Question

3. A security is currently trading at $96. The six-month forward price of this security is $100. It will pay a coupon of $6 in three months. The relevant interest rate is 10% p.a. (continuously compounding). No other payouts are expected in the next six months. Show the exact strategy you will use to make an arbitrage profit. State the profit and show all cash flows arising from the strategy.

Answer 1

Spot price of the security (S₀) = $ 96

Actual Forward price (F) = $ 100

Interest rate (r) = 10%

Tenure of forward (T) = 6 months = 0.5 years

Coupon payment time (t) = 3 months = 0.25 years

Time remaining after coupon (e)= 3 months = 0.25 yrs

Security will receive coupon of $ 6 in 3 months

Present value (PV) of coupon = Coupon / e^rt

Where, r = 10% = 0.1

t = 0.25

PV of coupon = 6 / e^0.1×0.25

^{​​​​​​}PV of coupon = 6/1.0253

PV of coupon = $ 5.8519

Under no arbitrage condition future price will be as follows,

Forward price = ( S₀ - PV of coupon ) × e^rT

Forward Price = ( 96 - 5.8519) × e^0.1×0.5

Forward Price = 90.1481 × 1.0513

Forward price = 94.7727

Therefore, theoretically forward price should be $ 94.77

Since actual forward price is higher that theoretical forward price our strategy should be,

Borrow amount at 10% and buy Spot at $ 96

And sell forward at $ 100

1st leg - spot

Initial cash outflow. = $ 96

Value after 6 months

= 96 × e^rT

= 96 × e^{​​​​​​0.1×0.5}

= 96 × 1.0513

= $100.9248

Outflow with interest after 6 months = $ 100.92

If actual spot at the end of 6 months turns out to be $ 102

Pay off wil be as follows

Loss in forward = 102 - 100 = $ 2

Profit on spot = 102 - 100.92 = $ 1.08

Future value of coupon

= 6 × e^re

= 6 × e^0.1×0.25  

= 6 × 1.0253

=$ 6.1518

Therefore payoff at the end of 6 months

= -2 + 1.08 + 6.15

= 5.23

Payoff at the end of 6 months = $ 5.23