quantitative finance

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01 February 2018
Problem-solving
Problem (a)
The expected value of the unhedged firm is:
0.5*200/(1+0.05)+0.5*(500-(500-300)*0.4)/(1+0.05)=95.24+200=295.24
Problem (b)
Let’s buy put options for gold pricing at 500 with premium P.
The expected value of the hedged firm is:
(500-(500-300)*0.4)/(1+0.05)-P=420-P
Problem (c)
The safe debt is the one that pays off not more than 200 in a year. The value of the firm is:0.5*(200-200)/(1+0.05)+0.5*(500-200-(500-200-300)*0.4)/(1+0.05)=142.86
Problem (d)
The safe debt is the one that pays off not more than 200 in a year. Let’s buy put options for gold pricing at 500 with premium P. The value of the firm is:
(500-200-(500-200-300)*0.4)/(1+0.05)-P=285.71-P
Problem (e)
The firm’s value is:
0.5*(200-250-20)/1.05+0.5*(500-250)/1.05=119.05-33.33=85.71
The return of the investment may be either: -(33.33/250)=-13.3%, or 119.05/250=47.6%
The yield to maturity is:
ytm=((0.05+0.5*(-0.133-0.05))+ (0.05+0.5*(0.476-0.05))/2=(-0.0415+0.263)/2=11.1%
Problem (g)
We have 2 scenarios for each case:value 200 and no gamble – expected result is 200-250-20=-70
Value 200 and gamble: unlucky gamble=200-250-20-20=-90, lucky gamble:200-250+1000-(1000+200-250-300)*0.4=690
Expected result is: 0.005*690-0.995*90=-86.1
Equity holders will choose not to gamble
Value 500 and no gamble – expected result is 500-250=250
Value 500 and gamble: unlucky gamble=500-250-20=230, lucky gamble:500-250+1000-(1000+500-250-300)*0.

4=870
Expected result is:0.005*870+0.995*230=233.2
Expected YTM in case a is -20%, in case b – 100%.
Problem (h)
In case of hedging there are 2 scenarios:
No gambling, expected return is: 500-P
Gambling: unlucky gamble=500-250-20-P=230-P, lucky gamble:500-250+1000-P-(1000+500-250-300-P)*0.4=870-1.4P
Expected return is: (230-P)*0.995+(870-1.4P)*0.005=233.2+1.002PThe equity holders will probably not choose to gamble