29 Yields
Interest rates can vary according to the interval of time, or term, to which they apply. The term structure of interest rates can be described by the following:
- yield curve
- spot rate curve
- forward rates
29.1 Yield Rate
The present value of a stream of cash flows is the sum of the discounted values of those cash flows, where the cash flows are discounted at the yield rate \(i\) corresponding to the pattern of cash flows: \[PV = \sum_{t > 0} \frac{CF_t}{(1+i)^t}\]
Example: The table below provides the times until maturity, prices, and coupon rates of 10 bonds. Each of the bonds makes annual coupon payments and has a face amount of \(\$100\). Calculate the annual effective yield of each bond.
| Maturity | Bond Price | Annual Coupon |
|---|---|---|
| 1 | 105.8824 | 8.000% |
| 2 | 104.8186 | 5.000% |
| 3 | 97.1714 | 2.000% |
| 4 | 85.4804 | 0.000% |
| 5 | 106.5850 | 6.000% |
| 6 | 110.1514 | 7.000% |
| 7 | 77.6705 | 2.000% |
| 8 | 104.6116 | 7.000% |
| 9 | 98.3537 | 6.500% |
| 10 | 114.0472 | 9.000% |
Solution: We can find the yield rate by using the BA II Calculator. For example, the first bond we can find the yield from the following keystrokes:
1 [N] 105.8824 [+/-] [PV] 8 [PMT] 100 [FV] [CPT] [I/Y]
-> [I/Y] = 2.000
Repeat for the rest of the bonds, we have the yield rates:
| Maturity | Bond Price | Annual Coupon | Yield Rate |
|---|---|---|---|
| 1 | 105.8824 | 8.000% | 2.000% |
| 2 | 104.8186 | 5.000% | 2.500% |
| 3 | 97.1714 | 2.000% | 3.000% |
| 4 | 85.4804 | 0.000% | 4.000% |
| 5 | 106.5850 | 6.000% | 4.500% |
| 6 | 110.1514 | 7.000% | 5.000% |
| 7 | 77.6705 | 2.000% | 6.000% |
| 8 | 104.6116 | 7.000% | 6.250% |
| 9 | 98.3537 | 6.500% | 6.750% |
| 10 | 114.0472 | 9.000% | 7.000% |