# 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 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).