where f = f . 2 2 ≤ | | | i| iX=1 This simpliﬁes to the convex constraint

1 1Tf + f 1 + 2ℓ. | − | | |≤ This additional constraint makes the minimization problem more difﬁcult. It can be solved numerically by using primal-dual algorithms from convex optimization. Rather than studying this general problem, in the next section we will consider the simpler problem of restricting to portfolios that contain only long positions, which is equivalent to taking ℓ = 0 above. Exercise. Consider the following groups of assets: (a) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2009; (b) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2007; (c) S&P 500 and Russell 1000 and 2000 index funds in 2009; (d) S&P 500 and Russell 1000 and 2000 index funds in 2007.

Assume that µsi is the US Treasury Bill rate at the end of the given year, and that µcl is three percentage points higher. On a single graph plot the points (σi, mi) for every asset in groups (a) and (c) along with the efﬁcient frontiers for group (a), for group (c), and for groups (a) and (c) combined taking into account the risk-free assets. Do the same thing for groups (b) and (d) on a second graph. (Use daily data.) Comment on any relationships you see between the objects plotted on each graph.

f f Exercise. Find st and ct for the six asset groupings in the preceding exercise. Compare the analogous groupings in 2007 and 2009. 3. Modeling Long Portfolios

Because the value of any portfolio with short positions has the potential to go negative, many investors do not want to hold a short position in any risky asset. In that case the optimization problem must be restricted to so-called long portfolios by imposing the constraints f 0. Because these are inequality constraints, the problem is not solved≥ simply by Lagrange multipliers. Here we treat it in the case where f (µ ) 0 for some µ . f 0 ≥ 0 f The frontier portfolio distribution f(µ) can be expressed as µ µ0 1 f (µ)= f (µ )+ − V− m µ 1 . f f 0 ν 2 − mv as Because 1TV 1 m µ 1 = b µ a = 0, and because 1 and m − − mv − mv are not co-linear, we see that the second term above has both positive and f 0 negative entries whenever µ = µ0. Because f(µ0) , the set of µ for which f (µ) 0 is satisﬁed must6 be a closed interval≥ containing µ . f ≥ 0 Remark. It is often true that f 0 because the least risky position in a mv ≥ healthy market generally does not require any assets to be held short. In that case it is natural to take µ = µ . However, it is false that f 0 0 mv mv ≥ for every positive deﬁnite V. Indeed, for the case N = 2 one has

V v11 v12 V 1 1 v22 v12 = , − = 2 − , v12 v22! v v v v12 v11 ! 11 22 − 12 − while fmv is a positive multiple of

V 11 1 v22 v12 − = 2 − . v v v v11 v12! 11 22 − 12 − On one hand, V is positive definite if and only if v11 > 0, v22 > 0, and 2 V V 11 v11v22 > v12. On the other hand, if is positive definite then − has nonnegative entries if and only if v v and v v . You can find 11 ≥ 12 22 ≥ 12 positive definite matrices for which these conditions do not hold. Long Frontier. The set of points in the σµ-plane that correspond to long portfolios that have less volatility than every other long portfolio with the same return rate mean is called the long frontier. Because the distribution f of any long portfolio satisfies 1Tf = 1 and f 0 while its return rate ≥ mean is given by µ = mTf, we see that µ must satisfy the bounds

µ µ µ , mn ≤ ≤ mx where µ = min m : i = 1, , N , mn { i ··· } µ = max m : i = 1, , N . mx { i ··· } It should also be clear to you that every point in [µmn,µmx] is the return rate mean for some long portfolio. The long frontier is therefore given by

σ = σlf(µ) where the function σlf(µ) is only deﬁned over [µmn,µmx]. As we will see, this function is given by a ﬁnite list of formulas that are straightforward to obtain when N is not too large. Because f (µ ) 0, the distributions of those frontier portfolios that are f 0 ≥ long portfolios are given by f (µ) where µ [µ , µ ] with f ∈ 1 1 µ = max µ R : f (µ) 0 entrywise , 1 ∈ f ≥ µ = min nµ R : f (µ) 0 entrywiseo. 1 ∈ f ≥ The long frontier coincidesn with the frontier for µ [µ , µ o]. That is to say, ∈ 1 1 σ (µ)= σ (µ) for µ [µ , µ ] . lf f ∈ 1 1

We can extend σlf(µ) to the interval [µmn,µmx] by an iterative process. We will show how to do this for the right endpoint. The steps are analogous for the left endpoint. We initialize the iteration by setting

m = m , V = V , σ (µ)= σ (µ) , f (µ)= f (µ) . 0 0 f0 f f0 f Suppose we have extended σlf(µ) to an interval with right endpoint µk. If m V µk = µmx then we are done. Otherwise, let the vector k and matrix k m V be obtained from k 1 and k 1 by removing every entry with an index corresponding to an entry− of f −(µ ) that is zero. In other words, let m fk 1 k k − be the return rate mean vector and Vk be the return rate covariance matrix after we drop from consideration every asset corresponding to an entry of f (µ ) that is zero. (Typically only one asset will be dropped each time.) fk 1 k − Let σ = σ (µ) be the frontier of this reduced portfolio. The dimension fk of the associated frontier distribution f (µ) is less than that of f (µ) by fk fk 1 the number of zero entries of f (µ ). The entries of f (µ ) are− exactly fk 1 k fk k the positive entries of f (µ ).− Therefore σ (µ) satisﬁes fk 1 k fk − σ (µ )= f (µ )TV f (µ ) fk k fk k k fk k = f (µ )TV f (µ )= σ (µ ) . fk 1 k k 1 fk 1 k fk 1 k − − − − Because σ (µ) is associated with fewer assets, we also know that fk σ (µ) σ (µ) for every µ. fk fk 1 ≥ − Because these functions are equal at µ = µk, we conclude that moreover

σ′ (µ )= σ′ (µ ) . fk k fk 1 k − Now let R f 0 µ +1 = max µ : (µ) entrywise . k ∈ fk ≥ Because f (µ ) > 0, it isn clear that µ > µ . Finally, seto fk k k+1 k σ (µ)= σ (µ) for µ [µ , µ ] . lf fk ∈ k k+1 We have thereby extended σlf(µ) to an interval with right endpoint µk+1, whereby we can return to the beginning of the iteration. After applying the analogous iterative process to extend the left endpoint, you ﬁnd that σlf(µ) is expressed over [µmn,µmx] as the list function σ (µ) for µ [µ ,µ ] , fk ∈ k+1 k  σlf(µ)= σf(µ) for µ [µ1, µ1] ,  ∈ σ (µ) for µ [µ , µ ] . fk ∈ k k+1  This is strictly convex and continuously differentiable over [µmn,µmx]. Its second derivative will have a jump discontinuity at each µk and µk that lies in (µmn,µmx).

Remark. Here we will not show why the above algorithm for computing σlf(µ) works. The proof is far more complicated than others in this course. The algorithm is straightforward to implement when N is not too large. When either N is large or no µ0 exists then σlf(µ) can be approximated numerically using a primal-dual interior algorithm for convex optimization. Such algorithms are taught in some graduate courses on optimization. Long Frontier Portfolios. Associated with each of the distributions f (µ) fk and f (µ) of the reduced portfolios in the above construction we deﬁne fk the distributions f (µ) and f (µ) to be the N-vectors obtained by adding fk fk zero entries corresponding to assets that are not held by the respective reduced portfolios.

The distibutions associated with the long frontier portfolios are then given over [µmn,µmx] by the list function f (µ) for µ [µ ,µ ] , fk ∈ k+1 k f f lf(µ)= f(µ) for µ [µ1, µ1] ,  ∈ f (µ) for µ [µ , µ ] , fk ∈ k k+1  This is continuous and piecewise linear over [µmn,µmx]. Its ﬁrst derivative will have a jump discontinuity at each µk and µk that lies in (µmn,µmx). f Because lf(µ) is continuous and piecewise linear over [µmn,µmx] with nodes µk and µk in [µmn,µmx], it can be expressed in terms of the so- called nodal portfolio distributions given by f = f (µ ) , f = f (µ ) . k fk k k fk k Because f = f (µ ) , f = f (µ ) , k+1 fk k+1 k+1 fk k+1 by the two mutual fund property we have µ µ µ µ f (µ)= k+1 − f + − k f , fk µ µ k µ µ k+1 k+1 − k k+1 − k µ1 µ µ µ1 ff(µ)= − f 1 + − f1 , µ1 µ1 µ1 µ1 µ − µ − k+1 µ µk f (µ)= − f k + − f k+1 . fk µ µ µ µ k+1 − k k+1 − k General Portfolio with Two Risky Assets. Recall the portfolio of two risky assets with mean vector m and covarience matrix V given by m v v m = 1 , V = 11 12 . m2! v12 v22! Here we will assume that m1 < m2, so that µmn = m1 and µmx = m2. The frontier portfolios are

f 1 m2 µ R f(µ)= − for µ . m m µ m1! ∈ 2 − 1 − Clearly f (µ) 0 if and only if µ [m , m ] = [µ ,µ ]. Therefore f ≥ ∈ 1 2 mn mx f (µ)= f (µ) for µ [m , m ] , lf f ∈ 1 2 and the long frontier is determined by

σ (µ)= σ (µ)= f (µ)TV f (µ) for µ [m , m ] . lf f f f ∈ 1 2 q In this case there is no need to construct σlf(µ) by the foregoing algorithm. Simple Portfolio with Three Risky Assets. Recall the portfolio of three risky assets with mean vector m and covarience matrix V given by

m1 m d 1 r r m − V 2 = m2 =  m  , = s r 1 r . m3 m + d r r 1             Here m R, d, s R , and r ( 1, 1), where the last condition is ∈ ∈ + ∈ −2 equivalent to the condition that V is positive deﬁnite given s > 0. Its frontier parameters are

1 1 + 2r b σmv = = s , µmv = = m, sa s 3 a b2 d 2 νas = c = . s − a s s1 r − f 11 Its minimum volatility portfolio is mv = 3 , whereby we can take µ0 = m. Clearly [µ ,µ ] = [m d, m + d]. mn mx − Its frontier is determined by

1 + 2r 1 r µ m 2 σf(µ)= s v + − − for µ ( , ) , u 3 2 d ! ∈ −∞ ∞ u while the distributiont of the frontier portfolio with return rate mean µ is

1 µ m m+2d µ − 3 3 − 2d 2d − f  1   1  f(µ)= 3 = 3 .    2  1 µ m µ m+ d 3 + 2−d   − 3     2d      The frontier portfolio holds long postitions when µ (m 2d, m + 2d). ∈ − 3 3 Therefore [µ , µ ] = [m 2d, m + 2d] and the long frontier satisfies 1 1 − 3 3 σ (µ)= σ (µ) for µ [m 2d, m + 2d] . lf f ∈ − 3 3 The distribution weight of first asset vanishes at the right endpoint while that of the third vanishes at the left endpoint. 2 In order to extend the long frontier beyond the right endpoint µ1 = m + 3d to µmx = m + d we reduce the portfolio by removing the first asset and set

m2 m 2 1 r m1 = = , V1 = s . m3! m + d! r 1! Then

1 1 1 r 1 1 V1− = − , V1− 1 = 1 , s2(1 r2) r 1 ! s2(1 + r) − − whereby

T 1 2 T 1 2m + d a1 = 1 V− 1 = , b1 = 1 V− m1 = , 1 s2(1 + r) 1 s2(1 + r) 2 T 1 2m(m + d) d c1 = m V− m1 = + . 1 1 s2(1 + r) s2(1 r2) − The associated frontier parameters are 1 1+ r b = = = 1 = + 1 σmv1 s , µmv1 m 2d, sa1 s 2 a1 2 b1 d 2 νas = vc1 = , 1 u − a 2s s1 r u 1 − whereby the frontier of the reducedt portfolio is given by

1 2 1+ r 1 r µ m 2d σ (µ)= s v + − − − . f1 u 2 2 1 u 2d ! u Similarly, to extend beyondt the left endpoint we remove the third asset and ﬁnd that the frontier of the reduced portfolio is given by

1 2 1+ r 1 r µ m + 2d σ (µ)= s v + − − . f1 u 2 2 1 u 2d ! u t By putting these pieces together we see that the long frontier is given by

2 1+ r 1 r µ m+1d + − 2 for [ 2 ] s v − 1 µ m d, m 3d , u 2 2 2d ! ∈ − −  u  t  1 + 2r 1 r µ m 2 2 2 σlf(µ)= s + − − for µ [m d, m + d] ,  s 3 2 d ∈ − 3 3  1 2 1+ r 1 r µ m d 2 s + − − −2 for µ [m + d, m + d] .  v 1d 3  u 2 2 2 ! ∈  u  t This is strictly convex and continuously differentiable over [m d, m + d]. − Its second derivative is defined and positive everywhere in [m d, m + d] − except at the points µ = m 2d where it has jump discontinuities. We ± 3 have 5 + 4r σ (m 2d)= s , σ (m d)= s. lf ± 3 s 9 lf ± Finally, the long frontier distributions are given by m µ d− µ m+d 2  −  for µ [m d, m d] ,  d ∈ − − 3   0     1 µ m  3 2−d  −1 2 2 f (µ)=   for µ [m d, m + d] , lf  3 3 3  1 µ m ∈ −  + −  3 2d     0   m+d µ for µ [m + 2d, m + d] .  d −  3  ∈  µ m  −d      Notice that the distribution weights do not depend on either s or r. They are continuous and piecewise linear over [m d, m+d]. Their first derivatives − are defined everywhere in [m d, m + d] except at the points µ = m 2d − ± 3 where they have jump discontinuities. Exercise. Find a 2 2 positive definite matrix V such that the vector V 11 × − has a negative entry.

Exercise. Consider the following groups of assets: (a) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2009; (b) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2007; (c) S&P 500 and Russell 1000 and 2000 index funds in 2009; (d) S&P 500 and Russell 1000 and 2000 index funds in 2007. For group (a), group (c), and groups (a) and (c) combined, determine if f (µ ) 0 for some µ . If so, add plots of the associated long frontiers f 0 ≥ 0 to the graph you produced for these assets in the last exercise of the last section. (Use daily data.) Do the same thing for groups (b) and (d). Explain any relationships you see between the objects plotted on each graph. For which of these groupings is fmv 0? Compute f (µ) for each of these ≥ lf groupings, identifying the nodal portfolios. 4. Modeling Long Portfolios with a Safe Investment

We now consider investors who will not hold a short position in any asset. Such an investor will not borrow to invest in a risky asset, so the safe investment is the only risky-free asset that we need to consider. We will suppose that µ µ . We will also suppose that σ (µ ) > 0 and si ≤ mx lf′ mx σ (µ ) 0, which is a common situation. (This will be the case when lf′ mn ≤ f 0.) We will use the capital allocation line construction to obtain the mv ≥ efﬁcient long frontier for portfolios that might include the safe investment.

Efﬁcient Long Frontier. The tangent line to the curve σ = σlf(µ) at the point (σmx,µmx) will intersect the µ-axis at µ = ηmx where

σlf(µmx) ηmx = µmx . − σlf′ (µmx) We will consider the cases µ η and µ < η separately. si ≥ mx si mx In the case when µ η the efﬁcient long frontier is simply given by si ≥ mx µmx µsi µef(σ)= µsi + − σ for σ [0, σmx] . σmx ∈

In the case when µsi < ηmx there is a tangency portfolio (σst,µst) such that σlf′ (µst) > 0. Because σlf(µ) is an increasing, continuously differentiable function over [µst,µmx] with image [σst, σmx], it has an increasing, 1 continuously differentiable inverse function σlf− (σ) over [σst, σmx] with image [µst,µmx]. The efﬁcient long frontier is then given by µ µ µ + st − si σ for σ [0, σ ] , si σ ∈ st µef(σ)=  st  1 σ− (σ) for σ [σ , σ ] . lf ∈ st mx  Because σlf(µ) can be expressed as a list function, you can also express 1 σ− (σ) as a list function. We illustrate this below for the case f 0. lf mv ≥ Suppose that f 0 and set µ = µ . Then σ (µ) has the form mv ≥ 0 mv lf 2 µ µmv σ (µ)= σ (µ) σ 2 + − k for µ [µ , µ ] , lf f v mvk k k+1 k ≡ u νas ! ∈ u k where σ , µ , andt ν are the frontier parameters associated with mvk mvk ask th the vector m and matrix Vk that determined σ (µ) in the k step of our k fk iterative construction of ( ). In particular, = , = , σlf µ σmv0 σmv µmv0 µmv and ν = ν because m = m and V = V. as0 as 0 0

1 Then σlf− (σ) has the form

σ 1(σ)= µ + ν σ2 σ 2 for σ [σ , σ ] , lf− mvk ask mvk k k+1 r − ∈ where σk = σlf(µk) and σk+1 = σlf(µk+1). Finally, we must ﬁnd the tangency portfolio (σst,µst). The tangent line to the long frontier at the point (σk, µk) intercepts the µ-axis at µ = ηk where 2 2 σ (µ ) νas σmv η = µ lf k = µ k k . k k − σ (µ ) mvk − µ µ lf′ k k − mvk These intercepts satisfy η < η η when µ < µ µ . If k k+1 ≤ mx k k+1 ≤ mx we set η = then for every µ < η there is a unique j such that 0 −∞ si mx η µ < η . j ≤ si j+1 For this value of j we have the tangancy parameters

µ µ 2 ν σ 2 mvj − si asj mvj νst = νas v1+ , σst = σmv v1+ . j u ν σ j u µ µ u asj mvj ! u mvj si! u u − t t Therefore when µsi < ηmx the efﬁcient long frontier is given by

µ + ν σ for σ [0, σ ] , si st ∈ st µ (σ)= µ + ν σ2 σ 2 for σ [σ , σ ] , ef  mvj asj mvj st j+1  − ∈ µ + ν q σ2 σ 2 for σ [σ , σ ] and k>j. mvk ask − mvk ∈ k k+1   q  Efficient Long Frontier Portfolios. Recall that the distibutions associated with the efficient long frontier portfolios without the safe investment are given over [µmv,µmx] by µ µ µ µ f (µ)= k+1 − f + − k f for µ [µ , µ ] , lf µ µ k µ µ k+1 ∈ k k+1 k+1 − k k+1 − k f f f where µ0 = µmv, 0 = mv, while k is the nodal portfolio distribution associated with µ for any k 1. k ≥ The return rate mean and distribution for the safe tangency portolio are 2 2 νas σmv µ = µ + j j , st mvj µ µ mvj − si µ µst µ µ f = f (µ )= j+1 − f + st − j f . st lf st µ µ j µ µ j+1 j+1 − j j+1 − j The distribution of risky assets for any efficient long frontier portfolio is σ f for σ [0, σ ] , σ st ∈ st  st µ µ (σ) µ (σ) µ  j+1 − ef f ef st f f (σ)=  st + − j+1 for σ [σst, σj+1] , ef  µ µ µ µ ∈  j+1 − st j+1 − st µ µ (σ) µ (σ) µ k+1 − ef f + ef − k f for σ [σ , σ ] ,  k k+1 k k+1  µk+1 µk µk+1 µk ∈  − −  where k > j in the last case. Remark. If we had also added a credit line to the portolio then we would have needed to find the credit tangency portfolio and added the appropriate capital allocation line to the efficient long frontier. Typically there are two kinds of credit lines an investor might consider. One available from your broker usually requires that some of your risky assets be held as collateral. One downside of using this kind of credit line is that when the market goes down then your broker can force you either to add assets to your collateral or to sell assets in a low market to pay off your loan. Another kind of credit line might use real estate as collateral. Of course, if the price of real estate falls then you again might be forced to sell assets in a low market to pay off your loan. For investors who hold short positions in risky assets, these risks are hedged because they also make money when markets go down. Investors who hold only long positions in risky assets and use a credit line can find themselves highly exposed to large losses in a market downturn. It is not a wise position to take — yet many do in a bubble. General Portfolio with Two Risky Assets. Recall the portfolio of two risky assets with mean vector m and covarience matrix V given by m v v m = 1 , V = 11 12 . m2! v12 v22! Here we will assume that m1 < m2, so that µmn = m1 and µmx = m2. The long frontier associated with just these two risky assets is given by

2 2 µ µmv σlf(µ)= vσmv + − for µ [m1, m2] , u νas ! ∈ u where the frontier parameterst are

2 2 v11v22 v12 (m2 m1) σmv = v − , νas = v − , uv11 + v22 2v12 uv11 + v22 2v12 u − u − t (v v )m + (v t v )m µ = 22 − 12 1 11 − 12 2 . mv v + v 2v 11 22 − 12 The minimum volatility portfolio is

1 v22 v12 fmv = − . v + v 2v v11 v12! 11 22 − 12 − We will assume that v12 v11 and v12 v22, so that fmv 0 and µ = fT m [m , m ]≤. The efﬁcient long≤ frontier associated≥ with just mv mv ∈ 1 2 these two risky assets is then given by (σlf(µ),µ) where µ [µmv, m2]. We now show how this is modiﬁed by the inclusion of a safe inves∈ tment.

The parameters associated with the construction of σlf(µ) are

µ0 = µmv , µ1 = µmx = m2 , σ0 = σmv , σ1 = σmx = σ2 = √v22 . The µ-intercept of the tangent line through (σmx,µmx) = (σ2, m2) is 2 2 σlf(µmx) νas σ2 v22m1 v12m2 η = µ = m2 = − . mx mx − σ (µ ) − m µ v v lf′ mx 2 − mv 22 − 12 We will present the two cases that arise in order of increasing complexity: η µ and µ < η . mx ≤ si si mx

When η µ the efﬁcient long frontier is determined by mx ≤ si m2 µsi µef(σ)= µsi + − σ for σ [0, σ2] . σ2 ∈

When µsi < ηmx the tangency portfolio parameters are given by

2 2 µmv µsi νas σmv νst = νmvv1+ − , σst = σmvv1+ , u νas σmv ! u µmv µ ! u u si t − and the efﬁcient long frontier is determined by t

µsi + νst σ for σ [0, σst] , µef(σ)= ∈ µ + ν σ2 σ 2 for σ [σ , σ ] .  mv as − mv ∈ st 2 q  Simple Portfolio with Three Risky Assets. Recall the portfolio of three risky assets with mean vector m and covarience matrix V given by

m1 m d 1 r r m − V 2 = m2 =  m  , = s r 1 r . m3 m + d r r 1       The efﬁcient long frontier  associated with just these three risky assets is given by (σ (µ),µ) where µ [m, m + d] and lf ∈ 1 + 2r 1 r µ m 2 s + − − for µ [m, m + 2d] ,  3 s 3 2 d ∈ σlf(µ)=  1 2  1+ r 1 r µ m d s + − − − 2 for µ [m + 2d, m + d] . v 2 2 1 ∈ 3 u 2d !  u  t We now show how this is modiﬁed by the inclusion of a safe investment. In the construction of σlf(µ) we found that 2 µ0 = m, µ1 = m + 3d, µ2 = µmx = m + d,

1 + 2r 5 + 4r σ = s , σ = s , σ = σmx = s. 0 s 3 1 s 9 2 The frontier parameters for σ (µ) were f0 1 + 2r d 2 σmv = s , µmv = m , νas = , 0 s 3 0 0 s s1 r − while those for σ (µ) were f1

1+ r 1 d 2 σmv = s , µmv = m + d , νas = . 1 s 2 1 2 1 2s s1 r − Because µ m , for µ [m, m + 2d] , −2 3 21 r  d ∈ σ (µ)σ′ (µ)= s − 1 lf lf µ m 2d 2 2  − − , for µ [m + d, m + d] ,  1 2 ∈ 3 4d   we can see that the µ-intercepts of the tangent lines through the points (σ1, µ1) and (σ2, µ2) = (σmx,µmx) are respectively σ (µ ) 5 + 4r 1 + 2r η = µ lf 1 = m + 2d d = m d, 1 1 − σ (µ ) 3 − 3 3r − 1 r lf′ 1 − − σ (µ ) 1 r η = µ lf mx = m + d d = m d. mx mx − σ (µ ) − 1 r − 1 r lf′ mx − − We will present the three cases that arise in order of increasing complexity: η µ , η µ < η , and µ < η . mx ≤ si 1 ≤ si mx si 1 When η µ the efﬁcient long frontier is determined by mx ≤ si µmx µsi µef(σ)= µsi + − σ for σ [0, σmx] . σmx ∈

When η µ < η the tangency portfolio parameters are given by 1 ≤ si mx 2 2 µmv µ νas σmv ν = ν 1+ 1 − si , σ = σ 1+ 1 1 , st as1v st mv1v u νas σmv ! u µmv µ ! u 1 1 u 1 − si and the efﬁcientt long frontier is determined by t

µsi + νst σ for σ [0, σst] , µef(σ)= ∈ µ + ν σ2 σ 2 for σ [σ , σ ] .  mv1 as1 − mv1 ∈ st mx q  When µsi < η1 the tangency portfolio parameters are given by 2 2 µmv µ νas σmv ν = ν 1+ 0 − si , σ = σ 1+ 0 0 , st as0v st mv0v u νas σmv ! u µmv µ ! u 0 0 u 0 − si and the efﬁcientt long frontier is determined by t

µsi + νst σ for σ [0, σst] ,  ∈ µ + ν σ2 σ 2 for σ [σ , σ ] , µef(σ)=  mv0 as0 mv0 st 1  − ∈  q µ + ν σ2 σ 2 for σ [σ , σ ] . mv1 as1 − mv1 ∈ 1 mx   q   Remark. The above formulas for µef(σ) can be made more explicit by replacing σ , µ , ν , σ , µ , ν , σ , µ , and σ with mv0 mv0 as0 mv1 mv1 as1 mx mx 1 their explicit expressions in terms of m, d, s and r. Exercise. Consider the following groups of assets: (a) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2009; (b) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2007; (c) S&P 500 and Russell 1000 and 2000 index funds in 2009; (d) S&P 500 and Russell 1000 and 2000 index funds in 2007. Add plots of the associated efﬁcient long frontiers to the graphs that you produced for the last exercise. Comment on any relationships you see between the objects plotted on each graph.

f Exercise. Find st on the efﬁcient long frontier for the six asset groupings f in the preceding exercise. Compare st for the analogous groupings in 2007 and 2009.