Spot Rates, Forward Rates, and Bootstrapping

The spot rate of a bond is the current yield for a given term. Market spot rates for certain terms equal the yield to maturity of zero-coupon bonds with those terms. The spot rate increases as the term increases, but this pattern deviates frequently. So bonds with longer maturities generally have higher yields. A graph of the spot rates for different maturities forms the yield curve, and the shape of this curve often determines the effectiveness of certain bond strategies, especially those to lower interest rate risk, such as bond immunization strategies. Moreover, some holders of coupon bonds want 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 estimate the market price for a bond with a certain term. Often, however, not enough zero-coupon bonds are selling in the market to clearly indicate what actual bonds prices would be 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, 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 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 = the present value of all 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 Calculated Using a Constant Yield to Maturity
Bond Price = C1
(1+YTM)1		 + C2
(1+YTM)2		 + ... + Cn
(1+YTM)n		 + P
(1+YTM)n
  • C = coupon payment per period
  • P = par value of bond
  • n = number of years until maturity
  • YTM = 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 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 called implied forward rates, because they are implied by bond market prices in the same way that implied volatility is determined by option market prices.

Bootstrapping

Treasuries are the ideal type of bond to use to construct a yield curve because they lack 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 equivalent to the Treasury bonds. 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-Yield Equation Using Different Rates for Each Cash Flow
Bond Price = C1
(1+r1)1		 + C2
(1+r2)2		 + ... + Cn
(1+rn)n		 + P
(1+rn)n
  • r = interest rate per period
    • If 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 called the bootstrapping technique, because the prior calculated spot rates are used to calculate later spot rates in successive steps.

Example: Bootstrapping

Two 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:

  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. Present Value of First Coupon Payment + Present Value of Final Payment = 98
    2. 3/1.0404 + 103/(1 + y/2)2 = 2.88 + 103/(1 + y/2)2 = 98
    3. 103/(1 + y/2)2 = 98 − 2.88 = 95.12
    4. 103 = 95.12 × (1 + y/2)2
    5. (1 + y/2)2 = 103/95.12 = 1.082883
    6. Present Value of Final Bond Payment = 103/1.082883 = 95.12
    7. Market Price of Bond = $2.88 + $95.12 = $98

So, according to these market prices, the spot rate for the current 6-month term annualized is 8.0808% and the forward rate for the 2nd 6-month term annualized is 8.2883%.

Conclusion

The bootstrapping technique is simple in concept, 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. Continuously compounded interest rates are used to obtain more accurate results. 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.