4.4. THE MARKET VALUE OF AN OUTSTANDING FORWARD CONTRACT 145
4.4.4 Corollary 3: The Forward Rate and the Risk-Adjusted Ex-
pected Future Spot Rate
The zero-value property of forward contracts discussed above has another, and quite
fundamental, interpretation. Suppose that the clp/nokfour-year forward rate
equals 110, implying that you can exchange one future nokfor 110 futureclp
andvice versawithout any up-front cash flow. This must mean that the market
perceives these amounts as beingequivalent(that is, having the same value). If this
were not so, there would have been an up-front compensation to make up for the
difference in value.
Since any forward contract has a zero value, the present values ofclp1 four
years andnok110 four years must be equal anywhere; that is, the equivalence of
these amounts holds for any investor or hedger anywhere. However, the equivalence
property takes on a special meaning if we pick theclp (which is the currency
in which our forward rate is expressed) as the home currency: in that particular
num ́eraire, theclpamount is risk-free, or certain. In terms ofclp, we can write
the equal-value property as:
PVt(S ̃T) = PVt(Ft,T), (4.20)where PVt(.) is the present-value operator. In a way, equation [4.20] is just the zero-
value property: the present value of the uncertain future cash inflowS ̃T generated
by the contract cancels out against thePVof the known future outflow,Ft,T. We
can lose or gain, but these prospects balance out in present-value terms, from our
time-tviewpoint. But the related, second interpretation stems from the fact that
in home currency, the forward price on the right-hand side of Equation [4.20] is a
risk-free, known number whereas the future spot rate on the left is uncertain. That
is, at timetan amount ofFt,T Pesos payable atT is not justequivalentto one
unit of foreign currency payable atT; this amount of future home currency is also
acertain, risk-free amount. For this reason, we shall say that in home currency, the
forward rate is the time-tcertainty equivalentof the future spot rate,S ̃T.
Example 4.15
In our earlierclp/nokexamples, the certainty equivalent of one Norvegian Crown
four years out isclp110. You can offer the market a sureclp110 atTand get one
Crown (with risky valueS ̃T) in return; but equally well you can offer the market
one Crown (with risky valueS ̃T) and get a sureclp110 in return.
The notion of the certainty equivalent deserves some elaboration. Many in-
troductory finance books discuss the concept of an investor’s subjective certainty
equivalent of a risky income. This is defined as the single known amount of income
that is equally attractive as the entire risky distribution.
Example 4.16
Suppose that you are indifferent between, on the one hand, a lottery ticket that pays