160 CHAPTER 4. UNDERSTANDING FORWARD EXCHANGE RATES FOR CURRENCY
DoItYourself problem 4.6
Consider a FFunder which you will depositjpy 1b in nine months and receive
1.005b in twelve. The effective risk-free rates for these maturities arert,T 1 =0.6%
andrt,T 2 =0.81, respectively. Value each of thepn’s that replicate the two legs of
theFF. Compute the net value.
The generalisation is obvious. Below, we take a notional deposit amount of 1 (at
T 1 ):
PV, att, of a unitFF =promised in-
flow atT 2
1 +rt,T 2
︸ ︷︷ ︸
“asset”pn−
promised out-
flow atT 1
1 +rt,T 1
︸ ︷︷ ︸
“liability”pn=1 +rft 0 ,T 1 ,T 2
1 +rt,T 2−
1
1 +rt,T 1. (4.45)
In one special case we can consider the expiry moment (t=T 1 ):
DoItYourself problem 4.7
Derive, from this general formula, our earlier cash-settlement equation,
PV, atT 1 , of a unitFF =rft 0 ,T 1 ,T 2 −rT 1 ,T 2
1 +rT 1 ,T 2. (4.46)
The other special case worth considering is the value at initiation (t 0 =t). We
know that this value must be zero, like for any standard forward contract, so this
provides a way to relate the forward rate to the two spot rates, all att:
DoItYourself problem 4.8
Derive, from the general formula, the relation between the time-tspot and forward
rates:
(1 +rt,T 1 )(1 +rt,Tf 1 ,T 2 ) = 1 +rt,T 2 ; (4.47)⇔rft,T 1 ,T 2 =1 +rt,T 2
1 +rt,T 1− 1. (4.48)
The left-hand side of the first equality, Equation [4.47], has an obvious interpre-
tation: it shows the gross return from a synthetic deposit started right now (t) and
expiring atT 2 , made not directly, but replicated by making at-to-T 1 spot deposit
which is rolled over (i.e. re-invested, here including the interest earned) via aT 1 -to-
T 2 forward deposit. So the money is contractually committed for the totalt-to-T 2
period, and the total return is fixed right now—two ingredients that also character-
ize at-to-T 2 deposit. In that light, Equation [4.47] just says that the direct and the
synthetict-to-T 2 deposits should have the same return.