Chapter 20, Problem 18C

Summary Introduction

(a)

To calculate:

By using a misestimated beta of $0.5$ , ascertain the standard deviation of the (now imperfect) hedged portfolio.

Introduction:

Standard deviation is a measure to calculate the deviation from the mean which is also called as a measure of dispersion. It helps in analyzing the performance of the fund.

Answer to Problem 18C

The standard deviation for the (imperfect) hedge portfolio is $6.129$ .

Explanation of Solution

Given:

  $\begin{array}{c}\text{Misstated beta}=0.50\\ \text{Original beta}=0.75\\ \text{Standard deviation of market}=5\%\\ \text{Residual standard deviation}=6\%\\ \text{Portfolio value}=\$6\text{ million}\end{array}$

For calculating the standard deviation, the following formula is to be used:

  $\text{Standard deviation}=\sqrt{\left[{\left(\text{Difference in beta}\right)}^2\times {\left(\text{Standard deviation}\right)}^2\right]+{\left(\text{Residual standard deviation}\right)}^2}$

By using the formula, the standard deviation is:

  $\begin{array}{c}\text{Standard deviation}=\sqrt{\left[{\left(\text{Difference in beta}\right)}^2\times {\left(\text{Standard deviation}\right)}^2\right]+{\left(\text{Residual standard deviation}\right)}^2}\\ =\sqrt{\left[{\left(0.75-0.50\right)}^2\times {\left(5\right)}^2\right]+{\left(6\right)}^2}\\ =\sqrt{\left(0.0625\times 25\right)+36}\\ =\sqrt{37.5625}\\ =6.129\end{array}$

Summary Introduction

(b)

To calculate:

By taking the expected market return value of $1\%$ and a standard deviation of market $5\%$ , ascertain the probability of incurring losses in the next month and compare the same with the probability calculated in problem $16$ .

Introduction:

Standard deviation is a measure to calculate the deviation from the mean which is also called as a measure of dispersion. It helps in analyzing the performance of the fund.

Answer to Problem 18C

The probability for getting a negative return is $0.3342$ which is almost same to the probability of $0.3385$ of the previous problem.

Explanation of Solution

Given:

  $\begin{array}{c}\text{Misstated beta}=0.50\\ \text{Original beta}=0.75\\ \text{Standard deviation of market}=5\%\\ \text{Residual standard deviation}=6\%\\ \text{Portfolio value}=\$6\text{ million}\end{array}$

Based on the previous problem i.e $16$ , the probability for incurring losses is as follows:

The expected return for zero beta market was calculated by following formula:

  $\begin{array}{c}\text{Expected rate}=\text{Risk-free rate}+\text{Alpha}\\ =\text{0}\text{.5\%}+\text{2}\text{.0\%}\\ =\text{2}\text{.5\%}\end{array}$

So the rate of return is $2.5\%$ .

The monthly returns are distributed normally given in the question. So the rate of return for zero beta is

  $\begin{array}{l}\text{Z= }\frac{\text{rate of return for zero market position}}{\text{Residual standard deviation}}\\ \text{Z}=\frac{-2.5\%}{6.0\%}\\ \text{Z}=-0.4167\end{array}$

Thus, the probability of getting a negative return is:

  $\begin{array}{c}\text{Probability}=\text{N}\left(-0.417\right)\\ =0.3385\end{array}$

Now, in the present problem,

Number of contracts to be calculated which is as follows:

  $\begin{array}{c}\text{Number of contracts}=\frac{\text{Portfolio value}\times \text{Beta risk}}{\text{Contract multiplier}\times \text{Current stock price}}\\ =\frac{\$6000000\times 0.50}{\$50\times 2000}\\ =30\end{array}$

As the portfolio is unhedged, the rate of return should be computed fresh by adding the dolar value and future position.

The computation of dollar value of the stock portfolio:

  $\begin{array}{l}\text{Dollar value}=\text{Portfolio value}\times \left(1+\text{return of portfolio}\right)\\ =\$6000000\times \left(1+{\text{r}}_{\text{p}}\right)\\ =\$6000000\times \left[1+0.005+0.75\left({\text{r}}_{\text{m}}-0.005\right)+0.2+e\right]\\ =\$6127500+\left(\$4500000\times {\text{r}}_{\text{m}}\right)+\left(\$6000000\times e\right)\end{array}$

Now, the value of future position:

  $\begin{array}{c}\text{Future position}=\text{Number of contracts}\times \text{Contract multiplier}\times \left({\text{F}}_0-{\text{F}}_1\right)\\ =30\times \$50\times \left({\text{F}}_0-{\text{F}}_1\right)\\ =\$1500\times \left[\left({\text{S}}_0\times 1.005\right)-{\text{S}}_1\right]\\ =\$1500\times {\text{S}}_0\left[1.005-\left(1+{\text{r}}_{\text{m}}\right)\right]\\ =\$1500\times \left[2000\times \left(0.005-{\text{r}}_{\text{m}}\right)\right]\\ =\$15000-\left(\$3000000\times {\text{r}}_{\text{m}}\right)\end{array}$

The total value of dollar plus future is as follows:

  $\begin{array}{c}\text{Total value}=\$6142500+\left(\$1500000\times {\text{r}}_{\text{m}}\right)+\left(\$6000000\times e\right)\\ =\$6142500++\left(\$1500000\times 0.01\right)+\left(\$6000000\times e\right)\\ =\$6157500+\left(\$6000000\times e\right)\end{array}$

Now, the new rate of return for the imperfect hedge portfolio is:

  $\begin{array}{c}\text{Rate of return}=\left(\frac{\$6157500}{\$6000000}\right)-1\\ =0.2625\\ =2.625\%\end{array}$

The monthly returns are distributed normally given in the question. So the rate of return for zero beta is

  $\begin{array}{c}\text{Z value}=\frac{\text{Rate of return for market position}}{\text{Residual standard deviation}}\\ =\frac{-2.625}{6.129}\\ =-0.4283\end{array}$

Thus, the probability for negative return is to be:

  $\begin{array}{c}\text{Probability}=\text{N}\left(-0.4283\right)\\ =0.3342\end{array}$

Thus, it can be said that it almost same to the probability computed before for the previous problem.

Summary Introduction

(c)

To calculate:

By taking the data of problem $17$ and using a misestimated beta of $0.5$ , ascertain the probability of incurring losses in the next month and compare the same with the probability calculated in part (b).

Introduction:

Standard deviation is a measure to calculate the deviation from the mean which is also called as a measure of dispersion. It helps in analyzing the performance of the fund.

Answer to Problem 18C

The probability for getting a negative return is $0.02916$ .

Explanation of Solution

Given:

  $\begin{array}{c}\text{Misstated beta}=0.50\\ \text{Original beta}=0.75\\ \text{Standard deviation of market}=5\%\\ \text{New Residual standard deviation}=0.6\%\\ \text{Portfolio value}=\$6\text{ million}\end{array}$

For calculating the standard deviation, the following formula is to be used:

  $\text{Standard deviation}=\sqrt{\left[{\left(\text{Difference in beta}\right)}^2\times {\left(\text{Standard deviation}\right)}^2\right]+{\left(\text{Residual standard deviation}\right)}^2}$

By using the formula, the standard deviation is:

  $\begin{array}{c}\text{Standard deviation}=\sqrt{\left[{\left(\text{Difference in beta}\right)}^2\times {\left(\text{Standard deviation}\right)}^2\right]+{\left(\text{Residual standard deviation}\right)}^2}\\ =\sqrt{\left[{\left(0.75-0.50\right)}^2\times {\left(5\right)}^2\right]+{\left(0.6\right)}^2}\\ =\sqrt{\left(0.0625\times 25\right)+0.36}\\ =\sqrt{1.9225}\\ =1.3865\end{array}$

Now, the new rate of return for the imperfect hedge portfolio is:

  $\begin{array}{c}\text{Rate of return}=\left(\frac{\$6157500}{\$6000000}\right)-1\\ =0.2625\\ =2.625\%\end{array}$

The monthly returns are distributed normally given in the question. So the rate of return for zero beta is

  $\begin{array}{c}\text{Z value}=\frac{\text{Rate of return for market position}}{\text{Residual standard deviation}}\\ =\frac{-2.625}{1.3865}\\ =-1.8933\end{array}$

Thus, the probability for negative return is to be:

  $\begin{array}{c}\text{Probability}=\text{N}\left(-1.8933\right)\\ =0.02916\end{array}$

Summary Introduction

(d)

To determine:

The reason for explaining the fact that the misestimated beta affects more to $100$ stocks portfolio than a one stock portfolio.

Introduction:

Standard deviation is a measure to calculate the deviation from the mean which is also called as a measure of dispersion. It helps in analyzing the performance of the fund.

Explanation of Solution

The reason is the level of volatility to the portfolio. The more there is stock in portfolio with improper hedging, the more it contains volatility.

Conclusion

Thus, the reason that misestimated beta affects $100$ stocks portfolio more is due to the high level of volatility.