Portfolio theory and risk management

Homework set 2

Filip Lindskog

General information

The homework set gives at most 3 points which are added to your result onthe exam. You may work individually or in groups of at most two persons.To obtain the points you/the group must present the solution nicely in areport which clearly shows how the problems were solved. I will considerboth correctness and the quality of the report important when I evaluateyour work. The report, printed on paper, must be handed in on time inorder to be accepted.

The solutions to the homework set must be handed in no later thanFriday 1/10 10:15.

Exercises

Exercise 1. Consider the numerical evaluation of mean-variance approachesto optimal investments in Section 4.1.1 in the lecture notes. Change themean vector µ to values that you consider realistic (for yearly returns) sothat the theoretical Sharpe ratio becomes bigger, approximately one. Dothe corresponding computations and the plots corresponding to Figures 4.1,4.2, and 4.3.

Exercise 2. Consider the mean-variance approach to optimal investmentspresented in Section 4.1 in the lecture notes. Suppose there is also a liability(subtract the random variable L from the value of the portfolio at the endof the time period).

(a) Determine and interpret the optimal solution to the problem correspond-ing to (4.4) in the lecture notes (now with a liability).

(b) Suppose that you are not allowed to take a short position in the bond.Determine and interpret the optimal solution to the problem correspondingto (4.4) in the lecture notes (now with a liability and no short sales of thebond).

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1 Evaluating the methods on simulated data

Consider a vector R of percentage returns for two risky asset whose meanvector and covariance matrix are given by

µ =

(1.0251.075

)and Σ =

(σ2

1σ1σ20.5

σ1σ20.5 σ22

),

where σ1 = 0.3 and σ2 = 0.5. Suppose that there also exists a risk-free assetwith percentage return Rrf = 1.

Suppose that we want to invest according to the solution to one of thethree versions of the investment problem that we have analyzed analytically.The first investment criterion is

maximize cV0(µTw + wRrf

Rrf ) − 1

2wTΣw

subject to 1Tw + wRrf≤ V0 (Trade-off)

The second investment criterion is

maximize µTw + wRrf

Rrf

subject to wTΣw ≤ σ20V 2

0(MaxExp)

and 1Tw + wRrf≤ V0

with σ0 = 0.3, i.e. we want the standard deviation of the percentage returnfor the portfolio to be smaller or equal to the smallest of those of the riskyassets. The third investment criterion is

minimize wTΣw

subject to µTw + wRrf

Rrf ≥ µ0V0 (MinVar)

and 1Tw + wRrf≤ V0.

The parameters c = 1.995897 and µ0 = 1.045092 are chosen so that the op-timal solutions to the three investment problems coincide. We may withoutloss of generality set V0 = 1 which means that the solution w is the positionin the risky assets per unit of initial capital. The common theoretical solu-tion w to the investment problems (Trade-off), (MaxExp), and (MinVar) isgiven by

w = cΣ−1(µ − Rrf1)

= σ0

Σ−1(µ − Rrf1)√(µ − Rrf1)TΣ−1(µ − Rrf1)

= (µ0 − Rrf )Σ−1(µ − Rrf1)

(µ − Rrf1)TΣ−1(µ − Rrf1)

≈(

0.0740.577

)

2

and wRrf= 1 − 1Tw ≈ 0.349. However, we do not know µ and Σ and

therefore not the optimal solution (w, wRrf).

Suppose that R is normally distributed with mean µ and covariancematrix Σ and that we have observed 200 outcomes of independent copiesof R. From these observations we can compute estimates µ and Σ andobtain estimates (w, wRrf

) by replacing µ and Σ by µ and Σ in the aboveexpressions for the solutions to (Trade-off), (MaxExp), and (MinVar). Inorder to determine the accuracy of these estimates we repeat this scheme3000 times and plot the estimated weights w for the solutions to the threeinvestment problems.

The first three plots in Figure 1 are scatter plots of the 3000 portfolioweights in the risky assets for the three versions of the investment problemsin the above order. In total, 3000 samples of 200 independent copies of R

were generated. Each of the samples generated an estimate of (µ,Σ) whichin turn generated one point w for each of the three versions of the investmentproblem.

Notice that for (MaxExp) the risk constraint and the estimate Σ forcethe solution w to be a point on the ellipse wTΣw = σ2

0. Since the estimates

Σ vary across the 3000 samples the points w of the scatter plot form a pointcloud that is concentrated near the ellipse wTΣw = σ2

0. The points of the

scatter plots for (Trade-off) and (MinVar) are more spread-out, especiallyfor the problem (MinVar). Notice in particular that many of the the solu-tions w for (MinVar) based on the simulated samples are very far from thetheoretical solution w = (0.074, 0.577)T . The reason for this is that manyof the estimated values µ − Rrf1 are very close to 0 causing the weightsto “explode” due to the values very close to 0 in the denominator that for(MinVar), unlike (MaxExp), are not canceled by the same small values inthe nominator.

Each of the remaining three plots in Figure 1 are scatter plots of the3000 points

(σ(w), µ(w)) = (√

wTΣw,µTw + (1 − 1Tw)Rrf ),

for the three versions of the investment problem, where each w is a pointof the corresponding scatter plot in Figure 1. That is, the pairs of plots are(1,4), (2,5), and (3,6). For a given vector w of portfolio weights, σ(w) andµ(w) are the standard deviation and expected value of the percentage returnfor that portfolio. Note that (σ(w), µ(w)) = (σ0, µ0). Since w is a functionof the empirical mean vector µ and covariance matrix Σ, (σ(w), µ(w)) is arandom vector.

The standard deviations and expected values (σ(w), µ(w)) for the es-timated solutions to (MaxExp) are much closer to the theoretical value(σ0, µ0) than for (MinVar), and also closer than for (Trade-off). We findthat whereas the problem (MaxExp) is rather robust to noise perturbing

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the parameter µ, this is not at all so for the problem (MinVar). However,it is interesting to note that the empirical Sharpe ratios

µ(w) − Rrf

σ(w)=

(µ − Rrf1)TΣ−1(µ − Rrf1)√(µ − Rrf1)TΣ−1ΣΣ−1(µ − Rrf1)

coincide for the three versions of the investment problem.Figure 2 shows the same thing as Figure 1 when the theoretical mean

vector µ was set to µ = (1.1, 1.2)T . Here, the difference in accuracy for thesolutions to the three versions of the investment problem based on simulateddata is much smaller. The reason is that here we do not find estimatesµ − Rrf1 ≈ 0 and therefore no exploding weights w due to “division byzero”.

Let us look a bit closer at the accuracy of the estimation of means. Forsake of clarity we consider the univariate case. Consider the simplest pos-sible univariate case; given a sample {R1, . . . , Rm} of independent randomvariables with common mean E[Rk] = µ and variance Var(Rk) = σ2 weconsider the problem of estimating µ. Set µ = (R1 + · · · + Rm)/m, i.e. thestandard estimator. Then E[µ] = µ and

Var(µ) = E[(µ − µ)2] = E

(

1

m

m∑

k=1

(Rk − µ)

)2

=1

m2

m∑

k=1

E[(Rk − µ)2] +1

m2

∑

j 6=k

E[(Rj − µ)(Rk − µ)]︸ ︷︷ ︸=0

=σ2

m.

Hence, the estimator µ has standard deviation σ/√

m. In the simulationstudy above we have, for the first component, µ = 1.025, σ = 0.3 andm = 200. In particular,

µ − Rrf = 0.025 ≈ 0.021 ≈ σ/√

m =√

Var(µ − Rrf ).

2 Investments in the presence of liabilities

We now consider optimal investments in the presence of liabilities. We con-sider the trade-off version of the optimal investment problem, high expectedpayoff is good and high variance of payoff is bad and a constant c ≥ 0 deter-mines the trade-off we aim for between the two. The optimization problemreads

maximize cE[h0 + hTS1 − L] − 1

2Var(h0 + hTS1 − L)

subject to h0B0 + hTS0 ≤ V0.(1)

4

−2 −1 0 1 2

−0.

50.

00.

51.

01.

52.

0

−1.0 −0.5 0.0 0.5 1.0

−0.

50.

00.

5

−10 −5 0 5 10 15 20

−10

−5

05

0.0 0.2 0.4 0.6 0.8 1.0

1.00

1.05

1.10

1.15

0.26 0.28 0.30 0.32 0.34 0.36

0.96

0.98

1.00

1.02

1.04

0 1 2 3 4 5 6

0.4

0.6

0.8

1.0

1.2

1.4

Figure 1: The first three plots show empirical optimal portfolio weightsin risky assets based on 3000 samples of size 200 for the (Trade-off)-,(MaxExp)-, and (MinVar) version for µ = (1.025, 1.075)T . The remainingplots show the corresponding standard deviation-mean pairs.

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0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

−0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.2 0.3 0.4 0.5

1.05

1.10

1.15

1.20

1.25

0.26 0.28 0.30 0.32 0.34 0.36

1.10

1.11

1.12

1.13

1.14

1.15

1.16

0.2 0.3 0.4 0.5 0.6 0.7

1.10

1.15

1.20

1.25

1.30

Figure 2: The first three plots show empirical optimal portfolio weightsin risky assets based on 3000 samples of size 200 for the (Trade-off)-,(MaxExp)-, and (MinVar) version for µ = (1.1, 1.2)T . The remaining plotsshow the corresponding standard deviation-mean pairs.

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We choose to formulate it as the convex optimization problem

minimize 1

2

(hTΣS1

h + σ2

L − 2hTΣL,S1

)− c(h0 + hT

µS1− µL

)

subject to h0B0 + hT

0S0 ≤ V0.

The same arguments as in the case of no liability leads to the necessary andsufficient conditions

h = Σ−1

S1

(cµS1

− λ1S0 + ΣL,S1

),

λ1B0 − c = 0,

λ1(h0B0 + hTS0 − V0) = 0,

h0B0 + hTS0 ≤ V0,

and therefore the optimal solution

h = cΣ−1

S1

(µS1

− 1

B0

S0

)+ Σ−1

S1ΣL,S1

,

h0 =1

B0

(V0 − hTS0

).

We observe that the solution corresponds to the minimum variance hedgeplus the optimal investment position without a liability. If the initial cap-ital V0 is insufficient to take this position, then this problem is solved byborrowing money (a short position in the risk-free bond).

A relevant question is: what is the right trade-off between hedging theliability and speculating in case borrowing money is not possible? Thisproblem is the problem above with the inclusion of the constraint h0 ≥ 0(or equivalently −h0 ≤ 0). The necessary and sufficient conditions for anoptimal solution becomes

h = Σ−1

S1

(cµS1

− λ1S0 + ΣL,S1

),

λ1B0 − c − λ2 = 0,

λ1(h0B0 + hTS0 − V0) = 0,

λ2h0 = 0,

λ1, λ2, h0 ≥ 0,

h0B0 + hTS0 ≤ V0.

If λ2 = 0, then we have the solution above, with h0 ≥ 0. Therefore theinteresting case is when λ2 > 0 which implies h0 = 0. In this case hTS0 = V0

since hTS0 < V0 would correspond to throwing away money rather thaninvesting them in a risk-free bond, which is clearly sub-optimal. The optimalsolution is

h = cΣ−1

S1

(µS1

− 1

B0

S0

)− λ2

B0

Σ−1

S1S0 + Σ−1

S1ΣL,S1

,

hTS0 = V0.

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Combining the two equations give

cST

0 Σ−1

S1µS1

− c

B0

ST

0 Σ−1

S1S0 −

λ2

B0

ST

0 Σ−1

S1S0 + ΣT

L,S1Σ−1

S1S0 = V0.

Since ΣS1is positive definite, so is Σ−1

S1and therefore ST

0Σ−1

S1S0 > 0. This

means that if the position corresponding to the optimal solution withoutborrowing restriction is too expensive then taking λ2 > 0 large enough givesa modified position that is affordable without borrowed money. We observethat the solution now is to take the position corresponding to the minimumvariance hedge of the liability and the optimal speculative position in therisky assets but then to adjust the position if it turns out to be too expensive.

Summing-up we have found that the mean-variance approach to optimalinvestments in the presence of liabilities, and possibly borrowing restric-tions, provides a rather simple solution and intuitive solution. The solutionconsists of computing the variance minimizing hedge and the optimal solu-tion without a liability and taking the position which is the sum of the twopositions. This may require borrowing money. If this option is not available,then h0 = 0 and the position in the risky assets is modified by subtractingthe position

λ2

B0

Σ−1

S1S0

for a number λ2 > 0 such that the cost of the position in the risky assets isthe initial capital V0.

Example 1. Suppose that the risky assets can be divided into a set ofhedging instruments (bonds say) and a set of pure investment assets (stockssay), where the values of the assets of the latter kind are uncorrelated withthe liability. Write

Sk =

(Si

k

Shk

)for k = 1, 2 and h =

(hi

hh

).

This means that

ΣS1=

(Σ

Si1

0

0 ΣSh

1

)and Σ−1

S1=

(Σ−1

Si1

0

0 ΣSh

1

−1

)

and therefore the solution to the optimal investment problem with a liabilityand no risk-free borrowing reads

hi = cΣ−1

Si1

(µ

Si1

− 1

B0

Si0

)− λ2

B0

Σ−1

Si1

Si0,

hh = cΣ−1

Sh1

(µ

Sh1

− 1

B0

Sh0

)− λ2

B0

Σ−1

Sh1

Sh0 + Σ−1

Sh1

ΣL,Sh1

,

hiTSi0 + hhTSh

0 = V0.

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This means that we can divide the original investment problem with a lia-bility into two simpler problems, one with the liability and only the hedginginstruments as risky assets, and one without a liability and only the pureinvestment assets as risky assets.

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