Portfolios

Question 1

If the asset is the only risky asset held by the investor…

Answer 1

…Sharpe measure is the appropriate measure.

Sharpe measure = (Rp - Rf) / stdv

Question 2

If the stock is one of many stocks in an actively managed portfolio, then …

Answer 2

… the Treynor’s measure is preferred.

Treynore measure = (Rp - Rf) / beta

Question 3

What is Jensen’s alpha?

Answer 3

Jensen’s alpha:
αP = rP − [ ̄rf+ βP(r M − r f)]

Jensen’s alpha is the average return on the portfolio over and above that predicted by the CAPM, given the portfolio’s beta and the average market return.

Question 4

Based on current dividend yields and expected capital gains, the expected rates of return on
portfolios A and B are 12% and 16%, respectively. The beta of A is .7, while that of B is 1.4.

The T-bill rate is currently 5%, whereas the expected rate of return of the S&P 500 index is 13%.

The standard deviation of portfolio A is 12% annually, that of B is 31%, and that of the S&P 500 index is 18%.

a. If you currently hold a market index portfolio, would you choose to add either of these portfo-
lios to your holdings? Explain.

b. If instead you could invest only in T-bills and one of these portfolios, which would you
choose?

Answer 4

a. The alphas for the two portfolios are:

αA = 12% – [5% + 0.7 × (13% – 5%)] = 1.4%

αB = 16% – [5% + 1.4 × (13% – 5%)] = –0.2%

Ideally, you would want to take a long position in Portfolio A and a short position in Portfolio B.b.

If you will hold only one of the two portfolios, then the Sharpe measure is the appropriate criterion:

Sa = (0.12 - 0.05) / 0.12 = 0.583

Sb = (0.16 - 0.05) / 0.31 = 0.355

Using the Sharpe criterion, Portfolio A is the preferred portfolio

Question 5

What is the information ratio?

Answer 5

It measures abnormal return per unit of risk that in principle could be diversified away by holding a market index portfolio.

excess return per unit of nonsystematic risk.

Information ratio: αP /σ(eP)

Question 6

When calculating the sharp ratio we divide with:

• residual SD
• SD of excess returns

Answer 6

• SD of excess returns

Question 7

If you add only one stock to a well-diversified fund what measure do you use?

Answer 7

Alpha

Question 8

The intercept of the scatter diagram is a measure of…

Answer 8

…stock selection ability

Question 9

Timing ability is indicated by the … of the plotted line

Answer 9

curvature (slope, b)

Question 10

The steeper slope shows that the manager maintained …

Answer 10

… higher portfolio sensitivity to market swings (i.e., a higher beta)

Question 11

The total value added by the manager is the difference between…

Answer 11

… the sum of the weighted returns of the manager minus the sum of of weighted returns of the market

Σwiri - Σwimi

Question 12

Conventional wisdom says that one should measure a manager’s investment performance over an entire market cycle. What arguments support this contention? What arguments contradict it?

Answer 12

Support:
Different market conditions favour different strategies.
For instance, a manager focused on high beta may excel in up markets but struggle in down markets.
Evaluating a manager’s performance over a full market cycle provides a more accurate assessment. Additionally, more data points over a cycle enhance the reliability of measurements.

Contradict:
Waiting for an entire market cycle may not be necessary if we adjust for beta. If beta is properly controlled, we can assess performance without the need for a complete cycle.

Question 13

If a portfolio ranks high in Treynor measure but low in the Sharpe ratio this might be due to…

Answer 13

lack of diversification.

The Sharpe ratio measures excess return per unit of total risk, while the Treynor ratio measures excess return per unit of systematic risk.

Since the portfolio performed well on the Treynor measure and so poorly on the Sharpe Measure, it seems that the fund carries a greater amount of unsystematic risk, meaning it is not well-diversified and systematic risk is not the relevant risk measure.

Question 14

The Treynor ratio measures excess return per unit of … risk.

Answer 14

systematic

Question 15

This would be the only stock in your portfolio, use:

Answer 15

Sharp ratio

Question 16

You would mix this stock with a passive index fund, use:

Answer 16

Alpa

Question 17

Investor D is a passive investor and therefore holds the market portfolio. After studying alarge universe of stocks, he is convinced of the following characteristics for stock A: thealpha equals 0.5% and σ(eA) is equal to 15%. βAequals 1. For the market portfolio, Destimated an expected excess return of 6% with a σ of 17%. Which proportion of theportfolio should be allocated to the passive portfolio, and which proportion to stock A? Whatis the resulting Sharpe ratio of the portfolio? Did it increase relative to the Sharpe ratio of thepassive portfolio?

Answer 17

from the formula table:

initial weight wa0 = (a / sd^2) / (Erm / sdm^2)

if b not 1

wa* = wa0 / (1 + (1 - b)*wa

allocate wa in active , 1-wa in passive.

the change in Sharpe ratio is:

1. calculate the sharp ratio
2. calculate the information ratio as info ratio = a / sd

total portfolio Sharpe raio = sh^2 + info^2 = g

root_g = total portfolio sharp ratio

Sharpe ratio - root_g = change in sharp ratio