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Tutorial 3
Question 1
You've observed the following returns on Crash-n-Burn Computer's stock over the past five
years: 7 percent, 12 percent, 11 percent, 38 percent, and 14 percent.
a. What was the arithmetic average return on Crash-n-Burns stock over this five-year period?
b. What was the variance of Crash-n-Burns return over his period? The standard deviation?
Solution
a. To find the average return, we sum all the returns and divide by the number of returns, so:
Arithmetic average return = (0.07 0.12 + 0.11 + 0.38 + 0.14)/5 = 11.60%
b. Using the equation to calculate variance, we find:
Variance = 1/4[(0.07 0.116)
2
+ (0.12 0.116)
2
+ (0.11 0.116)
2
+ (0.38 0.116)
2
+
(0.14 0.116)
2
] = 0.032030 3.2% (the unit for variance is not %, but %
squared)
So, the standard deviation is:
Standard deviation = (0.03230)
1/2
= 0.1790 or 17.90% 3.2
Question 2
For the above question, suppose the average inflation rate over this period was 3.5 percent and
the average rate T-bill rate over the period was 4.2 percent.
a. What was the average real return on Crash-n-Burns stock?
b. What was the average nominal risk premium on Crash-n-Burns stock?
Solution
a. Exact: (1 + R) = (1 + r)(1 + h) Fischer equation
Real return = r = (1.1160/1.035) 1 = 7.83%
Approx.: Real return = 11.60% 3.5% = 8.10%
b. The average nominal risk premium is simply the average return of the asset, minus the
average risk-free rate:
Nominal RP = Nominal return Nominal risk-free rate
= 11.60% 4.20% = 7.40%
(This is the approx. nominal RP, which is also equal to the approx. real RP)
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Question 3
Given the information in the problem just above, what was the average real risk-free rate over
this time period? What was the average real risk premium?
Solution
Exact: Real risk-free rate = (1.042/1.035) 1 = 0.68%
Approx.: Real risk-free rate = 4.2% 3.5% = 0.70%
To calculate the average real risk premium, we can subtract the average real risk-free rate from
the average real return. So, the average real risk premium was:
Exact: Real RP = Real return Real risk-free rate
= 7.83% 0.68% = 7.15%
Alternatively, the exact real risk premium can be computed using the approximate
expected real risk premium, divided by one plus the inflation rate, so:
Real RP = 7.40%/1.035 = 7.15%
Approx.: Real RP = Real return Real risk-free rate
= 8.10% 0.70% = 7.4%
Alternatively,
Real RP = Nominal return Nominal risk-free rate
= 11.60% 4.2% = 7.4%
(Note: nominal minus nominal gives real, because inflation is removed in the process)
Nominal Rate
Inflation
Approx.
Real Rate
Exact
Real Rate
Stocks
Return
11.60% 3.50% 11.60% 3.50% =
8.10%
(1+11.60%)/(1+3.50%) 1 =
7.83%
Risk-free
Rate
4.20% 3.50% 4.20% 3.50% =
0.70%
(1+4.20%)/(1+3.50%) 1 =
0.68%
Stocks
Risk
Premium
11.60% 4.20% =
7.40%
Do not use
(1.1160/1.042) 1
8.10% 0.70% =
7.40%
Do not use
(1.0810/1.007) 1
7.83% 0.68% =
7.15%
OR
7.40%/(1+3.5%) =
7.15%
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Question 4
A stock has had the following year-end prices and dividends:
Year Price Dividend
1 $60.18
2 73.66 $0.60
3 94.18 0.64
4 89.35 0.72
5 78.49 0.80
6 95.05 1.20
What are the arithmetic and geometric returns for the stock?
Solution
To calculate the arithmetic and geometric average returns, we must first calculate the return for
each year. The return for each year is:
R
1
= ($73.66 60.18 + 0.60) / $60.18 = 23.40%
R
2
= ($94.18 73.66 + 0.64) / $73.66 = 28.73%
R
3
= ($89.35 94.18 + 0.72) / $94.18 = 4.36%
R
4
= ($78.49 89.35 + 0.80) / $89.35 = 11.26%
R
5
= ($95.05 78.49 + 1.20) / $78.49 = 12.63%
The arithmetic average return was:
R
A
= (0.2340 + 0.2873 0.0436 0.1126 + 0.2263)/5 = 11.83%
Arithmetic average measures the likely return in a typical year.
The geometric average return was:
R
G
= [(1 + 0.2340)(1 + 0.2873)(1 0.0436)(1 0.1126)(1 + 0.2263)]
1/5
1 = 10.58%
Geometric mean usually measures the average compounded annual return over a specific
historical period.
That is one of the reasons why using arithmetic return is more appropriate for estimating
future return.
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Question 5
Consider the following information on three stocks:
State of
Economy
Probability of
State of
Economy
Rate of Return if State Occurs
Stock A Stock B Stock C
Boom 0.35 0.24 0.36 0.55
Normal 0.50 0.17 0.13 0.09
Bust 0.15 0.00 -0.28 -0.45
a. If your portfolio is invested 40 percent each in A and B and 20 percent in C, what is the
portfolio expected return? The variance? The standard deviation?
b. If the expected T-bill rate is 3.80 percent, what is the expected risk premium on the portfolio?
c. If the expected inflation rate is 3.50 percent, what are the approximate and exact expected
real returns on the portfolio? What are the approximate and exact expected real risk
premiums on the portfolio?
Solution
a. Boom : E(R
p
) = 0.4(0.24) + 0.4(0.36) + 0.2(0.55) = 35.00%
Normal : E(R
p
) = 0.4(0.17) + 0.4(0.13) + 0.2(0.09) = 13.80%
Bust : E(R
p
) = 0.4(0.00) + 0.4(0.28) + 0.2(0.45) = 20.20%
And the expected return of the portfolio is:
E(R
p
) = 0.35(35%) + 0.50(13.8%) + 0.15(20.20%) = 16.12%
2
p
= 0.35(0.35 0.1612)
2
+ 0.50(0.138 0.1612)
2
+ 0.15(0.202 0.1612)
2
= 0.03253
p
= (0.03253)
1/2
= 0.1804 or 18.04%
b. The risk premium is the return of a risky asset, minus the risk-free rate. T-bills are often used
as the risk-free rate, so:
RP
i
= E(R
p
) R
f
= 16.12% 3.80% = 12.32%
c. Approximate
Real return = 16.12% 3.5% = 12.62%
Real risk premium = Nominal return Nominal risk-free rate
= 16.12% 3.8% = 12.32%
(note: nominal minus nominal gives real, because inflation is removed in the process)
Exact
To find the exact real return, we use the Fisher equation:
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1 + E(R
i
) = (1 + h)[1 + e(r
i
)]
1.1612 = (1.0350)[1 + e(r
i
)]
Real return = e(r
i
) = (1.1612/1.035) 1 = 12.19%
Real risk-free rate = (1 + 3.8%) / (1 + 3.5%) 1 = 0.2898%
Real risk premium = Real return Real risk-free rate
= 12.19% 0.2898% = 11.90%
Alternatively: The exact real risk premium can be computed using the approximate real
risk premium, divided by one plus the inflation rate, so:
Exact expected real risk premium = 12.32%/1.035 = 11.90%
Nominal Rate
Inflation
Approx.
Real Rate
Exact
Real Rate
Portfolios
Return
16.12% 3.50% 16.12% 3.50% =
12.62%
(1+16.12%)/(1+3.50%) 1 =
12.19%
Risk-free
(T-Bills)
3.80% 3.50% 3.80% 3.50% =
0.30%
(1+3.80%)/(1+3.50%) 1 =
0.2898%
Portfolio
Risk
Premium
16.12% 3.8% =
12.32%
Do not use
(1.1612/1.038) 1
12.62% 0.30% =
12.32%
Do not use
(1.1262/1.003) 1
12.19% 0.2898% =
11.90%
OR
12.32%/(1+3.5%) =
11.90%
