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Another way to calculate implied spot and forward rates is with discount factors. In fact, this is how yield curve analysis is carried out in practice using spreadsheets. A discount factor is by definition the present value of one unit of currency at some future date. A financial institution that has a multitude of loans, bonds, and derivative contracts to value needs discount factors that correspond to each future date for which cash is received or paid out. Here I keep it simple and just use the four annual payment securities in Table 5.1 to illustrate the discount factor bootstrapping process.

The 1-year bond has a coupon rate of zero and is priced at 97.0625 per 100 of par value. This one is easy: The price of zero-coupon bond is its discount factor. So, the 1-year discount factor, denoted DF1, is simply

0.970625. The 2-year bond in Table 5.1 has a coupon rate of 3.25% and is priced at 100.8750. The 2-year discount factor is the solution for DF2 in this equation.

The bootstrapping process proceeds as in the section above where the implied spot rates are obtained. The difference is that now the algebra is much easier.

The 3-year and 4-year bonds have coupon rates of 4.50% and 4.00% and prices of 102.7500 and 99.3125, respectively. Working your way out the yield curve sequentially gets the next two annual discount factors.

The output from the previous step becomes an input in the next step.

Once you have the discount factors, valuing fixed-income bonds is straightforward. Remember the 9%, 4-year bond in the previous section? Its price turned out to be 117.6342 when each payment is discounted using the implied spot rates. Now we just need to multiply the scheduled payments by the discount factors.

Note the minor difference in the fourth decimal. This answer, 117.6341, is actually the more accurate because the spot rates entail more rounding. If both bootstrapping procedures are put on a spreadsheet, the results are identical.

Another application of the discount factors is to get the 4-year par yield, which is the coupon rate on a bond that has a (flat) price equal to par value. It's the solution for PMT in this equation.

The key point is that spot rates and discount factors contain the same information. Discounting with spot rates is more intuitive (and that's why I lead with it in the chapter) but using discount factors is more easily implemented on a spreadsheet.

If you start with the discount factors, you can always solve for the corresponding spot rates. Here are the calculations for the 2-year, 3-year, and 4-year zero-coupon rates, z1, z2, and z3:

These are the same as above.

In addition, the implied forward rates are ratios of the discount factors.

Here are the 1 × 2, 2 × 3, and 3×4 IFRs:

Once again, the minor differences between these and those calculated above are due to rounding in the spot rates. These results are slightly more accurate.

To see how the spot rates, forward rates and discount factors are interrelated, think about how the 3 × 4 is calculated with the 3-year and 4-year spot rates.

This is equivalent to this equation:

The numerator is DF3, the present value of 1 discounted back to date 0 using the 3-year spot rate; the denominator is DF4. The ratio between the discount factors minus one is the 3×4 implied forward rate.

The calculations in this section are simplified because the underlying bonds make annual payments and the annual rates have a periodicity of 1. To generalize using the notation of equations 5.3 and 5.4, the discount factor for year A given the annual spot rate Rate0×A and its periodicity, PER, is:


The spot rate given the discount factor is:


The implied forward rate between year A and year B given the discount factors and the periodicity is:


Suppose that 4-year and 5-year zero-coupon bonds are priced at 89.75 and 86.25 (percent of par value), respectively. What is the 4×5 implied forward rate quoted on a semiannual bond basis? Using equation 5.11 it is 4.0176% (s.a.).

Another method is to solve for the 4-year and 5-year spot rates using Equation 5.10.

Equation 5.4 can be used to get the 4×5 implied forward using the spot rates.

Note that BYears – AYears = 5-4=1. Once again there is a small difference in the fourth decimal and the result using discount factors is slightly more accurate. The main difference, however, is how much easier it is to work with discount factors, especially on a spreadsheet.

Discount rates truly are the keys to the kingdom of yield curve analysis and fixed-income valuation.


Many textbooks focus on the least important (and least interesting) aspects of yield curve analysis – the classic theories of the term structure of interest rates. Still, the expectations, segmented markets, and liquidity preference theories do serve to direct attention to the drivers of bond yields. Fortunately, there are many applications of implied spot and forward curves that are theory-free. To the extent that you believe in no-arbitrage pricing, you can move seamlessly between the observed yield curve on coupon bonds and the implied spot and forward curves.

Implied spot and forward rates are incredibly important to financial market participants. Who would not be interested in techniques that help you make maturity-choice decisions, identify arbitrage opportunities, and value debt securities and interest rate derivatives? The bond math calculations covered in this chapter are essential for fixed-income professionals (and even for amateurs).

Now we are ready to take on the other side of the trade-off between risk and return. So far we have focused mostly on measures of return, such as money market rates, bond yields to maturity, horizon yields, after-tax rates, and in this chapter implied spot and forward rates. Next we take a mathematical plunge into risk analysis – the widely heralded bond duration statistic and its not-so-famous companion, convexity.

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