# Spot Rates, Forward Rates, and Bootstrapping

The spot rate is the current yield for a given term. Market spot rates for certain terms are equal to the yield to maturity of zero-coupon bonds with those terms. Generally, the spot rate increases as the term increases, but there are many deviations from this pattern. So bonds with longer maturities will generally have higher yields. A graph of the spot rates for different maturities forms the yield curve, and the shape of this curve often determine the effectiveness of certain bond strategies, especially those to lower interest rate risk, such as immunization. Moreover, some holders of coupon bonds want to be able to strip the bonds into a series of zero-coupon bonds, either to mitigate risk by more closely matching the duration of assets to liabilities or to earn a profit by selling the zeros. Profit can also be made by reconstituting the zero-coupon bonds back into the original bond, if the sum of the zeros is cheaper than the reconstituted bond. Selling zeros or reconstituting the zeros depending on market prices is a form of arbitrage, a means of earning a riskless profit. However, whether it would be profitable to issue zeros, strip coupons, or reconstitute coupons depends on the spot-rate curve, or the yield curve, which allows the investor to determine at what price a given bond with a certain term would sell for. Often, however, there are not enough zero-coupon bonds selling in the market to give a clear indication of what bonds would actually sell for at a given maturity. How can spot rates be determined for maturities where market information is lacking?

Closely related to the spot rate is the forward rate, which is the interest rate for a certain term that begins in the future and ends later. So if a business wanted to borrow money 1 year from now for a term of 2 years at a known interest rate today, then a bank can guarantee that rate through the use a forward rate contract using the forward rate as interest on the loan. Forward rate contracts, which are a common type of derivative, are based on forward rates. Forward rates are also necessary for evaluating bonds with embedded options. But since forward rates are future spot prices for interest rates, which is unknowable, how are forward rates determined?

Spot rate curves and forward rates that are implied by market prices can be determined from the market prices of coupon bonds through a process called bootstrapping.

## Forward Rates

The price of a bond is equal to the present value of all of its cash flows. The usual technique is to use a constant yield to maturity (YTM) in calculating the present value of the cash flows. However, the bond price equation can be used to calculate the forward rates as implied by the current market prices of different coupon bonds.

 Bond Price = C1(1+YTM)1 + C2(1+YTM)2 + ... + Cn(1+YTM)n + P(1+YTM)n C = coupon payment per periodP = par value of bondn = number of years until maturityYTM = yield to maturity

A coupon bond can be considered as a group of zero-coupon bonds with a zero corresponding to each coupon payment and to the final principal repayment. In this way, each cash flow should be discounted at the interest rate that is appropriate for the period in which the cash flow will be received. The value of the zero-coupon bonds must equal the coupon bond; otherwise, an arbitrageur could strip the bond and sell the zeros for a profit, as they sometimes do.

The forward rates thus calculated are not forecasts of future interest rates, since future interest rates are unknown. Rather, the forward rates are simply calculated from current bond prices; hence, they are sometimes referred to as implied forward rates, because they are implied by the market prices of the bonds in the same way that implied volatility is determined by market option prices.

## Bootstrapping

Treasuries are the ideal bond to use in constructing a yield curve because they are devoid of credit risk, so Treasury prices depend more on market interest rates. Treasuries define a risk-free yield curve, but the market prices also imply forward rates, which are yields for certain periods in the future.

Because Treasury notes and bonds are generally issued as coupon bonds, their prices cannot simply be used to construct the spot rate curve or to calculate forward rates. Instead, a theoretical spot rate curve and implied forward rates are constructed through the process of bootstrapping which calculates the forward rates by considering the value of the zero coupon bonds that are equivalent to the Treasury bond. The calculated forward rates can then construct the spot-rate curve by adding the yields for each term to the desired maturity.

The bootstrapping technique is based on the price-yield equation using different rates for each of the 6-month terms, as determined by market prices:

 Bond Price = C1(1+r1)1 + C2(1+r2)2 + ... + Cn(1+rn)n + P(1+rn)n r = interest rate per periodIf r = an annual yield, but the term is for ½ year, then divide by 2

The interest rate is 1st calculated for the 6-month bond that has a known market price, which has only a single payment, consisting of the coupon payment and the principal repayment, at its maturity. After the rate is calculated for the 1st period with the 6-month bond, then that rate is used to calculate the rate for the 2nd period of a 1-year bond, and so on, until all the rates for the desired number of terms for which there are market prices available have been determined. This is referred to as the bootstrapping technique, because the prior calculated spot rates are used to calculate later spot rates in successive steps.

### Example: Bootstrapping

These 2 6% coupon bonds with no credit-default risk and a nominal par value of \$100 have the following clean market prices (no accrued interest) and times left to maturity. Note that the annualized yield is divided by 2 because each term only covers ½ year:

• 6-month bond: \$99
• 1-year bond: \$98
• y = annualized yield to maturity
1. Determine the yield for the 6 month bond using the market price of \$99. At the end of 6 months it will pay a coupon of \$3 plus the principal repayment, for a total of \$103:
1. 99 = 103/(1+y/2)
2. 99 × (1+y/2) = 103
3. 1+y/2 = 103/99 = 1.0404
4. y/2 = 1.0404 – 1 = .0404
5. y = .0404 × 2 = .0808 = 8.08%
2. Determine the 2nd-term yield for the 1-year bond by using the market price of \$98 for the bond and the yield for the 1st term calculated in step 1:
1. 3/1.0808 + 103/(1 + y/2)2 = 2.7757 + 103/(1 + y/2)2 = 98
2. 103/(1 + y/2)2 = 98 – 2.7757 = 95.2243
3. 103 = 95.2243 × (1 + y/2)2
4. (1 + y/2)2 = 103/95.2243 = 1.0817
5. 1 + y/2 = √1.0817 = 1.0400
6. y/2 = 1.0400 – 1 = .04
7. y = 0.04 × 2 = .08 = 8%

So, according to these market prices, the spot rate for the current 6-month term is 8.08% and the forward rate for the 2nd 6-month term is 8%.

## Conclusion

The bootstrapping technique is a simple technique, but finding the real yield curve and smoothing it out requires more complicated mathematics, because bond prices are not only affected by interest rates but also by other factors, such as credit risk, taxes, liquidity, and the simple variance in supply and demand for each maturity. More sophisticated mathematical techniques are used to determine more realistic rates, but these are beyond the scope of this article. Nonetheless, bootstrapping does illustrate how forward rates can be calculated from current bond prices, which can then be pieced together to fill in the gaps in the spot-rate curve.