How does this formula for the price of a bond in terms of forward rates work?

Question

I am currently reading Chapter 3 of Tuckman's 'Fixed Income Securities' and it states that we can write the price of a bond using its term structure in terms of forward rates but with periods of unspecified length as follows:

$P = \frac{c}{1+f(1)}$$+\frac{c}{(1+f(1))(1+f(2))}$+...+$\frac{1+c}{1+f(1))(1+f(2))...(1+f(T))}$, where $c$ is the coupon and $f(1),f(2)...$ equal forward rates.

But I am not sure how the 'periods of unspecified length' reconcile with the equation above - from what I know, where f(t) = the forward rate from year t-0.5 to year t and assuming semiannual compounding:

$P = \frac{c}{2}[\frac{1}{1+\frac{f(0.5)}{2}}+\frac{1}{(1+\frac{f(0.5)}{2})(1+\frac{f(1)}{2})}+...+\frac{1}{(1+\frac{f(0.5)}{2})(1+\frac{f(1)}{2})...(1+\frac{f(T)}{2})}]+\frac{1}{(1+\frac{f(0.5)}{2})(1+\frac{f(1)}{2})...({1+\frac{f(T)}{2}})}$.

Answers would be very much appreciated!

Answer 1

Assume today is $t$, and the 1st coupon pays at time $T_1$, the 2nd one at $T_2$, etc. Then your term structure of spot rates would be $R_1 = R(T_1) = f(t,T_1)$ for the 1st maturity, and $R_2 = R(T_2)$ for the 2nd maturity, and so on... Note that by no arbitrage $1+R_2 = (1+f(t,T_1)) (1+ f(T_1,T_2))$. Here I denote by $f(x, y)$ today's value of a forward rate which is valid from $x$ to $y$ (you could (or should) of course write this in a cleaner fashion as $f(t,x,y)$ to indicate it's the time $t$ observed forward, but let's keep the notation simple to illustrate your point of the interval length from $x$ to $y$). Thus, in brief, your term structure can be equivalently written in terms of forward rates.

In a concrete example: say coupons occur every 6 months. Then you discount your first coupon cashflow with the current 6m spot rate (which is $f(0m,6m)$) by dividing through $1+R_{6m}$. The second one you discount at the 12m spot rate, dividing through $1+R_{12m}$, which - by the logic described above - can be written as $(1+f(0m,6m))(1+f(6m,12m))$. And so on.

So this connects to @noob2's comment, which indicates that each of the individual forwards is more likely to be of the "same unit" length (namely 6m in my example). The link to coupon frequency is not a 1:1 mapping, i.e., you could also express the 12m rate in terms of the quarterly forwards 0x3, 3x6, 6x9, and 9x12, but still use this term structure to value coupons that occur semi-annually.

HTH.