Question

Consider a stock that will pay a dividend of D dollars in one year, which is when a futures contract matures.

Consider the following strategy: Buy the stock, short a futures contract on the stock, and borrow ${\mathit{S}}_{\mathbf{0}}$dollars, where ${\mathit{S}}_{\mathbf{0}}$is the current price of the stock.

a. What are the cash flows now and in one year? (Hint: Remember the dividend the stock will pay.)

b. Show that the equilibrium futures price must be${\mathit{F}}_{\mathbf{0}}\mathbf{=}{\mathit{S}}_{\mathbf{0}}\left(1+r\right)$to avoid arbitrage.

c. Call the dividend yield d = D / ${\mathit{S}}_{\mathbf{0}}$, and conclude that ${\mathit{F}}_{\mathbf{0}}\mathbf{=}{\mathit{S}}_{\mathbf{0}}\left(1+r-d\right)$.

Short answer

Answer

a. 0 and (1 + r)$F_0+D-S_0\left(1+r\right)$.

b. As below

c. As below

Step-by-step solution

Calculation of cash flow now and in one year

Action	Initial cash flow	Cash flow at time T
Buy stock	$-S_0$	$S_T+D$
Short future	0	$F_0-S_T$
Borrow	$S_0$	$-S_0\left(1+r\right)$
Total	0	$F_0+D-S_0\left(1+r\right)$

Validation on equilibrium price

Since the initial investment is zero, but the final cash flow is not zero. So to avoid arbitrage opportunity, the equilibrium future price will be final cash flow equated to zero.

Hence $F_0=S_0\left(1+r\right)-D$

Validation of F0=(1+r-d)

Since D = (d x $S_0$)

$F_0=S_0$(1 + r) – D

After substitution

$F_0=S_0$(1 + r) – d x $S_0$

$F_0=S_0$(1 + r-d)