Chapter 24, Problem 21PS

a.

Summary Introduction

To compute: The Sharpe ratios for the Miranda fund and the S&P 500.

Introduction:

Sharpe ratio: It is one of risk measurement ratios. When a performance of an investment has to be done and that too in comparison with a risk-free asset, Sharpe ratio is utilized.

Answer to Problem 21PS

The Sharpe ratio for Mirinda Fund is 0.2216 and S&P 500 is -0.5568.

Explanation of Solution

Given information:

Risk-free interest rate=2%

One- year Trailing Returns
	Miranda Fund	S&P 500
Return	10.20%	-22.50%
Standard deviation	37.00%	44.00%
Beta	1.10	1.00

Let us calculate the Sharpe ratio using the formula:

  $Sharperatio=\frac{R_P-R_f}{Standarddeviation}$

Let us now substitute the given values in the formula.

	Mirinda Fund	S&P 500
Sharpe Ratio	$\begin{array}{l}=\frac{10.20\%-2\%}{37.00\%}\\ =\frac{8.20\%}{37.00\%}\Rightarrow 0.2216\end{array}$	$\begin{array}{l}=\frac{-22.50\%-2\%}{44.00\%}\\ =\frac{-24.50\%}{44.00\%}\Rightarrow -0.5568\end{array}$

Therefore, the Sharpe ratio for Mirinda Fund is 0.2216 and S&P 500 is -0.5568.

b.

Summary Introduction

To compute: The M2 measures for Miranda and S&P 500.

Introduction:

M² measure: It is also termed as M-square measure. It is used to calculate the return which has to be adjusted for the risk associated with a portfolio along with benchmark.

Answer to Problem 21PS

The M2 of Mirinda Fund (portfolio P) would be 34.25%.

Explanation of Solution

Giveninformation:

Risk-free interest rate=2%

One- year Trailing Returns
	Miranda Fund	S&P 500
Return	10.20%	-22.50%
Standard deviation	37.00%	44.00%
Beta	1.10	1.00

The formula to calculate the M²is as follows:-

  $M^2=\overline{R_P}-\overline{R_M}$

Where,

  $\overline{R_P}=Returnonadjustedportfolio$

  $\overline{R_M}=marketreturn$

One point to be kept in mind is that to get the position of volatility which is similar to the market, the funds have to be combined with the T-Bills.

Let us now calculate the adjusted portfolio.

  $\begin{array}{l}Adjustedportfolio=\frac{StandarddeviationofS\&P500}{StandarddeviationofMirindaFund}\\ =\frac{44.00\%}{37.00\%}\Rightarrow 1.1892\end{array}$

Now, we have to calculate the position in T-Bills:

  $\begin{array}{l}PositioninT-Bills=1-Adjustedportfolio\\ =1-1.1892\Rightarrow -0.1892\end{array}$

Therefore, when we check for the weight of Mirinda fund in adjusted portfolio, it will be 1.1892 and when amount is borrowed at a rate which is equal to 0.1892 will result in risk-free rate.

Therefore, the end result would be as follows:

Volatility of return of portfolio=Volatility in return on market index.

Now, let us calculate the M²of Mirinda Fund:

  $M^2=\overline{R_P}-\overline{R_M}$

  $\begin{array}{l}=(1.1892\times 10.20\%)+(-0.1892\times 2\%)-(-22.50\%)\\ =0.1212-0.0037+0.225\\ =0.3425\end{array}$

This value has to be converted into percentage by multiplying it with 100.

Therefore, the M2 of Mirinda Fund (portfolio P) would be 34.25%

c.

Summary Introduction

To compute: The Treynor’s measure ratios for the Miranda fund and the S&P 500.

Introduction:

Treynor’s ratio: It is also termed as Treynor’s measure in some instances. It is one of the measures used to calculate the risk adjusted return. It calculates the returns on the basis of systematic risk which differentiates this ratio with other related ratios.

Answer to Problem 21PS

The Sharpe ratio for Mirinda Fund is 7.45% and S&P 500 is -24.50%.

Explanation of Solution

Given information:

Risk-free interest rate=2%

One- year Trailing Returns
	Miranda Fund	S&P 500
Return	10.20%	-22.50%
Standard deviation	37.00%	44.00%
Beta	1.10	1.00

The formula to be used to calculate the Treynor’s measure is as follows:

  $Treynor’sratio=\frac{R_P-R_f}{\beta }$

Where

R_P=Return on portfolio

R_F= Risk free rate of return

ß = Systematic risk

	Mirinda Fund	S&P 500
Treynor’s measure	$\begin{array}{l}=\frac{10.20\%-2\%}{1.10}\\ =\frac{8.20\%}{1.10}\Rightarrow 7.45\%\end{array}$	$\begin{array}{l}=\frac{-22.50\%-2\%}{1.00}\\ =\frac{-24.50\%}{1.00}\Rightarrow -24.50\%\end{array}$

Therefore, the Sharpe ratio for Mirinda Fund is 7.45% and S&P 500 is -24.50%.

d.

Summary Introduction

To compute: The Jenson measure for the Miranda Fund.

Introduction:

Jenson Measure: This measure is used to calculate the average rate of return. The return calculated is on the investment made in a portfolio.

Answer to Problem 21PS

The Jenson measure of Mirinda fund is 35.15%.

Explanation of Solution

Given information:

Risk-free interest rate=2%

One- year Trailing Returns
	Miranda Fund	S&P 500
Return	10.20%	-22.50%
Standard deviation	37.00%	44.00%
Beta	1.10	1.00

The formula is as follows:

  $\begin{array}{l}Jensonmeasure=R_P-[r_f+\beta \times (r_m-r_f)]\\ =10.20\%-[2\%+1.10\times (-22.50-2\%)]\\ =10.20\%+24.95\%\\ =35.15\%\end{array}$

Therefore, the Jenson measure of Mirinda fund is 35.15%.