International Finance: Putting Theory Into Practice

4.7. APPENDIX: THE FORWARD FORWARD AND THE FORWARD RATE AGREEMENT 165

Figure 4.7:Term Structures: Forward, Spot, and Two Types of Yields

0.02

0.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

1 2 3 4 5 6 7

forward

spot, p.a.

IRR annuity

yield at par

Here we look at the special caseCj = 1,∀j, the constant unit-cash-flow stream,
the right-hand side of the above equation. Let us first find thepvof the constant
stream. Since we already know thePVof a single unit payment made atn, thePV
of a stream paid out at times 1, ... ,nis simply the sum. This specialPVis denoted
asa 0 ,n, from “annuity”. We compute its value for variousnas

a 0 ,n=

∑n

j=1

PV 0 ,j. (4.54)

Next we find the yield that equates thisPVto the discounted cash flows. When the
cash flows all equal unity, the left-hand side of Equation [4.53] is equal to (1−(1 +
y)−n)/y, but theythat solves the constraint must still be found numerically, using
e.g. a spreadsheet tool. In the table the result is found under the labelR 0 ,n. Note
how this yield is an analytically non-traceable mixture of all spot rates. The hump
is flattened out even more, and its peak pushed back one more period.

TheTSof yields at par for bullet loansis defined as a yield that sets thePV
of a bullet loan equal to par. But it is known that to get a unit value the yield must
be set equal to the coupon rate. So we can now rephrase the question as follows:
how do we set the coupon ratecsuch that thePV’s of the coupons and the principal
sum to unity?

c 0 ,n: c 0 ,n×a 0 ,n

︸ ︷︷ ︸

PV of coupons

+ PV 0 ,n× 1

︸ ︷︷ ︸

PV of amortization

= 1⇒c 0 ,n=

1 −PV 0 ,n

a 0 ,n

. (4.55)

Again, this is numerically much closer to the spot rates than the yield on constant-
annuity loans, and the reason obviously is that the bullet loan is closer to a zero-