Question

A hedge fund with net asset value of $61 per share currently has a high water mark of $67. Suppose it is January 1, the standard deviation of the fund’s annual returns is 31%, and the risk-free rate is 3%. The fund has an incentive fee of 10%.

a. What is the value of the annual incentive fee according to the Black-Scholes formula? (Treat the risk-free rate as a continuously compounded value to maintain consistency with the Black-Scholes formula.) (Do not round intermediate calculations. Round your answer to 3 decimal places.)

  
b. What would the annual incentive fee be worth if the fund had no high water mark and it earned its incentive fee on its total return? (Do not round intermediate calculations. Round your answer to 3 decimal places.)

  
c. What would the annual incentive fee be worth if the fund had no high water mark and it earned its incentive fee on its return in excess of the risk-free rate? (Do not round intermediate calculations. Round your answer to 3 decimal places.)

  

d. Recalculate the incentive fee value for part (b) assuming that an increase in fund leverage increases volatility to 41%. (Do not round intermediate calculations. Round your answer to 3 decimal places.)

Answer 1

Incentive fees is paid once the NAV achieves the water mark NAV. If NAV remains below water mark NAV, then no incentive is paid.

This can be mathematically expressed as:

Incentive fees = 10% x NAV in excess of water mark NAV = 10% x max (S - K, 0) where S is NAV at the end of 1 year, K is the water mark NAV. What we see as the term max (S - K, 0) is nothing but a payoff from the call option. We can therefore value the incentive fees = 10% x Vaue of the call option with water mark as strike price.

Recall the Black Scholes formula for the call option:

Part (a)

Please see the snapshot below from the model. The last row highlighted in yellow is your answer. Figures in parenthesis, if any, mean negative values. All financials are in $. Adjacent cells in blue contain the formula in excel I have used to get the final output.

Part (b)

No high water mark, this means K = same as current NAV = 61

Part (c)

Return in excess of risk free rate, r of 3%

Hence, strike price, K = S0 x ert = 61 x e3% x 1 =  62.86

Part (d)

K = 61 and σ = 41.00%