Question

Suppose you are a fund manager. The daily risk-free rate is 0.02/252 as continuously compounding rate. The daily expected net returns of your fund is 0.08/252. The volatility of log returns of your fund is 0.25/sqrt(252) daily. Your fund is currently worth $200mil. Your bonus depends on your 1-year (252 days) performance. If the value of your fund turns out less than $200mil a year later, you have no bonus. If the value of your fund turns out between $200mil and $210mil a year later, you will receive $2mil bonus. If the value of your fund turns out between $210mil and $220mil a year later, you will receive $3mil bonus. If the value of your fund turns out larger than $220mil a year later, you will receive $5mil bonus. What is the present value of this bonus scheme?

1) $1mil

2) $1.5mil

3) $2mil

4) $2.5mil

5) $3mil

Answer 1

So in order to calculate the Expected value of bonus we need to calculate the probabilities of the fund value

Probability of fund value < 200 million. Z= (X- µ)σ Where X is the random value, µ is the expected value and σ volatility of returns. σ = 200*0.25 = 50 million. µ = 200*1.08 = 216 million.

Z = -0.32. Probability (Z = -0.32) = 0.3744

Similarly if we calculate the probability (200 million < Fund value < 210 million) = 0.0778

Probability (210 million < Fund value < 220 million) = 0.0796

Probability (Fund value > 220 million) = 0.4681

Probability of generating return less than 200 million			0.374484
Probability of generating more less than 220 million			0.468119
Probability of generating between 200 and 210 million			0.077757
Probability of generating between 210 and 220 million			0.07964

Therefore, we can calculate the expected value of bonus = p=1np*Bonus

Range	Probability	Bonus
< 200	0.37448417	0
200 - 210	0.46811863	2
210 - 220	0.07775741	3
> 220	0.0796398	5
Expected value of bonus	1.56770847

Therefore, using the formula the expected value of bonus = 1.5677 million

We know that the risk free rate = 2%. Using the risk free rate as the discount factor, we can calculate the NPV = E(Bonus)/(1+r) where r is 0.02 and E(bonus) = 1.5677.

Therefore NPV of bonus = 1.5677/1.02 = 1.5369 or approximately 1.54 million. The nearest option available is b) 1.5 million