Formalize: Forward Contracts on Investment Assets with Known Yield
(Two Dividend Dates)

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How many shares of a stock do you need to buy at t = 0 such that you will end up having exactly one share of the stock at t = T, but without accumulating dividends? The stock will pay dividend twice at known dividend rates of q₁ at t = t1 < t2, q₂ at t = t2 <T.

The answer to this question is buy (1 - q₁) (1 - q₂) shares at t = 0.

Note:

1. The dollar amounts of dividend are: D_t1 = q₁ · S_t1 and D_t2 = q₂ · S_t2 at t = t1 and t2, respectively.
2. The cost of obtaining a share of the stock that pays S_T at t = T is: S₀ (1 - q₁) (1 - q₂)
3. Therefore,  F₀ = S₀ (1 - q₁) (1 - q₂)(1+r)^T or F₀ = S₀ (1 - q₁) (1 - q₂)e^rT

Proof:

1. At t = 0, buy  (1 - q₁) · (1 - q₂) units
2. At  t = t1, receive the dividend: (1- q₁)(1- q₂) q₁S_t1
   Invest dividends: buy additional shares at the ex-div price of S_t1 (1 - q₁)/share:
   ⇒ No of shares: (1 - q₁)(1 - q₂) q₁ S_t1 / {S_t1 (1 - q₁)} = (1 -  q₂) q₁ share
   ⇒ On ex-div date, t1, the total number of shares is
       (1 - q₁) (1 - q₂) + (1 - q₂) q₁ units
3. At t = t2, receive the dividend & buy additional shares:
   ⇒ {(1 - q₁)(1 - q₂) + (1 - q₂) q₁} q₂ S_t2 / {S_t2 (1 - q₂)} =  q₂ shares
   ⇒ Then the total number of shares at t2 is:
   (1 - q₁) (1 - q₂) + (1 - q₂) q₁ + q₂ = 1 share ...Q.E.D.