Chapter 21, Problem 45PS

Summary Introduction

(A)

Adequate Information:

The portfolio you manage is currently worth $100 million, and you promise to provide a minimum return of 0%. The equity portfolio has a standard deviation of 25% per year, and T-bills pay 5% per year.

  $\begin{array}{l}\text{Current value of portfolio}\left({\text{S}}_{\text{o}}\right)=\$100\text{ million}\\ \text{Floor promised to clients}\text{. 0\% return}\left(X\right)=\$100\text{ million}\\ \text{Volatility}\left(\sigma \right)=0.25\\ \text{Risk-free rate}\left(r\right)=0.05\\ \text{Horizon of program}\left(T\right)=4years\\ \text{Dividend yield }\left(\delta \right)=0\end{array}$

To calculate:

The fraction of the portfolio, fraction in equity and the stock portfolio if falls by 3%.

Introduction:

Portfolio involves financial asset grouping viz. currencies, commodities, bonds, stocks and cash equivalents. It also comprises of fund counterparts such as closed funds, exchange traded funds and mutual funds. Non-publicly tradable securities such as private, art and real estate investment are also part of portfolios.

Explanation of Solution

Fractions of portfolio invested in T-bills and equity can be calculated based on the Black-Scholes
model. The value of $N\left(d_1\right)$defines the normal distribution of portfolio returns. Weighted given
to T-bills and equity can be known for value of $N\left(d_1\right)$

Formula for $d_1$ as per Black-Scholes model is as follows:

$\begin{array}{l}{d}_1=\frac{In\left(\frac{S_o}{X}\right)+\left(r-\delta +\frac{{\sigma }^2}{2}\right)T}{\sigma \sqrt{T}}\\ S_o=\text{Currnt stock price}\\ N\left(d_1\right)\text{and }N\left(d_2\right)=\text{Cumulative normal distribution functions}\\ X=\text{Exercise pricer}\\ r=\text{ Annualised risk free rate}\\ d=\text{ Annual dividend yield of underlying stock}\\ s=\text{Annualised standard deviation of share returns}\\ T=\text{ Time to expiry }\\ In\left(\frac{S_o}{X}\right)=\text{Natural logarithmic value of}\frac{S_o}{X}\\ e=\text{Exponential having the constant of }2.71828\\ \delta =\text{Annual dividend yield of underlying stock}\\ \sigma =\text{Annualised standard deviation of share returns or Volatility}\end{array}$

Substitute values to calculate $d_1$as follows:

$\begin{array}{l}{S}_o=\$100\text{ million},X=\$100\text{ million},r=0.05,\sigma =0.25,\delta =0,\text{and}T=0\\ d_1=\frac{In\left(\frac{S_o}{X}\right)+\left(r-\delta +\frac{{\sigma }^2}{2}\right)T}{\sigma \sqrt{T}}\\ =In\left(\frac{\$100}{\$100}\right)+\left(0.05-0+\frac{{0.25}^2}{2}\right)4\\ =\frac{In\left(1\right)+\left(0.05-0+0.03125\right)4}{0.5}\\ =\frac{In\left(1\right)+0.325}{0.5}\\ =\frac{0+0.325}{0.5}\\ =0.65\end{array}$

Calculate the Black-Scholes hedge ratio $N\left(d_1\right)$of call option using Ms-Excel"NORMSDIST" Function as follows:

$N\left(d_1\right)$can be calculated using Excel "NORMSDIST" function. The Excel steps to calculate
"NORMSDIST" function are as follows:

• First, go to the Menu bar of Excel and select 'Formulas' option
• Select Insert Function '(fx)
• Then select a category as Financial
• Then select "NORMSDIST" and click OK
• Then the Function Argument window will open. Now, input the given data in the required field
• Click OK
• The formula will display the final answer as 0.7422

Therefore, $N\left(d_1\right)$ value is 0.7422

Calculate the put delte as follows:

  $\begin{array}{l}\text{Put}\text{delta}=N\left(d_1\right)-1\\ =0.7422-1\\ =-0.2578\end{array}$

Hence, place 25.785 of the portfolio in T-bills and 74.22% $\left(\text{100}-\text{25}\text{.78\%}\right)$inequity

Conclusion

Hence, place 25.785 of the portfolio in T-bills and 74.22% $\left(\text{100}-\text{25}\text{.78\%}\right)$ inequity and the stock portfolio if falls by 3% will be $\$\mathrm{1.1763.}$

Summary Introduction

(B)

Adequate information:

  $\begin{array}{l}\text{Current value of portfolio}\left({\text{S}}_{\text{o}}\right)=\$97\text{ million}\\ \text{Floor promised to clients}\text{. 0\% return}\left(X\right)=\$100\text{ million}\\ \text{Volatility}\left(\sigma \right)=0.25\\ \text{Risk-free rate}\left(r\right)=0.05\\ \text{Horizon of program}\left(T\right)=4years\\ \text{Dividend yield }\left(\delta \right)=0\end{array}$

To calculate:

Fractions of portfolio divided into bills and equity If stock prices fall by 3% on the first day of trading. It means $S_o$ (current value of the portfolio) falls to $97

Introduction:

Portfolio involves financial asset grouping viz. currencies, commodities, bonds, stocks and cash equivalents. It also comprises of fund counterparts such as closed funds, exchange traded funds and mutual funds. Non-publicly tradable securities such as private, art and real estate investment are also part of portfolios.

Explanation of Solution

Fractions of portfolio invested in T-bills and equity can be calculated based on the Black-Scoles model. The value of $N\left(d_1\right)$ defines the normal distribution of portfolio returns. Weighted given to T-bills and equity can be known for value of $N\left(d_1\right)$

Substitute values to calculate $d_1$as follows

$\begin{array}{l}{S}_o=\$97\text{ million},X=\$100\text{ million},r=0.05,\sigma =0.25,\delta =0,\text{and}\text{T}=4\\ d_1=\frac{In\left(\frac{S_o}{X}\right)+\left(r-\delta +\frac{{\sigma }^2}{2}\right)T}{\sigma \sqrt{T}}\\ =In\left(\frac{\$97}{\$100}\right)+\left(0.05-0+\frac{{0.25}^2}{2}\right)4\\ =\frac{In\left(0.97\right)+\left(0.05-0+0.03125\right)4}{0.5}\\ =\frac{In\left(0.97\right)+0.08125\times 4}{0.5}\\ =\frac{-0.0305+0.325}{0.5}\\ =\frac{0.2945}{0.5}\\ =0.589\end{array}$

Calculate the Black-Scholes hedge ratio $N\left(d_1\right)$of call option using Ms-Excel"NORMSDIST" Function as follows:

Hence, $N\left(d_1\right)$value is 0.7221.

Calculate the put delta as follows:

  $\begin{array}{l}\text{Difference }=\left(\begin{array}{l}\text{Amount of T-bills as per sub-part }\left(\text{b}\right)\\ -\text{Amount of T-bills as per sub-part }\left(\text{a}\right)\end{array}\right)\\ =\$97\text{ million}\times 27.79\%)-\left(\$100\text{ million}\times 25.78\%\right)\\ =\$26.9563-\$25.78\\ =\$1.1763\end{array}$

Conclusion

Hence, place 27.79% of the portfolio in T-bills and 72.21% $\left(\text{100}-\text{27}\text{.79\%}\right)$in equity. The amount of T-bills is $26.96 million $\left(\text{\$97 million}\times \text{0}\text{.2779}\right)$ .The manager must sell $1.1763 million of equity and use the proceeds to buy bills.