CALCULATING AND USING IMPLIED SPOT (ZERO-COUPON) RATES
The implied spot curve is arguably the second most important calculation in yield curve analysis after the forward curve. This curve will be the sequence of spot (or zero-coupon) rates that are consistent with the prices and yields on coupon bonds. Building the implied spot curve is a great example of “bootstrapping” in that the result of one calculation is used in the subsequent one. This is not going to involve a specific formula; instead it is a process best learned by working through an example.
Suppose that we observe price and yield data on four actively traded benchmark securities for the same risk class, for instance, government bonds (see Table 5.1).
We need some simplifying assumptions to illustrate bootstrapping. We require a starting place in the money market where we observe the pricing on a short-term zero-coupon bond. Here the 1-year represents the starter zero. But keep in mind that in practice we would use the price on a T-bill or commercial paper or a time deposit that pays LIBOR. The 1-year bond is priced at 97.0625 to yield 3.0264%.
TABLE 5.1 Example of Observed Prices and Yields
Maturity (Years) |
Coupon Rate |
Price (% of Par Value) |
Yield to Maturity |
1 |
0% |
97.0625 |
3.0264% |
2 |
3.25% |
100.8750 |
2.7941% |
3 |
4.50% |
102.7500 |
3.5181% |
4 |
4.00% |
99.3125 |
4.1902% |
The 2-year, 3-year, and 4-year bonds are assumed make annual coupon payments to simplify the example. Assume that we are on a coupon date and the payment has just been made so that there is no accrued interest. Based on these coupon rates and the prices, we observe an interesting U-shaped yield curve because the lowest yield is on the 2-year bond. The internal rates of return follow equation 3.4 from Chapter 3 and are street convention yields for a periodicity of 1 (i.e., they are effective annual rates).
The implied 2-year spot rate (the 0×2) turns out to be 2.7903%; it is the solution for z in this expression.
The first cash flow is discounted by the 0×1 spot rate of 3.0264%. That's why we need a starter zero taken from the money market. The algebra problem is to find the 2-year spot rate such that when the second cash flow is discounted by that rate, the sum is the price of the bond.
The implied 3-year spot rate (the 0 × 3) is 3.5476%.
Our 0×2 result is now an input into the equation and is used to discount the second cash flow. This is what is meant by bootstrapping the curve. Note that later calculations will be susceptible to any errors made earlier in the process.
We continue working our way out the yield curve to get the implied 0×4 spot rate. This uses the initial 0×1 starter zero and the 0×2 and 0×3 implied rates.
Clearly, this repetitive sequence of calculations is suited perfectly for a computer. You probably can see how you would build that spreadsheet, given the dates and amounts of the scheduled future cash flows on the underlying benchmark bonds. In general, you would need to use the total current market value including accrued interest on the left side of the equation.
These equations demonstrate why a yield-to-maturity statistic can be interpreted as a “present value average” of the zero-coupon rates, as we saw in Chapter 3. Consider the 4-year bond in this example. Its price is the sum of the discounted cash flows using either the yield to maturity or the sequence of spot rates.
Here 4.1902% is a “weighted average” of 3.0264%, 2.7903%, 3.5476%, and 4.2525%, with most of the weight on the last spot rate because it corresponds to the largest cash flow.
The implied spot curve can be used to derive a related description of the market, again assuming no arbitrage. This is the par curve – the sequence of yields such that the bond for each time to maturity trades at par value. For example, the 4-year par yield comes from the solution for PMT:
Each cash flow is discounted using the corresponding implied spot rate. So, while the actual 4%, 4-year bond is trading at a small discount to yield 4.1902%, we now deduce that a hypothetical 4.1876%, 4-year bond would be priced at par value.
Another clever application of the implied spot curve is in calculating the credit spread over benchmark bonds. Suppose we are analyzing a 4-year, level-payment, fully amortizing bank loan. The bank loan is priced at par value and makes an annual payment of 28.2 (percent of par value), including both principal and interest. Its yield to maturity is 4.9982%.
What is the credit spread, assuming the four bonds in this section are Treasuries? Using the 4-year par yield, it is 81.06 basis points: 4.9982% – 4.1876% = 0.8106%. Using the actual 4-year bond, the spread is 80.80 basis points: 4.9982% – 4.1902% = 0.8080%. The problem with these is that the “average life” of this amortizing bank loan is less than four years – the 4-year Treasury yield is not the right benchmark.
A better way to assess the compensation for the credit risk (as well as any difference in liquidity and taxation) is to calculate the static spread, also known as the zero-volatility spread or just Z-spread. It is the uniform (hence, static) spread over the benchmark implied spot rates. It is the solution for the static spread, denoted ss, in this expression:
Here you need trial-and-error search (or Excel and Solver) to find that ss = 0.014048. The static spread for this fully amortizing bank loan is 140.48 basis points.