Calculating theoretical spot rates of treasury bonds beginning with treasury bills

Question

In Introduction to Fixed Income Analytics by Frank Fabozzi, p. 41, there is an example how to calculate the theoretical spot rate of a 1.5 year treasury bond with a 3.5% annual interest and semiannual coupons.

\begin{array} {|c|c|c|c|c|} \hline \text{Period} & \text{Years} & \substack{\text{Annual Yield to}\\\text{Maturity (BEY) (%)}} & \text{Price} & \substack{\text{Spot Rate}\\\text{(BEY) (%)}}\\ \hline 1 & 0.5 & 3.00 &-& 3.0000 \\ 2 & 1.0 & 3.30 &-& 3.3000 \\ 3 & 1.5 & 3.50 &100.00& \text{?}\\ \hline \end{array}

The first two rows are the zero-coupon treasury bills.

The cash flow of the 1.5 year treasury bond is obviously:

0.5 year: 0.035 × \$100 × 0.5 = \$1.75
1.0 year: 0.035 × \$100 × 0.5 = \$1.75
1.5 year: 0.035 × \$100 × 0.5 + 100 = \$101.75

He now claims that the present value of the cash flows is:

$$\mathrm{PV}(z_1, z_2, z_3) = \frac{1.75}{(1+z_1)^1} + \frac{1.75}{(1+z_2)^2} + \frac{101.75}{(1+z_3)^3}$$ where

$z_1 =$ one-half the annualized 6-month theoretical spot rate
$z_2 =$ one-half the annualized 1-year theoretical spot rate
$z_3 =$ one-half the annualized 1.5-year theoretical spot rate.

If we solve $\mathrm{PV}(3.00, 3.30, z_3) = 100$ for $z_3$, we are supposed to get the theoretical spot rate for the 1.5 year treasury bond as described above.

But why can he do the whole example on semiannual intervals? The 1-year treasury bill does not pay any semiannual coupons, right? Where does that compounding come from?

Why isn't it: $$\mathrm{PV}(z_1, z_2, z_3) = \frac{1.75}{1+z_1} + \frac{1.75}{2\cdot z_2} + \frac{101.75}{(1+z_3)^3}\; ?$$

OK, BEY is given as effective semiannual rate. The convention is simply defined that way. So the question has been resolved.

Answer 1

The question really is what is the discount factor for a payment in one year assuming semiannual compounding? Because then your present value is simply 1.75 times this discount factor.

If you have $k$ periods of compounding, a payment of \$1 in $n$ years worth today \begin{align*} \frac{1}{\left(1+\frac{r}{k}\right)^{n\cdot k}}, \end{align*} where $r$ is the annualised spot rate. In your case, $k=2$ for semiannual compounding and $n$ is firstly $0.5$, then $1$ and finally $1.5$. This gives rise to the following three discount factors \begin{align*} \frac{1}{\left(1+\frac{r_{0.5}}{2}\right)^{\frac{1}{2}\cdot 2}} &=\frac{1}{1+\frac{r_{0.5}}{2}}, \\ \frac{1}{\left(1+\frac{r_{1}}{2}\right)^{1\cdot 2}} &=\frac{1}{\left(1+\frac{r_{1}}{2}\right)^{2}},\\ \frac{1}{\left(1+\frac{r_{1.5}}{2}\right)^{\frac{3}{2}\cdot 2}} &=\frac{1}{\left(1+\frac{r_{1.5}}{2}\right)^{3}}, \end{align*}

Regarding the logic behind it: Your bond has three payments and you need to find their value today. So, you decompose the coupon-paying bond into different zero-coupon bonds (ZCB) with a face value of \$1 and you see that \begin{align*} T-Bond & = 1.75 \cdot ZCB(0.5y) + 1.75\cdot ZCB(1y) + 101.75 \cdot ZCB(1.5y) \\ &= 1.75\cdot \frac{1}{\left(1+\frac{r_{0.5y}}{k}\right)^{\frac{1}{2}\cdot k}} + 1.75\cdot \frac{1}{\left(1+\frac{r_{1y}}{k}\right)^{1\cdot k}} + 101.75 \cdot \frac{1}{\left(1+\frac{r_{1.5y}}{k}\right)^{\frac{3}{2}\cdot k}} \end{align*} So, you use the 1y spot rate because you got a payment in 1y and need to discount it back to today. What compounding you use, does not matter, so one can write the above equation for instance in terms of $e^{-r T}$ and still get the same result.