How to calculate average risk premium

Assignment 5

Text edition 7:

Chapter 12 - Questions and Problems - 5, 7, 8, 9, 12, 13.

5. Nominal versus Real Returns:

a. In nominal terms?

b. In real terms?

a) The nominal return is 10.23% from the table.

b) To find the real return, we use the fisher equation

(1 + R) = (1 + r) (1 + h)

(1 + 0.1023) = (1 + r) (1 + 0.0406)

1.1023 = (1 + r) (1.0406)

1 + r = 1.0593

r = 1.0593 – 1

r = 0.0593 or 5.93%

7. Calculating Returns and Variability:

Returns

Year X Y

1 6% 18%

2 24 39

3 13 -6

4 -14 -20

5 15 47

Return X

Arithmetic average returns, R = [R1 +R2 + R3 + R4 + R5]/N

= [0.06 + 0.24 + 0.13 - 0.14 + 0.15]/5

= 0.44/5

= 0.088 or 8.80%

Variance = 1/ (N – 1) [(R1 – R) 2 + (R2 – R) 2 + (R3 – R) 2 + (R4 – R) 2+ (R5 – R) 2]

= 1/ (5 – 1) [(0.06 – 0.088)2 + (0.24 – 0.088)2 + (0.13 – 0.088)2 + (-0.14 – 0.088)2+ (0.15 – 0.088)2]

= ¼ [0.08148]

= 0.02037

Standard deviation = √Variance

= √0.02037

= 0.1427 or 14.27%

Return Y

Arithmetic average returns, R = [R1 +R2 + R3 + R4 + R5]/N

= [0.18 + 0.39 + (-0.06) + (-0.20) + 0.47]/5

= 0.78/5

= 0.1560 or 15.60%

Variance = 1/ (N – 1) [(R1 – R) 2 + (R2 – R) 2 + (R3 – R) 2 + (R4 – R) 2+ (R5 – R) 2]

= 1/ (5 – 1) [(0.18 – 0.156)2 + (0.39 – 0.156)2 + (-0.06 – 0.156)2 + (-0.20 – 0.156)2+ (0.47 – 0.156)2]

= ¼ [0.32732]

= 0.08183

Standard deviation = √Variance

= √0.08183

= 0.2861 or 28.61%

8. Risk Premiums:

a. Calculate the arithmetic average returns for large-company stocks and T-Bills over this period.

b. Calculate the standard deviation of the returns for large-company stocks and T-Bills over this period.

c. Calculate the observed risk premium in each year for the large-company stocks versus T-Bills. What was the average risk premium over this period? What was the standard deviation of the risk premium over this period?

d. Is it possible for the risk premium to be negative before an investment is undertaken? Can the risk premium be negative after the fact?

Year

Large stock return

T-bill return

Risk premium

1970

-3.57%

6.89

-10.46

1971

8.01

3.86

4.15

1972

27.37

3.43

23.94

1973

0.27

4.78

-4.51

1974

-25.93

7.68

-33.61

1975

18.48

7.05

11.43

Total

24.63

33.69

-9.06

a) Large Company Stocks

Arithmetic average returns, R = [R1 +R2 + R3 + R4 + R5 + R6]/N

= 24.63/6

= 4.105%

T-bills

Arithmetic average returns, R = [R1 +R2 + R3 + R4 + R5 + R6]/N

= 33.69/6

= 5.615%

b) Large Company Stocks

Variance = 1/ (N – 1) [(R1 – R) 2 + (R2 – R) 2 + (R3 – R) 2 + (R4 – R) 2+ (R5 – R) 2 + (R6 – R) 2]

= 1/ (6 – 1) [(-3.57 – 4.105)2 + (8.01 – 4.105)2 + (27.37 – 4.105)2 + (0.27 – 4.105)2+ (-25.93 – 4.105)2+ (18.48 – 4.105)2]

= 1/5 [0.173885]

= 0.034777

Standard deviation = √Variance

= √0.034777

= 0.1865 or 18.65%

T-bills

Variance = 1/ (N – 1) [(R1 – R) 2 + (R2 – R) 2 + (R3 – R) 2 + (R4 – R) 2+ (R5 – R) 2 + (R6 – R) 2]

= 1/ (6 – 1) [(6.89 – 5.615)2 + (3.86 – 5.615)2 + (3.43 – 5.615)2 + (4.78 – 5.615)2+ (7.68 – 5.615)2+ (7.05 – 5.615)2]

= 1/5 [0.00165005]

= 0.00033001

Standard deviation = √Variance

= √0.00033001

= 0.0182 or 1.82%

c) Average observed risk premium = [R1 +R2 + R3 + R4 + R5 + R6]/N = -9.06/6 = -1.51%

Variance = 1/ (N – 1) [(R1 – R) 2 + (R2 – R) 2 + (R3 – R) 2 + (R4 – R) 2+ (R5 – R) 2 + (R6 – R) 2]

= 1/ (6 – 1) [(-10.46 – 1.51)2 + (4.15 – 1.51)2 + (23.94 – 1.51)2 + (-4.51 – 1.51)2+ (-33.61 – 1.51)2+ (11.43 – 1.51)2]

= 1/5 [0.1966694]

= 0.03933

Standard deviation = √Variance

= √0.03933

= 0.1983 or 19.83%

d) Before the fact, the risk premium will positive, investors demand compensation above the risk-free return to invest money. After the fact, the risk premium can be negative if assets nominal return is low and risk-free return is high unexpectedly.

9. Calculating Returns and Variability:

a. What was the arithmetic average return on Crash-n-Burn’s stock over this 5-year period?

b. What was the variance of Crash-n-Burn’s returns for this period? The standard deviation?

a) Arithmetic average return, R = [R1 +R2 + R3 + R4 + R5]/N

= [0.02 + (-0.08) + 0.24 + 0.19 + 0.12]/5

= 0.49/5

= 0.098 or 9.80%

b) Variance = 1/(N – 1) [(R1 – R)2 + (R2 – R)2 + (R3 – R)2 + (R4 – R)2+ (R5 – R)2]

= 1/ (5 – 1) [(0.02 – 0.098)2 + (-0.08 – 0.098)2 + (0.24 – 0.098)2 + (0.19 – 0.098)2+ (0.12 – 0.098)2]

= ¼ [0.06688]

= 0.01672

Standard deviation = √Variance

= √0.01672

= 0.1293 or 12.93%

12. Effects of Inflation:

T-bill rates were highest in initial period. During the period of high inflation, it was consistent in accordance with the Fisher effect.

13. Calculating Investment Returns:

Given that Coupon rate = 7%,

Price 1 year ago, P1 = $920

Required return on bond, I = 8%

Number of years, n = 6

Inflation rate, h = 4.2%

To find total real return, we have to find the nominal return based on the current price of bond. Now,

Coupon payment, C = 0.07 x 1000 = $70

P1 = C (PVIFA @ 8%, 6) + Face value (PVIF @ 8%, 6)

= $70 [(1.086 – 1)/ (0.08*1.086)] + 1000/1.086

= $70 (0.58687/0.12695) + 630.17

= $70 (4.6228) + 630.17

= $323.60 + $630.17

= $953.77

Nominal return, R = [(P1 – P0 + C]/P0

= [(953.77 – 920 + 70]/920

= 103.77/920

= 0.1128 or 11.28%

Using the fisher equation,

(1 + R) = (1 + r) (1 + h)

(1 + 0.1128) = (1 + r) (1 + 0.042)

(1.1128) = (1 + r) (1.042)

1 + r = 1.1128/1.042

1 + r = 1.0679

r = 1.0679 – 1

r = 0.0679 or 6.79%

The total real return on investment is 6.79%