Discounting Bond Cash Flows with Spot and Forward Rates
Summary
The document asks how to value a long-term coupon bond from a yield curve. The accepted answer explains that each coupon and the principal repayment should be discounted using the spot rate for its own payment date. It also notes that rates quoted in annual terms require a convention for scaling by the time to payment, such as compounding or a simple accrual adjustment. The bond’s yield to maturity is distinct: it is a single rate that, under the stated discounting convention, equates the discounted cash flows with the bond’s value.
A second response describes a related forward-rate representation. Forward rates for successive periods can be used to discount cash flows in a way consistent with the spot-rate valuation when the rates are constructed to avoid arbitrage. The document therefore distinguishes a spot curve, a single bond yield, and a sequence of forward rates. It gives conceptual guidance rather than a full pricing specification: compounding, payment timing, day-count conventions, and the precise curve inputs still need to match the bond and market being valued.
Key ideas
- Discount each bond cash flow using the spot rate corresponding to its payment date.
- A bond yield is a single rate that discounts all cash flows to the bond’s value.
- Annual rate quotes must be adjusted for the time until each payment under a chosen convention.
- A consistent sequence of forward rates can produce the same value as spot-rate discounting.
Tags
Full text
# Value of a 30 year bond using the Yield curve
# Value of a 30 year bond using the Yield curve
If I buy a $1 30 year bond with 4% coupon payment, would my cash flow be:
$$ V^{30}(t) = \frac{$1 \times0.04}{1 + R(t, 1)} + \frac{$1 \times0.04}{1 + R(t, 2)} + \cdots + \frac{$1 + $1 \times0.04}{1 + R(t, 30)} $$
or would it be:
$$ V^{30}(t) = \frac{$1 \times0.04}{1 + R(t, 30)} + \frac{$1 \times0.04}{1 + R(t, 30)} + \cdots + \frac{$1 + $1 \times0.04}{1 + R(t, 30)} $$
where $R(t, \theta)$ is the spot rate at $t$ over $\theta$. Sometimes people write $\theta = T-t$ where $T$ is the maturity date.
## Answer by David Duarte (score 4, accepted)
https://quant.stackexchange.com/a/54636
The value of the bond would be the first case, because you have to discount each cashflow with the relevant spot rate for that payment date.
Although, because rates are normally expressed in annual terms, you would have to adjust for the days: $(1+R)^{n}$ or $(1 + R \times n)$
What you might be confused with, is the yield of the bond, which would be the single rate that if used to discount all the cashflows would give you the bond value.
$$ V^{30}(t) = \frac{$1 \times0.04}{1 + Yld} + \frac{$1 \times0.04}{(1 + Yld)^2} + \cdots + \frac{$1 + $1 \times0.04}{(1 + Yld)^{30}} $$
## Answer by demully (score 1)
https://quant.stackexchange.com/a/54637
Almost both ;-)
If R is the spot 30 year yield, then: $NPV = \frac{coupon}{(1+R)^{t}}$, summed from $t=0$ to $t=30$.
This is almost the same as your second specification, albeit you do need to discount the coupon in year 20 by more than that in year 2. And it's the one they'll teach you as the standard model on any course.
But there is also an alternative model that is very like your first specification, that will come up with the same answer. The $R$ here here isn't the spot rate for the next $T$ years but the Ty3m forward rate, ie the 3m rate in $T$ years time. Of course, this is calculated precisely to avoid arbitrage; and so must, by definition, come up with the same answer as the normative model above. But by defining things thus and specifying these forward rates (which can be swapped), it becomes possible for the rates market to start to express very granular views about future interest rates in ways that are much more difficult with traditional bonds spanning longer periods.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.