Risk And Return: Measuring Returns

Consider the following distribution of returns of a stock:

Year	Return (r)
1	-10%
2	20%
3	30%

Question 1

What is the geometric mean return of the stock over 3 years? Round off your final answer to two digits after the decimal point. State your answer as a percentage rate, i.e ‘13.55’.

\((1 + r_g) = [(1+r_1) \times (1+r_2) \times\cdots\times (1+r_T)] ^ {1 \over T}\)

\(1 + r_g = [(0.9) \times (1.2) \times (1.3)] ^ {1 \over 3} = 1.1198\)

So \(r_g = 11.98\%\)

Question 2

What is the arithmetic mean return of the stock over 3 years? Round off your final answer to two digits after the decimal point. State your answer as a percentage rate, i.e ‘10.55’

The arithmetic average of n returns is given by \((r_1 + r_2 + \cdots + r_n) / n\)

\[r_a = {-10+20+30 \over 3} = 13.33\%\]

Question 3

Which of the following measures is most informative about the average past return of an investment?

☐ The arithmetic average return over the period of interest

☑ The geometric average return over the period of interest

☐ Both arithmetic and geometric average returns over the period of interest

☐ The standard deviation of the returns over the period of interest

The geometric average return tells us about the actual performance over the past sample period. It is the rate of return that would compound over the period to the same terminal value as the one obtained from the sequence of actual returns.

Question 4

In which of the following cases you should take into account compounding?

☐ When you calculate the arithmetic average return, you are computing the compounded average return.

☐ You are using compounded returns when you compute volatility.

☐ Calculating the arithmetic average or the geometric return always gives the same answer.

☑ When you calculate the geometric average return, you are computing the compounded average return.

Arithmetic and geometric average returns do not both take into account compounding. Geometric average return represents the average holding period return that would compound over the period to yield the same terminal value as the one obtained from the sequence of actual returns.

Question 5

If you want to have an indication of the expected rate of return for an investment, you would prefer to look at:

☑ The arithmetic average return over the period of interest

☐ The geometric average return over the period of interest

☐ Both arithmetic and geometric average returns over the period of interest

☐ The standard deviation of the returns over the period of interest

The arithmetic mean is an unbiased estimate of future expected returns.

Risk & Return: Measuring Risk

Question 1

Which of the following is a common measure of risk for returns?

☐ Expected return

☑ Standard deviation

☐ Inflation

☐ Geometric mean of returns

Standard deviation is a measure of the dispersion in returns.

Question 2

Based on the following distribution of returns for stock A, compute the standard deviation for stock A. Round off our final answer to two digits after the decimal point. State your answer as a percentage rate (such as 5.55)

Probability	Return on A
30%	10%
40%	5%
30%	30%

The standard deviation of the return is defined as the square root of variance, which is the expected value of the squared deviations from the expected return. The variance is calculated as:

\[\sigma^2_x = \sum\{[R_{x_i} - E(R_x)]^2 \cdot P_i\}\]

The expected return is calculated as the probability-weighted average of the returns. \(P(R_i)\) is the probability of each scenario and \(R_i\) the return in each scenario, where scenarios are labeled by \(i\). So we write the expected return as:

\[E(R) = \sum R_i \cdot P(R_i)\]

\(E(R_A) = 10\% \cdot 0.30 + 5\% \cdot 0.40 + 30\% \cdot 0.30 = 14\%\)

The variance for stock A is calculated as:

\[\sigma^2_A = \sum\{[R_{A_i} - E(R_A)]^2 \cdot P_i\}\]

\(\sigma^2_A = (10-14)^2 \cdot 0.30 + (5-14)^2 \cdot 0.40 + (30-14)^2 \cdot 0.30 = 114\%\)

Hence the standard deviation for A is equal to \(\sigma_A = \sqrt{114} = 10.68\%\)

Question 3

In the following graph we observe the probability distributions of assets A and B. Which of the following statements is correct?

☐ \(E(R_A) = E(R_B)\) and \(σ_A < σ_B\)

☐ \(E(R_A) < E(R_B)\) and \(σ_A = σ_B\)

☐ \(E(R_A) < E(R_B)\) and \(σ_A > σ_B\)

☑ \(E(R_A) = E(R_B)\) and \(σ_A > σ_B\)

The central tendencies – the means of the two distributions are equal to each other. On the other hand, the dispersion in the distribution of A is larger than that in B. Therefore, the standard deviation of A is greater than B’s standard deviation.

Question 4

When a distribution is skewed to the right:

☑ A. The extreme positive values dominate and the measure is positive

☐ B. The extreme negative values dominate and the measure is negative

☐ C. The standard deviation will underestimate risk

☐ D. both a and c are correct

Question 5

Kurtosis measure the degree of fat tails of a distribution. Which of the following answers is or are true? (More than one answer can be correct)

☑ The presence of fat tails implies that there is greater probability mass in extreme events in the tails

☑ Normal distribution has a kurtosis equal to 3

☐ Normal distribution has a kurtosis equal to 0

☐ The presence of fat tails implies that there is lower probability mass in extreme events in the tails

Fat tails imply that there is greater probability mass for extreme events in the tails than a normal distribution and the kurtosis measure for the standard normal distribution is 3.

Quiz 2.1 Risk & Return

Question 1

If your holding period return on a $100 investment was 12% at the end of the first year, -10% at the end of the second year and 5% at the end of the third year, what was your three-year holding period return?

☐ 4.45%

☐ 4.8%

☑ 5.84%

☐ 6%

☐ 7%

The three-year holding period return is computed as \([(1 + r_1) × (1 + r_2) × (1 + r_3)]−1\).

3 year return = (1 + 12%) ∗ (1 − 10%) ∗ (1 + 5%) − 1 = 5.84%

Question 2

Suppose the probabilities of a recession, a boom and no change in the current economic environment are 40%, 30%, and 30% respectively. Also suppose you will have an annual return on your investments of 10% in a recession, 40% in a boom and 20 % if there is no change. What is your expected annual return on your investment?

☐ 19%

☐ 20%

☐ 21%

☑ 22%

☐ 23%

Expected return = 0.4 ∗ 10 + 0.3 * 40 + 0.3 * 20 = 22%

Question 3

You bought a stock of company Alpha and held it over a five-year period. The annual returns of the stock are given by the following table. Based on the annual returns, we calculated the gross yearly returns. Are the calculations correct or are they false?

Year	Return = r	Gross return
1	-10%	0.9
2	-20%	0.8
3	30%	1.3
4	20%	1.2
5	15%	1.15

☑ The calculations are correct.

☐ The calculations are false.

We just add 1 to each return given in the table. So the gross return should be equal to 1 + r.

Question 4

What is the geometric mean return of the stock over 5 years?

☐ 4.23%

☑ 5.25%

☐ 30%

☐ 35%

\((1 + r_g) = [(1 + r_1)(1 + r_2) \cdots (1 + r_T)] ^ {1 \over T}\)

\(1 + r_g = [0.9 \times 0.8 \times 1.3 \times 1.2 \times 1.15] ^ {1 \over 5} = 1.0525\)

So \(r_g = 5.25\%\).

Question 5

What is the arithmetic mean return of the stock over five years?

☑ 7%

☐ 5.24%

☐ 7.5%

☐ 9%

☐ 35%

\(r_a = {−10 − 20 + 30 + 20 + 15 \over 5} = 7\%\)

Question 6

Suppose the probabilities of a recession, a boom and no change in the current economic environment are 40%, 30 %, and 30 % respectively. Also suppose you will have an annual return on your investments of 10% in a recession, 40% in a boom and 20 % if there is no change. What is the standard deviation of your annual return on your investment?

☐ 6.48

☐ 9.50

☑ 12.49

☐ 13.50

☐ 14.28

Expected return \(= 0.4 ∗ 10 + 0.3 * 40 + 0.3 * 20 = 22\%\)

Variance \(= 0.4 \times (10 − 22)^2 + 0.3 \times (40 − 22)^2 + 0.3 \times (20 − 22)^2 = 156\)

Stand.Dev. \(= \sqrt{156} = 12.49\)

Question 7

An analyst’s forecast for the end-of-year prices and dividend payments for company XYZ under various states of the economy are as follows:

State of the economy	Probability	Year-end price	Cash dividends
Crash	0.25	160	5
Poor	0.40	150	10
Good	0.30	160	20
Excellent	0.05	200	30

Suppose you bought one share of stock for $160. What’s the second highest rate of return you might get? Round off to one decimal. (i.e. “x.x”)

The correct answer that the good scenario provides us with the second highest return (12.5%).

% Return = \({\text{capital gain} + \text{dividend} \over \text{purchase price}}\) * 100

Under crash we have \({160-160+5 \over 160} \times 100 = 3.125\%\)

Under poor we have \({150-160+10 \over 160} \times 100 = 0.00\%\)

Under good we have \({160-160+20 \over 160} \times 100 = 12.50\%\)

Under excellent we have \({200-160+30 \over 160} \times 100 = 43.75\%\)

Question 8

Again, assuming that you bought one share of stock for $160, what is the expected annual return for this stock?

☐ -3.16%

☐ 0%

☐ 3.16%

☑ 6.72%

☐ 20%

Recall the expected return is the probability weighted average of the possible return outcomes. The rate of return you earn on your investment in each scenario is computed as

Rate of return = (capital gain + dividend) / purchase price.

Expected return = 0.25 x 3.125% + 0.4 x 0% + 0.3 x 12.5% + 0.05 x 43.75% = 6.72%

Question 9

A volatility strategy is:

☑ An investment strategy that collects a premium during stable periods, but has large losses during volatile times.

☐ An investment strategy that consists in diversifying a securities portfolio.

☐ None of the above.

Question 10

Suppose you have $100,000 to invest. Investing in equities will generate a gain of $50,000 with a probability of 60%, or a loss of $30,000 with a probability of 40%. Investing in the risk-free U.S. Treasury bills on the other hand will generate a sure gain of $5,000. Based on this data, what is the expected risk premium associated with investing in risky equities versus risk-free T-bills?

☐ 18%

☑ 13%

☐ 5%

☐ 12%

The expected risk premium associated with investing in risky equities versus risk-free Treasury bills is the expected return on equities minus the rate of return in Treasuries.

The expected rate of return in equities = 50% x 0.6 + -30% x 0.4 = 18%

The rate of return offered by Treasuries is 5%.

The risk premium is therefore 18% - 5% = 13%