# 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 doesn’t 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 tis face value. This relationship can be expressed as follows:

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

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:

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

The yield to maturity calculated above is the spot interest rate (s_{n}) 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:

$$ s_{15}=\left[\left(\frac{$100}{$38.50}\right)^\frac{1}{15\times2}-1\right]\times2=6.47\% $$

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:

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

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).