Forward Contract on Investment Assets with Known Dollar Income (continued)

Formalize: Forward Contracts on Investment Assets with Known Dollar Income

Forward Price:

Assume I = PV₀(the income during life of forward contract)

            ⇒       F(0,T) = (S₀ – I )e^rT                          [continuous]

Proof: Consider two portfolios:

1. Long forward at forward price F(0,T):
   t = 0: no CF
   t = T: pay F(0,T) & receive the asset
2. Synthetic Forward: Buy the asset spot with borrowed money
   t = 0: CF₀ = + S₀ (loan) – S₀ (pay for stock) = 0
                     + Asset (assume no storage cost)
   t = T: CF_T = - S₀ · e^rT (←loan pmt)
                    + I · e^rT  (←FV at t=T of income)
3. Since the two portfolios should be equivalent, we get:
   - F(0,T) = - S₀ · e^rT + I · e^rT  ⇒   F(0,T) = (S₀ – I )e^rT

Valuation of an Existing Forward Position:

Present Value of a Long Forward at t

Note: The following refers to the equation below found in section i. Discrete Compunding
The term, D/(1+r) (T-t), is PV of the dividend.
The term, F(0,T)/(1+r) (T-t), is PV of F(0,T).

1. Discrete compounding:
   At t = t:   Short sell at S_t
   At t = T:  Close the existing long forward: Pay F(0,T) and receive one share
               Close the short sell: Return the share & pay div (–D)
   → V_t (0,T)   = S_t – PV{D + F(0,T)}
                      = S_t – D/(1+r) ^(T-t) – F(0,T)/(1+r) ^(T-t)
2. Continuous compounding:
        V_t (0,T)  = S_t - De^-r(T-t) – F(0,T)e ^–r(T-t)