Spot Interest Rate

Spot interest rate for maturity of X years refers to the yield to maturity on a zero-coupon bond with X years till maturity. They are used to (a) determine the no-arbitrage value of a bond, (b) determine the implied forward interest rates through the process called bootstrapping and (c) plot the yield curve.

A zero-coupon bond is a debt instrument that pays its face value i.e. principal back at its maturity date. It does not make any other payments to the bond-holder. The yield on such an instrument is a direct measure of required return for the given maturity. The price of a zero-coupon bond equals the present value of its face value. This relationship can be expressed as follows:

$$ \text{P}=\frac{\text{FV}}{\left(\text{1}+\frac{\text{YTM}}{\text{m}}\right)^{\text{n}\times \text{m}}} $$

Where FV is the face value of the bond, YTM is the yield to maturity, m is the number of compounding periods per year and n is the number of years till maturity. By rearranging the above expression, we can work out the formula for yield to maturity on a zero-coupon bond:

$$ \text{s} _ \text{n}=\text{YTM}=\left[\left(\frac{\text{FV}}{\text{P}}\right)^\frac{\text{1}}{\text{n}\times \text{m}}-\text{1}\right]\times \text{m} $$

The yield to maturity calculated above is the spot interest rate (sn) for n years.

By determining spot interest rates corresponding to each cash flow of a bond and then discounting each cash flow using that period-specific yield, we can determine the no-arbitrage price of a bond.

Example: Spot Interest Rates and Yield curve

Let’s see how we can create the yield curve from the following current market prices of zero-coupon bonds with bi-annual compounding:

Current Price Maturity (in Years)
98.50 1
94.10 3
84.00 5
58.20 10
38.50 15
16.25 25

The following table shows relevant spot-rates:

Current Price Maturity (in Years) Spot Interest Rates
98.50 1 1.52%
94.10 3 2.04%
84.00 5 3.52%
58.20 10 5.49%
38.50 15 6.47%
16.25 25 7.40%

We illustrate how to determine the spot rate for the bond with 15 years till maturity as follows:

$$ \text{s} _ {\text{15}}=\left[\left(\frac{\text{\$100}}{\text{\$38.50}}\right)^\frac{\text{1}}{\text{15}\times\text{2}}-\text{1}\right]\times\text{2}=\text{6.47%} $$

If we plot the above schedule of spot interest rates with reference to their maturities, we get the yield curve:

Spot Interest Rate Yield Curve

Spot Interest Rate vs Yield to Maturity

Yield to maturity and spot interest rate in case of pure-discount bonds i.e. zero-coupon bonds are the same. However, in case of coupon-paying bonds, yield to maturity is the (somewhat) weighted average of the individual spot interest rates that apply to each cash flow of the bond.

Let’s say we have a 3- year bond with face value of $100 and annual coupon of $2.00. The spot interest rates for 1, 2 and 3 years are 1.50%, 1.75% and 1.95%.

The following equation describes the relationship between yield to maturity of the bond and the relevant spot interest rates:

$$ \frac{\text{\$2}}{({\text{1}+\text{YTM})}^\text{1}}+\frac{\text{\$2}}{({\text{1}+\text{YTM})}^\text{2}}+\frac{\text{\$100}+\text{\$2}}{({\text{1}+\text{YTM})}^\text{2}}\\=\frac{\text{\$2}}{({\text{1}+\text{1.50%})}^\text{1}}+\frac{\text{\$2}}{({\text{1}+\text{1.75%})}^\text{2}}+\frac{\text{\$100}+\text{\$2}}{({\text{1}+\text{1.95%})}^\text{3}} $$

What we have done is to find the no-arbitrage price of the bond using the spot interest rates (on the right-hand side of the equation). The left-hand side calculates the yield to maturity (i.e. internal rate of return) of the bond as the rate that equates its future cash flows to its no-arbitrage price.

The no-arbitrage price (i.e. right-hand side) works out to $102.33 which yields 1.94% (i.e. the left hand side).

by Obaidullah Jan, ACA, CFA and last modified on is a free educational website; of students, by students, and for students. You are welcome to learn a range of topics from accounting, economics, finance and more. We hope you like the work that has been done, and if you have any suggestions, your feedback is highly valuable. Let's connect!

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