CONNECT WITH LEARNSMART FOR BODIE: ESSE
CONNECT WITH LEARNSMART FOR BODIE: ESSE
11th Edition
ISBN: 2819440196222
Author: Bodie
Publisher: MCG
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Chapter 16, Problem 38C
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.

  Current value of portfolio(So)=$100 millionFloor promised to clients. 0% return(X)=$100 millionVolatility(σ)=0.25Risk-free rate(r)=0.05Horizon of program(T)=4 yearsDividend yield (δ)=0

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.

Expert Solution
Check Mark

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(d1)defines the normal distribution of portfolio returns. Weighted given
to T-bills and equity can be known for value of N(d1)

Formula for d1 as per Black-Scholes model is as follows:

d1=In( So X)+(rδ+ σ2 2)TσTSo=Currnt stock priceN(d1)and N(d2)=Cumulative normal distribution functionsX=Exercise pricerr= Annualised risk free rated= Annual dividend yield of underlying stocks=Annualised standard deviation of share returnsT = Time to expiry In( S oX)=Natural logarithmic value ofSoX e =Exponential having the constant of 2.71828δ=Annual dividend yield of underlying stockσ=Annualised standard deviation of share returns or Volatility

Substitute values to calculate d1as follows:

So=$100 million,X=$100 million,r=0.05,σ=0.25,δ=0,andT=0d1=In( So X)+(rδ+ σ2 2)TσT=In($100$100)+(0.050+ 0.25 22)4=In(1)+(0.050+0.03125)40.5=In(1)+0.3250.5=0+0.3250.5=0.65

Calculate the Black-Scholes hedge ratio N(d1)of call option using Ms-Excel"NORMSDIST" Function as follows:

CONNECT WITH LEARNSMART FOR BODIE: ESSE, Chapter 16, Problem 38C , additional homework tip  1

N(d1)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(d1) value is 0.7422

Calculate the put delte as follows:

  Putdelta=N(d1)1=0.74221=0.2578

Hence, place 25.785 of the portfolio in T-bills and 74.22% (10025.78%)inequity

Conclusion

Hence, place 25.785 of the portfolio in T-bills and 74.22% (10025.78%) inequity and the stock portfolio if falls by 3% will be $1.1763.

Summary Introduction

(B)

Adequate information:

  Current value of portfolio(So)=$97 millionFloor promised to clients. 0% return(X)=$100 millionVolatility(σ)=0.25Risk-free rate(r)=0.05Horizon of program(T)=4 yearsDividend yield (δ)=0

To calculate:

Fractions of portfolio divided into bills and equity If stock prices fall by 3% on the first day of trading. It means So (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.

Expert Solution
Check Mark

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(d1) defines the normal distribution of portfolio returns. Weighted given to T-bills and equity can be known for value of N(d1)

Substitute values to calculate d1as follows

So=$97 million,X=$100 million,r=0.05,σ=0.25,δ=0,andT=4d1=In( So X)+(rδ+ σ2 2)TσT=In($97$100)+(0.050+ 0.25 22)4=In(0.97)+(0.050+0.03125)40.5=In(0.97)+0.08125×40.5=0.0305+0.3250.5=0.29450.5=0.589

Calculate the Black-Scholes hedge ratio N(d1)of call option using Ms-Excel"NORMSDIST" Function as follows:

CONNECT WITH LEARNSMART FOR BODIE: ESSE, Chapter 16, Problem 38C , additional homework tip  2

Hence, N(d1)value is 0.7221.

Calculate the put delta as follows:

  Difference =(Amount of T-bills as per sub-part ( b)Amount of T-bills as per sub-part ( a))=$97 million×27.79%)($100 million×25.78%)=$26.9563$25.78=$1.1763

Conclusion

Hence, place 27.79% of the portfolio in T-bills and 72.21% (10027.79%)in equity. The amount of T-bills is $26.96 million ($97 million×0.2779) .The manager must sell $1.1763 million of equity and use the proceeds to buy bills.

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