Chapter 6, Problem 20PS

a

Summary Introduction

Adequate information:

U.S. market is risky portfolio

Risk-aversion coefficient A=4

Period representing the future expected performance = 1926 to 2015

	Average Annual Returns	S&P 500 Portfolio
Period	S&P 500 Portfolio	1-month T-bills	Risk premium	Standard deviation	Sharpe Ratio	Probability
1926-2015	11.67	3.58	8.10	20.48	0.40	--
1992-2015	11.10	3.52	7.59	18.22	0.42	0.94
1970-1991	10.91	7.48	3.44	16.71	0.21	0.50
1949-1969	15.35	2.28	13.08	17.66	0.74	0.24
1926 -1948	9.40	1.04	8.36	27.95	0.30	0.71

To compute: The fraction of portfolio to be allocated to T-bills and Equity.

Introduction:

Risk aversion coefficient: In today’s world, every aspect has a risk factor along with the benefits. When it comes to financial product, then it is obvious to have a risk. To measure such risk relative with its performance, we make use of standard deviation of returns. These are also known as volatility of returns. When a number from 1 (assuming to be lowest risk aversion) to 5 (assuming to be highest risk aversion) is attributed to an investor, this number is further assigned the letter A which is called the risk aversion coefficient.

Explanation of Solution

Here, we have to calculate the proportion of Y of the total investment. It is clear that the optimal position y related to a risky asset will be proportional to the risk premium while there is an inverse proportional relationship with that of the variance and degree of risk aversion.

  $y=\frac{E(r_p)-r_f}{A{\sigma }_p^2}$

When the entire period 1926-2015 is representing future expected performance, then A=4

  $\begin{array}{l}E(r_m)-r_f=8.10\%\\ {\sigma }_m=20.48\%\\ y=\frac{E(r_m)-r_f}{A{\sigma }_m^2}\end{array}$

By substituting the values, we get

  $y=\frac{8.10\%}{4\times {(20.48\%)}^2}$

We need to convert the percentages to normal values. After that, we get the equation as :

  $\begin{array}{l}y=\frac{0.081}{0.167772}\\ y=0.482798\end{array}$

Or 48.28% when converted to percentages

So, 48.28% have to be allocated to equity

If 100 is assumed to be the total, then

  $\begin{array}{l}T-Bills=Total-AllocationtoEquity\\ =100-48.28\%\\ =51.72\%\end{array}$

Therefore, 51.72% is to be allocated to T-bills.

Conclusion

So, 48.28% have to be allocated to equity and 51.72% is to be allocated to T-bills.

b

Summary Introduction

Adequate information:

U.S. market is risky portfolio

Risk-aversion coefficient A=4

Period representing the future expected performance = 1970 to 1991

To compute: The circumstances if period is representing the future expected performance is 1970 to 1991.

Introduction:

Risk aversion coefficient: In today’s world, every aspect has a risk factor along with the benefits. When it comes to financial product, then it is obvious to have a risk. To measure such risk relative with its performance, we make use of standard deviation of returns. These are also known as volatility of returns. When a number from 1 (assuming to be lowest risk aversion) to 5 (assuming to be highest risk aversion) is attributed to an investor, this number is further assigned the letter A which is called the risk aversion coefficient.

Explanation of Solution

When the entire period 1970-1991 is representing future expected performance, then A=4

E(r_m)-r_f=3.44%

s_M=16.71%

  $y=\frac{E(r_m)-r_f}{A{\sigma }_m^2}$

By substituting the values, we get

  $y=\frac{3.44\%}{4\times {(16.71\%)}^2}$

We need to convert the percentages to normal values. After that, we get the equation as :

  $\begin{array}{l}y=\frac{0.344}{4\times {(0.1671)}^2}\\ y=\frac{0.0344}{0.11169}\\ y=0.307995\end{array}$

or 30.80% when rounded off and converted to percentages

So, 30.80% have to be allocated to equity

If 100 is assumed to be the total, then

  $\begin{array}{l}T-Bills=Total-AllocationtoEquity\\ =100-30.80\%\\ =69.20\%\end{array}$

Therefore, 69.20% is to be allocated to T-bills.

Conclusion

So, 30.80% have to be allocated to equity and 69.20% is to be allocated to T-bills.

c

Summary Introduction

Adequate information:

U.S. market is risky portfolio

Risk-aversion coefficient A=4

Period representing the future expected performance = 1926 to 2015

Period representing the future expected performance = 1970 to 1991

To compute: The analysis of circumstances in both the given periods i.e, 1926-2015 and 1970-1991.

Introduction:

Risk aversion coefficient: In today’s world, every aspect has a risk factor along with the benefits. When it comes to financial product, then it is obvious to have a risk. To measure such risk relative with its performance, we make use of standard deviation of returns. These are also known as volatility of returns. When a number from 1 (assuming to be lowest risk aversion) to 5 (assuming to be highest risk aversion) is attributed to an investor, this number is further assigned the letter A which is called the risk aversion coefficient.

Explanation of Solution

It is clear that the risk premium in the period 1926-2015 is 8.10 ; whereas for the period 1970-1991 it is 3.44. So, when we compare both values, we find that the risk premium for the period 1970-1991 is less than the risk premium of the period 1926-2015.

Secondly, the Sharpe ratio for the period 1970-1991 is less and it has higher proportion of T-bills.

Conclusion

The results got from the calculation states that as market risk premium is probable to be less for the period 1926-2015, the market risk is supposed to higher and secondly, since the proportion of T-bills is more for the period 1970-1991, its Sharpe ratio will be less when compared to the period 1926-2015.