Chapter 18, Problem 21PS

Summary Introduction

(a)

To calculate:

The sharpe ratios for the fund and index i.e. Miranda fund and S&P $500$ .

Introduction:

Sharpe ratio is a ratio which helps in computing the reward-to-volatility ratio. This ratio gives an understanding of the incremental return which an investor will expect to earn on every $1\%$ increase in the value of standard deviation.

Answer to Problem 21PS

The sharpe ratio for Miranda fund is $0.2216$ and for the S&P $500$ is $-0.556$ .

Explanation of Solution

Given:

Risk free rate of Return or T-bill Rate, $\overline{{\text{r}}_{\text{f}}}=2\%$

One- Year Trailing Returns
	Miranda Fund	S&P 500
Return	10.2%	-22.5%
Standard Deviation	37%	44%
Beta	1.10	1.00

The sharpe ratio is calculated by the following formula:

  $\text{Sharpe Ratio}=\frac{\text{Expected return}-\text{Risk free return}}{\text{Standard deviation}}$

The sharpe ratio for Portfolio Miranda is:

  $\begin{array}{c}\text{Sharpe Ratio}=\frac{\text{Expected return}-\text{Risk free return}}{\text{Standard deviation}}\\ =\frac{\left(10.2-2\right)}{37}\\ =0.2216\end{array}$

The sharpe ratio for S&P $500$ is:

  $\begin{array}{c}\text{Sharpe Ratio}=\frac{\text{Expected return}-\text{Risk free return}}{\text{Standard deviation}}\\ =\frac{\left(-22.5-2\right)}{44}\\ =-0.556\end{array}$

Summary Introduction

(b)

To calculate:

The ${\text{M}}^2$ measure for the fund and index i.e. Miranda fund and S&P $500$

Introduction:

  ${\text{M}}^2$ is a measure which helps in ascertaining the adjusted return in regard to the associated risk with the portfolio.

Answer to Problem 21PS

The ${\text{M}}^2$ measure is $34.25\%$ .

Explanation of Solution

Given:

Risk free rate of Return or T-bill Rate, $\overline{{\text{r}}_{\text{f}}}=2\%$

One- Year Trailing Returns
	Miranda Fund	S&P 500
Return	10.2%	-22.5%
Standard Deviation	37%	44%
Beta	1.10	1.00

The ${\text{M}}^2$ is calculated by the following formula:

  ${\text{M}}^2={\text{r}}_{\text{P}}-{\text{r}}_{\text{M}}$

For computing ${\text{M}}^2$ , first adjusted portfolio should be calculated:

  $\begin{array}{c}\text{Adjusted Portfolio}=\frac{\text{Standard deviation of S\&P }500}{\text{Standard deviation of M fund}}\\ =\frac{44\%}{37\%}\\ =1.1892\end{array}$

Now, the position of T-bills:

  $\begin{array}{c}\text{Portfolio in T-bills}=1-\text{Adjusted portfolio}\\ =1-1.1892\\ =-0.1892\end{array}$

  ${\text{M}}^2$ for Portfolio Miranda is:

  $\begin{array}{c}{\text{M}}^2={\text{r}}_{\text{P}}-{\text{r}}_{\text{M}}\\ =\left[\left(1.1892\times 10.2\%\right)+\left(-0.1892\times 2\%\right)\right]-\left(22.50\%\right)\\ =34.25\%`\end{array}$

Thus, the ${\text{M}}^2$ measure is $34.25\%$ .

Summary Introduction

(c)

To calculate:

The treynor ratios for the fund and index i.e. Miranda fund and S&P $500$ .

Introduction:

Treynor ratio is a ratio which helps in computing the reward-to-volatility ratio. This ratio provides excess return over expected in regard to systematic risk i.e. beta.

Answer to Problem 21PS

The treynor ratio for Miranda fund is $7.45\%$ and for the S&P $500$ is $-24.50\%$ .

Explanation of Solution

Given:

Risk free rate of Return or T-bill Rate, $\overline{{\text{r}}_{\text{f}}}=2\%$

One- Year Trailing Returns
	Miranda Fund	S&P 500
Return	10.2%	-22.5%
Standard Deviation	37%	44%
Beta	1.10	1.00

The treynor ratio is calculated by the following formula:

  $\text{Treynor Ratio}=\frac{\text{Expected return}-\text{Risk free return}}{\text{Beta}}$

The treynor ratio for Portfolio Miranda is:

  $\begin{array}{c}\text{Treynor Ratio}=\frac{\text{Expected return}-\text{Risk free return}}{\text{Beta}}\\ =\frac{\left(10.2\%-2\%\right)}{1.10}\\ =7.45\%\end{array}$

The treynor ratio for S&P $500$ is:

  $\begin{array}{c}\text{Treynor Ratio}=\frac{\text{Expected return}-\text{Risk free return}}{\text{Beta}}\\ =\frac{\left(-22.5\%-2\%\right)}{1.00}\\ =-24.50\%\end{array}$

Summary Introduction

(d)

To calculate:

The Jensen measure i.e. alpha for the fund i.e. Miranda fund.

Introduction:

The alpha shows the performance of portfolio in relation to benchmark.

Answer to Problem 21PS

The Jensen measure for Miranda fund is $35.15\%$ .

Explanation of Solution

Given:

Risk free rate of Return or T-bill Rate, ${\text{r}}_{\text{f}}=2\%$

One- Year Trailing Returns
	Miranda Fund	S&P 500
Return	10.2%	-22.5%
Standard Deviation	37%	44%
Beta	1.10	1.00

The Jensen measure i.e. alpha is calculated by the following formula:

  $\text{α}={\text{r}}_{\text{p}}-\left[{\text{r}}_{\text{f}}+{\text{β}}_{\text{p}}\left({\text{r}}_{\text{M}}-{\text{r}}_{\text{f}}\right)\right]$

The Jensen measure i.e. alpha for Portfolio Miranda is:

  $\begin{array}{c}\text{α}={\text{r}}_{\text{p}}-\left[{\text{r}}_{\text{f}}+{\text{β}}_{\text{p}}\left({\text{r}}_{\text{M}}-{\text{r}}_{\text{f}}\right)\right]\\ =10.2\%-\left[2\%+1.10\left(-22.5\%-2\%\right)\right]\\ =35.15\%\end{array}$

Thus, the Jensen alpha of the fund id $35.15\%$