Class Resources Cribsheet1

Columbia University: Financial Engineering and Risk Management M. Haugh & G. Iyengar

Crib sheet for Assignment #1 1. Present value calculation (a) Compute the present value of the cash flows (c0 , c1 , c2 ) = (100, 200, −100) where time is in years and interest is compounded annually with an interest rate r = 12% per annum. c2 c1 + (1 + r) (1 + r)2 200 100 = 100 + − 1.12 (1.12)2 = 198.852 ≈ 198.85 = c0 +

PV

(b) Compute the present value of the cash flow (c0 , c1 , c2 ) = (100, 200, −100) where time is in months and interest is compounded monthly with an interest rate r = 12% per annum. c2 c1 + (1 + r/12) (1 + r/12)2 100 200 − = 100 + 1.01 (1.01)2 = 199.9902 ≈ 199.99 = c0 +

PV

2. Relation between spot rates, discount rates, and forward rates Suppose s2 = 10% and s4 = 15%. Then the discount rate d(0, 2) for two years is given by 1 1 = = 0.8264 ≈ 0.83. d(0, 2) = 2 (1 + s2 ) (1.1)2 The forward rate f2,4 to borrow for two years starting in year 2 is given by the solution of the equation (1 + s2 )2 (1 + f2,4 )2 = (1 + s4 )4 (1.1)2 (1 + f2,4 )2 = (1.15)4 (1 + f2,4 ) = (1.15)4 = 1.2023 (1.1)2

f2,4 = 0.2023 = 20.23% 1

3. Forward contract on a stock The forward price F for a forward contract on a non-dividend paying underlying with expiration T is given by S0 F = d(0, T ) where S0 is the current price and d(0, T ) is the discount rate from now (i.e. time 0) until expiration. Discount rate can be computed using the formula given above. 4. Swap valuation An investor holding a long position on a fixed-floating rate swap contract pays floating rate payments on a given principal N and receives fixed interest rate payments on the same principal N in years t = 1, 2, . . . , T . The P Vf ixed of the fixed rate payments at annual rate X is given by

T

P Vfixed = N X (d(0, 1) + d(0, 2) + . . . + d(0, T )) = N X

t=1

d(0, t).

The present value of the floating rate payments can be computed by noticing that this cash flow is the same as that of a floating rate bond up...