Question

Please only help me with 1.5(a).

1.3. You enter into a forward contract to buy 100 shares of Volkswagen stock in 1 year for 120/share. The current price of one share is 126.25. You paid 971 for this contract. In 1 year, the price of the stock is 128. The risk free rate over the year was 5%, compounded continuously.

a) What is the payoff of your forward contract? b) What is the profit of your forward contract?

1.4. Use values from the previous problem for this problem. In addition, assume that Volk- swagen pays no dividends. You decide to short sell 100 shares of Volkswagen stock. Before the sale, you are required to deposit 30% in a margin account earning the risk free rate and return the shares in 1 year. What is the rate of return on your investment (as a continuously compounded interest rate)? In case you are wondering, we are taking the money earned from the initial short sale and putting it under our mattress.

1.5. You enter into both arrangements from the previous problems. In addition, you take the money you earned from the short sale and deposit it into some account earning the risk free rate. (This arrangement is similar to reverse cash-and-carry).

1. a) What is the profit of your arrangement?

2. b) What is the rate of return of your arrangement (as a continuously compounded interest rate)?

Only help me with 1.5(a) please.

Answer 1

Long Forward Contract Size n = 100 shares of Volkswagen
Contract to buy at exercise price k = $120 per share
Current Price of Stock S₀ = $126.5
Price of the contract = $971
Price of stock after 1 year S₁ = $128
Risk Free Rate = 5% per annum compounded continuously

1.3(a) :
Pay-off from the Long forward contract = (S_T - K)*n
Where,
S_T = stock price at time T
K = exercise price
n = size of the contract

so, Pay-off at the end of 1 year =  (S₁ - K)*n = (128 - 120)*100 = $800

1.3(b) :
Profit from the forward contract at (t = 0)= PV of the pay-off from the contract - premium paid for the contract
PV of the pay-off = Pay-off at the end of T-years/e^rT
where,
r = risk free interest rate
T = time duration in years

PV of the pay-off = 800/e^0.05*1 = 800/1.051271 = 760.98

Hence,
Profit(loss) from the forward contract = 760.98 - 971 = -210.02

Note - remember that this is the Profit(loss) at time t = 0.
Profit(loss) at t = T = Profit(loss) at (t=0)*e^rT = -210.02*e^0.05*1 = = -210.02*1.051271 = -220.78

Ans: Profit(Loss) from contract at (t=0) = -210.02; at (t=1) = -220.78

1.4:
Short Sell 100 stocks of the Volkswagen share
Margin money = 30% of proceeds from short sell

Proceeds from the short sell = Stock Price at (t=0)*No. of shares = 126.5*100 = 12650
Margin Money = Proceeds from the short sell*Margin Money in % = 12650*0.30 = 3795
Value of margin money after investment period = Initial Margin Money*e^rT
Where,
r = risk free interest rate
T = time duration in years

so, Value of margin money after investment period = 3795*e^0.05*1 = 3795*1.051271 = 3989.57

Amount kept under the mattress = Total proceeds of short sell = 12650
* Money kept under the mattress earns no interest

At the end of 1 year, 100 shares will be bought to return the shares borrowed
Cost of buying 100 shares at (t=1) = No. of shares* Share Price(at t = 1) = 100*128 = 12800

Now,
Total Cashflow at(t = 0) CF₀
1. Money paid in the margin account(Cash outflow) = -3795

Total Cashflow at (t=1) CF₁
1. Total Margin Money with interest(cash inflow) = +3989.57
2. Money kept under the mattress(cash inflow) = + 12650
3. Money paid for purchase of shares(cash outflow) = -12800

Hence CF₀ = 3795
CF¹ = 3989.57 + 12650 - 12800 = 3839.57

Lets assume rate of return = R, then
0 = CF₀  + CF₁/e^R
0 = -3795 + 3839.57/e^R
3795 = 3839.57/e^R
e^R = 3839.57/3795 = 1.011745
R = 0.011677
or R = 1.1677

Hence return from this strategy = profitof 1.1677% compounded continuously.

1.5:
Short Sell 100 stocks of the Volkswagen share
Margin money = 30% of proceeds from short sell

AND

Enter into the forward contract of ques 1.3

Proceeds from the short sell = Stock Price at (t=0)*No. of shares = 126.5*100 = 12650
Margin Money = Proceeds from the short sell*Margin Money in % = 12650*0.30 = 3795
Value of margin money after investment period = Initial Margin Money*e^rT
Where,
r = risk free interest rate
T = time duration in years

so, Value of margin money after investment period = 3795*e^0.05*1 = 3795*1.051271 = 3989.57

Value of Proceed of short after 1 year = Total proceeds of short sell*e^rT = 12650*e^0.05*1 = 13298.58
* Money kept under the mattress earns no interest

At the end of 1 year, 100 shares will be bought to return the shares borrowed
Cost of buying 100 shares at (t=1) = No. of shares* Share Price(at t = 1) = 100*128 = 12800

Now,
Total Cashflow at(t = 0) CF₀
1. Money paid in the margin account(Cash outflow) = -3795

Total Cashflow at (t=1) CF₁
1. Total Margin Money with interest(cash inflow) = +3989.57
2. Margin Money Invested will get matured(cash inflow) = + 13298.58
3. Money paid for purchase of shares(cash outflow) = -12800

Hence CF₀ = 3795
CF¹ = 3989.57 + 13298.58- 12800 = 4488.53

Part 1.5(a)
Profit(loss) at (t=0) = CF₀ + PV of CF₁
PV of CF₁ = CF₁/e^rT = 4488.53/e^0.05*1 = 4488.53/1.051271 = 4269.263
Profit(loss) at (t=0) = -3795 + 4269.263 = $474.263

Profit(loss) at (t=1) = Profit(loss) at (t=0) * e^rT = 474.263*e^0.05*1 = 474.263*1.051271 = 498.58


Total Profit(loss) = Profit(loss) from Forward + Profit(loss) from Short Sell
at t=0, Profit = -210.02 + 474.263 = 264.34
at t=1, Profit = -220.78 + 498.58 = 277.80

Answer: Profit at t=0 is $264.34 or at t=1 is $277.80

Note: Student has asked only for 1.5(a) so 1.5(b) not answered.