International Finance: Putting Theory Into Practice

7.6. TEKNOTES 283

7.6 TekNotes

Technical Note 7.1 The value of a bond as a function of its yield to
maturity
The valuation formula is derived as follows. Let the face value be 1, the couponcper period,
and the first coupon due exactly 1 period from now. (The yield is denoted byR, notr, because
a compound per-period yield on a coupon bond should not be confused with an effective simple
return on a zero-coupon bond.) We start with (almost) a definition; use the annuity formula; add
and subtract 1; divide and multiply byR; and re-arrange:

PV =

Xn

t=1

c

(1 +R)t+

1

(1 +R)n

= c^1 −(1 +R)

−n

R + (1 +R)

−n

= c^1 −(1 +R)

−n

R + (1 +R)

−n−1 + 1

= c^1 −(1 +R)

−n

R +R

(1 +R)−n− 1

R + 1

= (c−R)^1 −(1 +R)

−n

R + 1 = (c−R)a(n,R) + 1. (7.12)

If the time to the next coupon is 1−frather than unity, thepvrises by a factor (1 +R)f.