Value of a 30 year bond using the Yield curve

Question

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 1

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 2

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.