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

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!

• Does Tuckman actually say "periods of unspecified length"? Because it looks to me like his equation is valid for periods where the length of the period is taken to be "one unit" by assumption and everything is done in terms of this (arbitrary) unit. – noob2 Dec 7 '20 at 18:15
• @noob2 Yes, that's what it says! So I'm quite confused - why would he have taken the length of the period to be 1? Not sure what the significance is given that there are semiannual coupons. Edit: can you explain further about this arbitrary unit? And how this would link with coupon payment frequency? – junior_pm Dec 7 '20 at 18:15

## 1 Answer

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.

• Hi, thank you for this. As for your last paragraph, in the example that I gave, it seems to suggest that there are no semiannual coupons in the first formula - is that true? I don't think I get 100% how the mapping works, or does the formula suggest that we discount each coupon payment (where coupons are paid at each unit of time t = 1, 2, 3...) using those forward rates f(1), f(2) etc? – junior_pm Dec 9 '20 at 16:57
• It might sound weird, but if you are very strict it does not even matter what "1" means as long as you are consistent. "1" could mean one year, or one semester (6m). For a cashflow at time "1" you need the corresponding time "1" zero rate to discount this cashflow (could be either 6m from now, or 1y from now - depending on how you define "1"). The same holds for "2", and so on. The forwards $f(1), f(2), ...$ etc. are just a different way to represent the yield curve that you use for discounting. – KevinT Dec 9 '20 at 18:18