(Please make sure to answer b) (20 points) You are considering the investment in
ID: 2775037 • Letter: #
Question
(Please make sure to answer b)
(20 points) You are considering the investment in stock XYZ. The stock price at the end of this quarter is $50. The stock pays no dividends.
(a) Suppose your forecast regarding the distribution of stock price at the end of next quarter is as follows:
Please fill out the table by computing the HPR in the next quarter, APR and EAR of this stock in each state of the market. Also compute the expectation and standard deviation of quarterly HPR on this stock. Suppose that the current rate of return on 30-day T-bills is 5% and will be constant over the next quarter, what is the quarterly risk premium and Sharpe ratio? (6 points) (Note: the quoted 30-day T-bills rate is annual rate of return APR.)
(b) In reality, investors do not know the true distribution of stock returns. So we use historical data to estimate the performance of this stock. We can observe the quarterly rate of returns data of stock XYZ in the past 25 years (which is 100 quarters). (i) If we only use the quarterly return data in the past four quarters: 12%, 5%, 7%, -8%, what are the arithmetic average, geometric average, and standard deviation of the sample rate of return? If we also observe the quoted 30-day T-bills rate in the past four quarters are 5.6%, 5.2%, 5.2%, 4.4%, what are the sample risk premium and Sharpe ratio? (5 points) (Note: the quoted 30-day T-bills rate is APR.) (ii) Discuss the advantage and limitation of previous time-series analysis using the most recent return data. (4 points) (iii) Now, we use all the 100 quarterly rate of returns data of stock XYZ. The statistics of quarterly excess returns are given in the following table: Statistic (Quarterly excess returns) Performance (%) Average 2.25% SD 6.12% LPSD 6.59% Skew -0.6 Kurtosis 0.94 VaR actual VaR normal Expected shortfall actual Expected shortfall normal -8.02 -7.82 -10.11 -9.81 Compute the Sharpe ratio and Sortino ratio. What does the distribution of excess returns look like, based on these sample statistics? (5 points)
State of the Market Probability Ending Price HPR (next quarter) APR EAR Boom 0.20 $55 Normal 0.55 $53 Recession 0.25 $46.5Explanation / Answer
a.
HPR is holding period return.
Holding period return = (selling price + dividend – buying price)/ buying price
Buying price is 50.
APR is annual percentage rate. Here the HPR is quarterly. So APR will be
APR = HPR x 4
EAR = (1+ HPR)^4 -1
State of mkt
Probability
Ending price
HPR
APR
EAR
Boom
0.20
55
(55-50)/50 =10%
10%x4 =40%
(1+0.10)^4 -1 = 46.41%
Normal
0.55
53
(53-50)/50 =6%
6%x4 =24%
(1+0.06)^4 -1 = 26.25%
Recession
0.25
46.50
(46.50-50)/50 =-7%
-7%x4 = -28%
(1-0.07)^4 -1 = 25.19%
State of mkt
Probability
HPR ( R)
P x R
R- ER
Px(R-E)^2
Boom
0.2
10%
0.02
6%
0.000832
Normal
0.55
6%
0.033
2%
0.00033
Recession
0.25
-7%
-0.0175
-11%
0.002783
ER
0.0355
0.003945
ER is expected return.
Standard deviation = (Px(R-E)^2)^0.50
=(0.003945)^0.50
=6.28%
Quarterly t bill rate Rf = 5%/4 = 1.25%
Quarterly risk premium = ER - Rf =3.55%-1.25% =2.3%
Sharpe ratio = ( ER-Rf)/SD
= (3.55%-1.25%)/6.28%
= 0.366
b.
Answer (i)
Arithmatic average of Sample Rate of Return = 4%
Geometric average of Sample Rate of Return = 3.73%
Standard Deviation of Sample Rate of Return = 4.743
Sample Risk Premium = - 1.37%
Sharpe Ratio for sample = - 0.29
Working
Quarterly returns for the past 4 quarters are 12%, 5%, 7%, -8%
Arithmatic average = (12+5+7-8)/4 = 16/4 = 4%
Geometric average = {(1+0.12 )*(1+0.05)*(1+0.07)*(1-0.08))^1/4 - 1
= {1.12*1.05*1.07*0.92}^1/4 - 1
= 1.15765^1/4 - 1= 1.03728 – 1 = 0.03728 or 3.73% (approximately)
Given Arithmatic Average is 4%, Standard deviation can be calculated as follows
Variance ={ (12-4)^2 +(5-4)^2 + (7-4)^2 + (-8+4)^2}/4 = {8^2 + 1^2 + 3^2 + (-4)^2}/4
= (64 + 1 + 9 + 16)/4 = 90/4 = 22.5
Standard Deviation = Square Root of (22.5) = 22.5^1/2 = 4.743
Observed T-Bill rates in the past 4 quarters are 5.6%, 5.2%, 5.2% and 4.4%
Geometric mean = {1.056 * 1.052 * 1.052 * 1.044}^1/4 -1 = 1.2201^1/4 – 1 = 1.05099 – 1 = 0.05099
Gemetric Mean = 0.05099 * 100 = 5.099 or 5.10% (rounded off)
Based on the above Return on stock = 3.73% Risk free return (APR) = 5.10%
Standard deviation of stock = 4.743
Risk Premium = Rm – Rf = 3.73 – 5.10 = -1.37%
Sharpe’s ratio can be calculated using the formula
Sharpe Ratio = (Stock Return – Risk free return) / Standard Deviation of Stock
= (3.73 – 5.10) / 4.743 = -0.2888 or- 0.29
Answer (ii)
Using the most recent time series data provides a correct picture if there is a substantial change in performance of the firm like losses in the recent time periods or abnormal growth in the latest time periods or there is a change in the capital structure. Also if a large amount of historical data is used, it may not provide a correct picture of the performance if any of the above factors are observed in the performance of the company. If none of these factors are present, then using a large amount of historical data will provide a better estimate of the growth of the firm.
Answer (iii)
Average Quarterly excess return = 2.25%
Standard Deviation = 6.12%
Risk Premium (excess return over risk-free rate) = 2.25%
Sharpe Ratio = 2.25%/6.12% = 0.3676
Sortino Ratio can be calculated using the formula
Sortino Ratio = Stock Return – Target Return /Downside Risk
Stock Return – Target return represents excess return.
Downside Risk can be calculated using a formula
Downside Risk = [{Sum of (Stock Return – Target Return)^2}*1/n * f(r)]^1/2
Where f(r) = 1 if return < target return
and f(r) = 0 if return r> or = target return
Given in the problem the variance of Expected Short fall = -8.02 vis-à-vis the Actual Shortfall = - 7.82. That is the actual short fall is lower than the expected / targeted short fall. This indicates that the actual return is less than targeted return. Therefore based on the above f(r) will be equal to 1. I/n is not taken into account as the excess returns of 2.25% were given for 100 quarters.
Downside Risk = [{2.25)*1]^1/2 = 1.5
Sortino Ratio = 2.5 / 1.5 = 1.6667
State of mkt
Probability
Ending price
HPR
APR
EAR
Boom
0.20
55
(55-50)/50 =10%
10%x4 =40%
(1+0.10)^4 -1 = 46.41%
Normal
0.55
53
(53-50)/50 =6%
6%x4 =24%
(1+0.06)^4 -1 = 26.25%
Recession
0.25
46.50
(46.50-50)/50 =-7%
-7%x4 = -28%
(1-0.07)^4 -1 = 25.19%
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