30 Spot Rates of Interest
30.1 Zero Coupon Bonds
A zero coupon bond, also referred as a discount bond, is a bond which has no coupons and has a single payment made at the time of maturity. A Treasury Bill (T-Bill) is an example of a zero coupon bond since it consists of a single payment made at the time of maturity.
30.2 Spot Rates
The yield to maturity on a zero coupon bond is called the spot rate of interest for that time to maturity. \(s_t\) is the spot rate for a t-year maturity zero coupon bond. The current price of a t-year zero coupon bond with a maturity amount of 1 is \(\frac{1}{(1+s_t)^t}\). The term structure is the set \(\{s_t\}_{t>0}\).
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 spot rates corresponding ot the cash flows: \[PV = \sum_{t>0}\frac{CF_t}{(1+s_t)^t}\] Unless we are told otherwise, we assume that spot rates are expressed as annual effective rates of interest.
Example: The table below provides the times until maturity, coupon rates, and annual effective yields. Each of the bonds makes annual coupon payments and has a face amount of \(\$100\). Calculate the 1-year, 2-year, 3-year, 4-year, and 5-year annual effective spot 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% |
Solution:
The 1-year bond has only one cash flow, so its yield is equal to the 1-year spot rate: \[\frac{108}{1.02}= \frac{108}{1+s_1} \implies s_1 = 0.02\] The 2-year spot rate is: \[\frac{5}{1.025} + \frac{105}{1.025^2} = \frac{5}{1.02} + \frac{108}{(1+s_2)^2} \implies s_2 = 0.02512\] The 3-year spot rate is:\[\frac{2}{1.03} + \frac{2}{1.03^2} + \frac{102}{1.03^3} = \frac{2}{1.02} + \frac{2}{1.02512^2} + \frac{102}{(1+s_3)^3} \implies s_3 = 0.03014\] The 4-year spot rate is exactly the yield rate of the zero-coupon bond: \[\frac{100}{1.04^4} = \frac{100}{(1+s_4)^4} \implies s_4 = 0.04\] The 5-year spot rate is: \[106.5850 = \frac{6}{1.02} + \frac{6}{1.02512^2} + \frac{6}{1.03014^3} + \frac{6}{1.04^4} +\frac{106}{(1+s_5)^5} \implies s_5 =0.04669\]
The remaining five spot rates can be calculated:
| Maturity | Annual Coupon | Yield | Spot |
|---|---|---|---|
| 1 | 8.000% | 2.000% | 2.000% |
| 2 | 5.000% | 2.500% | 2.512% |
| 3 | 2.000% | 3.000% | 3.014% |
| 4 | 0.000% | 4.000% | 4.000% |
| 5 | 6.000% | 4.500% | 4.669% |
| 6 | 7.000% | 5.000% | 5.260% |
| 7 | 2.000% | 6.000% | 6.142% |
| 8 | 7.000% | 6.250% | 6.767% |
| 9 | 6.500% | 6.750% | 7.359% |
| 10 | 9.000% | 7.000% | 7.903% |
We usually write the present value of an annuity in terms of a single yield, but we can also write it in terms of the spot rates: \[a_{\overline{n|}spots} = \frac{1}{1+s_1} + \frac{1}{(1+s_2)^2} + ... + \frac{1}{(1+s_n)^n}\] where the discount factor \(v^n_{spot}\) is defined as: \[v^n_{spot} = \frac{1}{(1+s_n)^n}\] Example: The table below provides the times until maturity, coupon rates, and annual effective yields for 2 bonds. Each of the bonds makes annual coupon payments and has a face amount of \(\$100\).
| Maturity | Annual Coupon | Yield |
|---|---|---|
| 5 | 6.00% | 4.50% |
| 6 | 7.00% | 5.00% |
You are given that the annual effective 5-year spot rate is 4.669%. Calculate the annual effective 6-year spot rate.
Solution:
We can calculate the price of the 5-year bond is 106.5850 and the price of the 6-year bond is 110.1514.
We have: \[106.5850 = 6\times a_{\overline{5|}spots} + 100 \times v^5_{spot} \implies a_{\overline{5|}spots} = 4.4976\] and: \[\begin{align}110.1514 &= 7\times a_{\overline{6|}spots} + 100 \times v^6_{spot} = 7\times a_{\overline{5|}spots} + 107 \times v^6_{spot} = 7\times 4.4776 + 107\times v^6_{spot} \\\implies v^6_{spot} &= 0.7352 \implies s_6 = 0.05260\end{align}\]
We can use the spot rates to find the coupon rate of an n-year bond that is priced at par:
\[F = Fr \times a_{\overline{n|}spots} + F\times v^n_{spot} \implies r = \frac{1-v^n_{spot}} {a_{\overline{n|}spots}}\]
Example: Consider a \(\$100\) par bond with coupon rate of \(4.0\%\) per annum paid semi-annually. The spot rates of interest in the market are given by:
| t | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 |
|---|---|---|---|---|---|---|---|---|---|---|
| \(s_t\) | 3.0 | 3.0 | 3.5 | 3.5 | 4.0 | 4.0 | 4.5 | 4.5 | 5.0 | 5.0 |
Find the price of the bond if it matures in 5 years.
Solution: The price can be calculated by \[P = 20\bigg(\frac{1}{1.03^{0.5}} + \frac{1}{1.03^{1}} + \frac{1}{1.035^{1.5}} + \frac{1}{1.035^{2}} + ... + \frac{1}{1.05^{5}}\bigg) + 100 \times \frac{1}{1.05^5} = 95.6141\]