Bootstrapping: Calculation of theoretical spot rate curve from a par yield curve
This educational publication is the special valuation series on fixed income securities. Its mandatory to read the Understanding Term Structure of Interest Rates posted under Insights here before reading Bootstrapping: Calculation of theoretical spot rate curve from a par yield curve.
There are multiple ways in which a theoretical spot rate curve can be derived from the par yield curve of Treasury securities. Some of them involve advanced statistical analysis while stripping coupon securities into zero-coupon bonds is not the best way to derive yield due to tax distortions. The most popular way is to use on the run treasury securities along with selected of the run treasury securities called Bootstrapping. US treasury frequently issued treasuries are Treasury Notes: 2, 5 & 10 and 30-year long term bonds. This leaves a lot of gaps that need to be filled through off-the run treasuries. One simple method is to use the linear interpolation method between two securities of different maturities. However, this method oversimplifies yields of securities which lie between two maturities.
# Forward Rates
A theoretical spot rate curve implies a term structure of forward rates. The theoretical spot rate curve helps to understand the market expectation of future short-term rates called forward rates. Many hypotheses explain the term structure of forward rates like pure expectations theory, liquidity theory, market segmentation theory, etc. However, a study done by Antti Ilanen found that forward rates had limited success in explaining future spot rates.
The three main influences on the yield curve are the following.
- Market expectation of future rate changes
- Bond risk premium
- Convexity bias
As of now, we will ignore convexity bias as we will come to this when we analyze bond valuation in depth. To understand the importance of forward rates take an example when an investor wants to invest over one year. The method of calculating a future interest rate either through a spot rate or yield curve is called a forward rate. The global macroeconomic outlook is very dynamic and affected by many factors to decisively predict forward rates for different maturities but market prices’ future expectation of interest rates into buckets of varying maturities. Forward rates are the interest rate charged between two periods in the future contracted today.
Understanding the importance is vital to have a grasp of the underpinnings of global economic conditions within the investment arena. The investor can invest the proceeds in a treasury of 1-year maturity or invest the capital in six-month security and roll over the investment again in six months. However, it would be prudent to understand how the interest rates would be prevailing in six months. If there is an onset of deep recession and the interest rates are going to go down, the investor is better off investing in 1-year security and locking his return. The investor would be even if the returns equal through both the alternatives.
Bootstrapping is the method used to construct a theoretical yield curve from a par yield curve of treasury securities. US Treasuries only issue zero coupon bonds for six-month and 1-year securities, so yields on them are spot rates. Refer to the table on the left. If 6 month and one-year spot rates are given, you can construct the spot rate curve for bonds of all term maturity provided you have the par yield curve for the same duration.
At par coupon rate = yield to maturity which is required for deriving spot rates for all durations. In the present example, calculate spot rate for 1.5- and 2-year time duration. Calculate spot rate for time period 3 i.e., a theoretical 1.5-year zero coupon bond from the given yield to maturity table
0.5- and 1-year bonds are Zero coupon bonds so consider their yields as spot rates. Consider all securities to have market price equal to its par. Consider Par / Face Value as 1000. For period 1.5 years, YTM is 5.60% annualized so cash flows period for the three periods are 28, 28 , 28+1000. The face value of the bond is 1000. Refer to the excel sheet for detailed calculations.
S2 : Since there are two time periods from the present to year 1, the suffix 2
The forward rate is calculated using the formula below. Refer to the excel sheet here for a comprehensive calculation of forward rates
Spot rate the geometric mean of one year forward rates
Example: An investor wants to invest for two years a sum of 100. If the spot rate S1= 5% (annualized) and the future rate between time period 1 and 2 is f1,2 is 5.90% annualized. What will be the expected return on the capital given the data.
Question: The one year forward rate are as follows. Find the annualized spot rate for time period 3. (Ans 5.9%)
Note: The excel sheet given is to build on practice problems will be following soon. The sheet also includes implied spot rate one year from now. We will be discussing theories affecting term structure of interest rates, work sheet problems and much more. Convexity concept will be discussed as we go forward. The implied spot rate column should be ignored for now.
For excel sheet refer to the original publication under Tutorial section here.