EBK ESSENTIALS OF INVESTMENTS
EBK ESSENTIALS OF INVESTMENTS
10th Edition
ISBN: 8220102800267
Author: Bodie
Publisher: YUZU
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Chapter 20, Problem 13PS
Summary Introduction

(a)

To calculate:

If the high water mark is $66 and asset value is $62 , then ascertain the annual incentive fee as per the Black-Scholes formula.

Introduction:

The Black-scholes model is used for determining the price of European call option by using the variation of price in financial instruments. This model uses stock price, option price, and time for ascertaining the call option price.

Expert Solution
Check Mark

Answer to Problem 13PS

The annual incentive fee is $2.3366 .

Explanation of Solution

Given:

  Net asset valueS0=$62High water markX=$66Risk free rate=4%Standard Deviation=0.50Incentive fee=20%Time=1 year

The black-scholes formula is as follows:

  C=S0Nd1XerTNd2

  Here,S0=Current stock priceNd1andNd2=Cumulative normal distributionX=Exercise pricer=Annualised risk free rates=Annualised standard deviationT=Time to expiry

For calculating the d1 and d2 , the formula is as follows:

  d1=In S 0 X + rδ+ σ 2 2 TσTd2=d1σT

  Here,In S 0 X=Natural logarithmic value of S 0 Xe=Exponential having the constant of 2.71828δ=Annual dividend yieldσ=Annualised standard deviation

Now the calculation of d1 of call option:

  d1=In S 0 X + rδ+ σ 2 2 TσT=In $62 $66 + 0.040+ 0.50 2 2 10.501=In 0.94+ 0.04+0.12510.50=0.0619+0.1650.50=0.2062

The value of Nd1 will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of Nd1 is 0.5817

Now, the calculation of d2 of call option:

  d2=d1σT=0.20620.501=0.20620.50=0.2938

The value of Nd2 will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of Nd2 is 0.3845

By substituting the values, value of call option is:

  C=S0Nd1XerTNd2=$62×0.5817$66×1e rT×0.3845=$36.0654$66×1 2.71828 0.04×1×0.3845=$36.0654$66×0.9608×0.3845=$11.6832

Thus, the value of call option is $11.6832

The computation of value of incentive fee is as follows:

  Annual incentive fee=Percentage of incentive×Call value=0.20×$11.6832=$2.3366

Thus, the annual incentive fee is $2.3366 .

Summary Introduction

(b)

To calculate:

If the high water mark is zero and asset value is $62 , then ascertain the annual incentive fee on total return as per the Black-Scholes formula.

Introduction:

The Black-scholes model is used for determining the price of European call option by using the variation of price in financial instruments. This model uses stock price, option price, and time for ascertaining the call option price.

Expert Solution
Check Mark

Answer to Problem 13PS

The annual incentive fee is $2.6505 .

Explanation of Solution

Given:

  Net asset valueS0=$62Risk free rate=4%Standard Deviation=0.50Incentive fee=20%Time=1 year

The value of X is changed to $62

The black-scholes formula is as follows:

  C=S0Nd1XerTNd2

For calculating the d1 and d2 , the formula is as follows:

  d1=In S 0 X + rδ+ σ 2 2 TσTd2=d1σT

Now the calculation of d1 of call option:

  d1=In S 0 X + rδ+ σ 2 2 TσT=In $62 $62 + 0.040+ 0.50 2 2 10.501=In1+ 0.04+0.12510.50=0+0.1650.50=0.33

The value of Nd1 will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of Nd1 is 0.6293

Now, the calculation of d2 of call option:

  d2=d1σT=0.330.501=0.330.50=0.17

The value of Nd2 will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of Nd2 is 0.4325

By substituting the values, value of call option is:

  C=S0Nd1XerTNd2=$62×0.6293$62×1e rT×0.4325=$39.0166$62×1 2.71828 0.04×1×0.4325=$39.0166$62×0.9608×0.4325=$13.2527

Thus, the value of call option is $13.2527

The computation of value of incentive fee is as follows:

  Annual incentive fee=Percentage of incentive×Call value=0.20×$13.2527=$2.6505

Thus, the annual incentive fee is $2.6505 .

Summary Introduction

(c)

To calculate:

If the high water mark is zero and asset value is $62 , then ascertain the annual incentive fee on return in excess of risk-free rate as per the Black-Scholes formula.

Introduction:

The Black-scholes model is used for determining the price of European call option by using the variation of price in financial instruments. This model uses stock price, option price, and time for ascertaining the call option price.

Expert Solution
Check Mark

Answer to Problem 13PS

The annual incentive fee is $2.4179 .

Explanation of Solution

Given:

  Net asset valueS0=$62Risk free rate=4%Standard Deviation=0.50Incentive fee=20%Time=1 year

The black-scholes formula is as follows:

  C=S0Nd1XerTNd2

For calculating the d1 and d2 , the formula is as follows:

  d1=In S 0 X + rδ+ σ 2 2 TσTd2=d1σT

The value of X has been changed which is:

  X=S0×e0.04=$62×e0.04=$62×2.718280.04=$64.5303

Now the calculation of d1 of call option:

  d1=In S 0 X + rδ+ σ 2 2 TσT=In $64.5303 $62 + 0.040+ 0.50 2 2 10.501=In 1.0408+ 0.04+0.12510.50=0.03999+0.1650.50=0.40998

The value of Nd1 will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of Nd1 is 0.6591

Now, the calculation of d2 of call option:

  d2=d1σT=0.409980.501=0.409980.50=0.09002

The value of Nd2 will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of Nd2 is 0.4641

By substituting the values, value of call option is:

  C=S0Nd1XerTNd2=$62×0.6591$64.5303×1e rT×0.4641=$40.8642$64.5303×1 2.71828 0.04×1×0.4641=$40.8642$64.5303×0.9608×0.4641=$12.0897

Thus, the value of call option is $12.0897

The computation of value of incentive fee is as follows:

  Annual incentive fee=Percentage of incentive×Call value=0.20×$12.0897=$2.4179

Thus, the annual incentive fee is $2.4179 .

Summary Introduction

(d)

To calculate:

If the high water mark is zero and asset value is $62 , then ascertain the annual incentive fee on total return as per the Black-Scholes formula with an increase in the risk to 0.60 .

Introduction:

The Black-scholes model is used for determining the price of European call option by using the variation of price in financial instruments. This model uses stock price, option price, and time for ascertaining the call option price.

Expert Solution
Check Mark

Answer to Problem 13PS

The annual incentive fee is $3.1159 .

Explanation of Solution

Given:

  Net asset valueS0=$62Risk free rate=4%Standard Deviation=0.60Incentive fee=20%Time=1 year

The value of X is changed to $62

The black-scholes formula is as follows:

  C=S0Nd1XerTNd2

For calculating the d1 and d2 , the formula is as follows:

  d1=In S 0 X + rδ+ σ 2 2 TσTd2=d1σT

Now the calculation of d1 of call option:

  d1=In S 0 X + rδ+ σ 2 2 TσT=In $62 $62 + 0.040+ 0.60 2 2 10.601=In1+ 0.04+0.1810.60=0+0.220.60=0.3667

The value of Nd1 will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of Nd1 is 0.6431

Now, the calculation of d2 of call option:

  d2=d1σT=0.36670.601=0.36670.60=0.2333

The value of Nd2 will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of Nd2 is 0.4078

By substituting the values, value of call option is:

  C=S0Nd1XerTNd2=$62×0.6431$62×1e rT×0.4078=$39.8722$62×1 2.71828 0.04×1×0.4078=$39.8722$62×0.9608×0.4078=$15.5797

Thus, the value of call option is $15.5797

The computation of value of incentive fee is as follows:

  Annual incentive fee=Percentage of incentive×Call value=0.20×$15.5797=$3.1159

Thus, the annual incentive fee is $3.1159 .

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