International Finance: Putting Theory Into Practice

164 CHAPTER 4. UNDERSTANDING FORWARD EXCHANGE RATES FOR CURRENCY

Table 4.3:Term Structures and their Linkages

rate orpvfactor forn= ...

1 2 3 4 5 6 7

forward ratep.p.,rf 0 ,n− 1 ,n 0.0300— 0.0350 0.0380 0.0400 0.0360 0.0300 0.0200

1 +rf 0 ,n− 1 ,n 1.0300— 1.0350 1.0380 1.0400 1.0360 1.0300 1.0200

1 +r 0 ,n= Πnj=1(1 +rf 0 ,j− 1 ,j) 1.0300— 1.0661 1.1066 1.1508 1.1923 1.2280 1.2526

̄r 0 ,n= (1 +r 0 ,n)^1 /n− 1 0.0300— 0.0325 0.0343 0.0357 0.0358 0.0348 0.0327

PV 0 ,n= 1/(1 +r 0 ,n) 0.9709— 0.9380 0.9037 0.8689 0.8387 0.8143 0.7984

pvannuity,a 0 ,n=

Pn

j=1PV^0 ,j 0.9709 1.9089 2.8126 3.6816 4.5203 5.3346 6.1330

R 0 ,n:[1−(1+R^0 ,n)

−n)

R 0 ,n =a^0 ,n 0.0300 0.0316 0.0330 0.0340 0.0346 0.0347 0.0342

c 0 ,n= (1−PV 0 ,n)/a 0 ,n 0.0300 0.0325 0.0342 0.0356 0.0357 0.0348 0.0329

Key Starting from an assumed set of forward rates I compute the set of ‘spot’ zero-coupon rates
(lines 3 and 4) and present value factors (line 5). This allows us to find thepvof a constant unit
annuity (line 6) and the corresponding yield. Finally I compute the yield at par for a bullet bond.
The math is described in the text.

rates into effective spot rates, using Equation [4.47]:

1 +r 0 ,n= Πnj=1(1 +r 0 f,j− 1 ,j). (4.50)

The rate on the left-hand side is the effective rate we have always used in this book.
But for the theory of term structures it is useful to convert the effective rate to a
per-period rate, which we denote by ̄r. The computation is

1 + ̄r 0 ,n:= (1 +r 0 ,n)^1 /n. (4.51)

The spot rates are the yields to maturity on zero-coupon bonds expiring atn. Note
how the per-period gross rates are rolling geometric averages—numerically close
to simple averages—of all gross forward rates between times 0 andn.^11 See how
the strong hump is very much flattened out by the rolling-averaging, and the peak
pushed ton= 5 instead ofn= 4 for the forward rates. A second alternative way to
work with the effective rate is to compute thepvof one unit ofhcpayable at time
n,
PV 0 ,n= 1/(1 +r 0 ,n). (4.52)

TheTSof yields for constant-annuity cash flowsis a differentTS. It is not
as popular as theTSof yields at par for bullet loans, see below, but it is convenient
to look at this one first. Any yield orinternal rate of returnis the compound “flat”
rate that equates a discounted stream of known future cash flowsCjto an observed
present value:

y:

C 1

1 +y

+

C 2

(1 +y)^2

+...+

Cn

(1 +y)n

= observed PV. (4.53)

(^11) A gross rate is 1 +r,rbeing the net rate we always use in this text.