porfilio allocation aug25,2015.pdf

8/20/2019 porfilio allocation aug25,2015.pdf

1/15

I. Matrix Algebra and Portfolio Math (44 points, 4 points each)

Let Ri denote the simple return on asset i (i = 1, , N ) with E [ Ri] = i, var( Ri) = 2i and

cov( Ri, R j) = ij. Define the ( N 1) vectors 1, , N R R R , 1( , , ) N ,

1, , N m m m , 1, , N x x

x , 1, , N y y y , 1, , N t t

t , 1, ,1 1 and the ( N

N ) covariance matrix

2

1 12 1

2

12 2 2

2

1 2

N

N

N N N

.

The N × 1 vectors m, x, y and t contain portfolio weights that sum to one. Using matrixalgebra, answer the following questions.

1. For the portfolios defined by the vectors x and y give the expression for the portfolio

returns, ( R p,x and R p,y), the portfolio expected returns ( p,x and p,y), the portfolio variances (2 2

, ,and p x p y ), and the covariance between R p,x and R p,y ( xy ).

2. For portfolio x, derive the N × 1 vector of marginal contributions to portfolio volatilitydefined by

, ( ) p x p

xMCR

x x.


2/15

3. Write down the optimization problem and give the Lagrangian used to determine theglobal minimum variance portfolio assuming short sales are allowed. Let m denote the vector of portfolio weights in the global minimum variance portfolio.

4. Write down the optimization problem and give the Lagrangian used to determine anefficient portfolio with target return equal to 0 assuming short sales are allowed. Let x denote

the vector of portfolio weights in the efficient portfolio.

5. Continuing with question 4, derive the first order conditions for determining the efficient portfolio x with target return 0 . Write these first order conditions as a system of linear

equations in the form A z = b and show how the portfolio x can be determined from this system.


3/15

6. Briefly describe how you would compute the efficient frontier containing only risky assets(Markowitz bullet) when short sales are allowed.

7. Write down the optimization problem used to determine the tangency portfolio, assuming

short sales are allowed and the risk free rate is given by f r . Let t denote the vector of portfolio

weights in the tangency portfolio.

8. Continuing with question 7, is there an analytical solution (i.e., matrix algebramathematical formula) for the tangency portfolio when short sales are allowed? If so, give thematrix algebra formula for this solution.


4/15

9. Write down the equations for the expected return ( e p ) and standard deviation (

e

p ) of

efficient portfolios consisting of the tangency portfolio and T-bills, where the T-bill rate (risk-

free rate) is given by f r and t denotes the vector of portfolio weights in the tangency portfolio.

10. The previous computations for efficient portfolio allowed for short sales (i.e., negativevalues for asset shares). However, in many practical situations short sales are not allowed. Givethree reasons why short sales may be prohibited.

11. Write down the optimization problem and give the Lagrangian used to determine anefficient portfolio with target return equal to 0 assuming short sales are not allowed. Let x

denote the vector of portfolio weights in the efficient portfolio.


5/15

II.

Efficient

Portfolios

(36

points,

4

points

each)

The graph below shows the efficient frontier (allowing short sales) computed from threeVanguard mutual funds: S&P 500 Index (vfinx), European Stock Index (veurx) and the US LongTerm Bond Index (vbltx).

Figure 1 Markowitz Bullet

Expected return and standard deviation estimates for specific assets are summarized in thetable below. These estimates are based on monthly simple return data over the five year (60month) period May 2009 – May 2014 (same data as in the class project but slightly different time period).

0.00 0.01 0.02 0.03 0.04 0.05 0 . 0

0 0

0 . 0

0 5

0 . 0

1 0

0 . 0

1 5

Efficient Frontier

Portfolio SD

P o r t f o l i o

E R

vfinx

veurx

vbltx

Global MinTangency

Evfinx

T-Bills


6/15

Table 1 Portfolio Statistics

Asset Mean

(E[R])

Standard

deviation

(SD(R))

Weight in

Global Min

Portfolio

Weight in

Efficient

Portfolio with

Mean =

1.48%

Weight in

Tangency

portfolio

VFINX 0.0148 0.0387 0.58 1.35 0.80

VEURX 0.0116 0.0574 -0.19 -0.69 -0.33

VBLTX 0.0083 0.0255 0.61 0.34 0.53

T-Bills 0.0010 0.0000

Global MinPortfolio

0.0114 0.0159

EfficientPortfolio withMean=0.0148

(labeled Evfinx)

0.0148 0.0232

TangencyPortfolio

0.0124 0.0166

Using the above information, please answer the following questions.

1. Compute annualized means and standard deviations from the monthly statistics in Table 1for the three portfolios vfinx, vbltx, and veurx using the square-root-of-time-rule (this is only anapproximation because we have simple returns) and put these results in the Table below. Also,compute the annualized T-Bill rate.

Asset Annualized Mean Annualized SDvfinx

vbltx

veurx

T-Bills

2. Using the annualized information from part 1, compute the annualized Sharpe ratios foreach of the three portfolios and put the results in the table below. Which portfolio is ranked bestusing the Sharpe ratio?

Asset Annualized Sharpe Ratio

vfinx

vbltx

veurx


7/15

3. The Sharpe ratios reported above are estimates. Briefly describe how you can computestandard errors and 95% confidence intervals for these estimated Sharpe ratios.

4. Find the efficient portfolio of risky assets only (i.e., a portfolio on the Markowitz bullet)

that has an expected monthly return equal to 0.014 (1.4%). In this portfolio, how much isinvested in vfinx, vbltx, and veurx? Show this portfolio on Figure 1.

5. How much should be invested in T-bills and the tangency portfolio to create an efficient portfolio with expected return equal to the average return on vfinx - 0.0148 (1.48%) ? What isthe standard deviation of this efficient portfolio? Indicate the location of this efficient portfolioon Figure 1.


8/15

6. In the efficient portfolio you found in part 5, what are the shares of wealth invested in T-Bills, vfinx, vbltx, and veurx?

7. Assuming an initial $100,000 investment for one month, compute the 5% value-at-risk based on the normal distribution for the global minimum variance portfolio.

8. The efficient frontier of risky assets shown in Figure 1 allows for short sales (see theweights in the portfolios listed in Table 1). Using the graph below indicate the location of theefficient frontier of risky assets that does not allow short sales.

0.00 0.01 0.02 0.03 0.04 0.05

0 . 0

0 0

0 . 0

0 5

0 . 0

1 0

0 . 0

1

5

Efficient Frontier

Portfolio SD

P o r t f o l i o E R

vfinx

veurx

vbltx

Global Min

T-Bills


9/15

9. Suppose you want to find the efficient portfolio of risky assets only that has an expectedmonthly return equal to 0.02 (2%) but that you are prevented from short selling. Is it possible tofind such an efficient portfolio? Briefly explain why or why not.

III. Risk Budgeting (20 points, 5 points each)

Table 2 below shows a risk budget report for the global minimum variance portfolio given in

Table 1. The global minimum variance portfolio has some special properties that are revealed inthe risk budget report below.

Table 2 Risk Budget Report for Global Minimum Variance Portfolio

, 0.0159 p m

Asseti

x i

i MCR iCR

iPCR ,i p ,i p

vfinx 0.585 0.0387 0.0159 0.00931 0.585

veurx -0.195 0.0574 0.0159 -0.00311 -0.195

vbltx 0.611 0.0255 0.0159 0.00972 0.611

1. The risk budget report for a portfolio shows the additive decomposition of portfolio volatilityinto contributions from the assets in the portfolio. In the global minimum variance portfoliowhich asset has the highest contribution to portfolio volatility? Which asset has the lowest

contribution? What is the relationship between an asset’s percent contribution to risk, iPCR ,

and its allocation weight,i

x ? Verify that the sum of the asset contributions to risk,i

CR , add to

portfolio volatility,, 0.0159 p m .


10/15

2. Suppose the risk manager wants to reduce the portfolio volatility. For which assets shouldallocations be reduced, and for which assets should allocations be increased to achieve this goal?

3. Give the definition of the “beta” (denoted ,i p ) of an asset’s return with respect to the

portfolio return. In lecture, we showed that , /i p i iPCR x . Using the information in the risk

budget report compute,i p

for each asset and put these values under the,i p

column in the table.

What relationship do you see?

4. Give the definition of the correlation (denoted ,i p ) of an asset’s return with respect to the

portfolio return. In lecture, we showed that , /i p i i MCR . Using the information in the risk

budget report compute ,i p for each asset and put these values under the ,i p column in the table.

What relationship do you see?


11/15

IV.

Statistical

Analysis

of

Efficient

Portfolios

(20

points

total)

The figures below show the simple returns of vfinx, veurx and vbltx along with 24-month rollingestimates of their average returns, standard deviations and pairwise correlations.


12/15


13/15

1. Briefly comment on the 24-month rolling estimates of the means, standard deviations and pairwise correlations. Which estimates appear to be constant and which do not? (8 points)


14/15

2. The CER model assumes that i and i and ij are constant over time. Given yourresponse to 1, is this a reasonable assumption for vfinx, veurx and vbltx. (4 points)

3. The following figure shows 24-month rolling weights in the global minimum variance portfolio (allowing short sales) constructed from vfinx, veurx and vbltx. Which CER

model estimates (i.e., ˆˆ ˆ, ,i i ij

) determine the global minimum variance weights? Do

you see evidence of substantial time variation in these weights? Can you explain any ofthis time variation by the time variation in the CER model estimates? (4 points)


15/15

4. The following figure shows 24-month rolling means and standard deviation of the globalminimum variance portfolio (allowing short sales) constructed from vfinx, veurx andvbltx. What has been the impact on these values of the time variation in the 24-month

estimates of i and i and ij? (4 points)

porfilio allocation aug25,2015.pdf

Documents