2: Bootstrapping of Yield Curves – Par, Zero, Forward

Mini Chapter Two

Bootstrapping of Yield Curves
- Par, Zero, Forward

Par curve

Par curve is the spot interest rate curve for coupon bearing instruments as traded in the market, interest rates on which discount the cash flows to a present value (PV) of 0. You can think of it as the YTM or IRR yields across tenors to signify a term structure of rates.

Zero curve

Zero curve is a theoretical expected yield curve as derived from the par curve trading in the market. It can be understood as a single period compounded return for a specific tenor. As an example consider the following par bond yield curve paying annual coupons for the sake of simplicity:

1y: 6.12% 2y: 6.52% 3y: 6.72% 4y: 6.87%

Now to find zero coupon yields:

• 1y par coupon would itself equal the 1y zero coupon given the annual (hence one time) payment. This would be the expected return for the 1y horizon that can discount the future value to equal the current par value. 

• As for the 2y zero rate:
  • Calculate the PV of the 1y coupon using 1y zero rate
  • Now calculate the 2y zero rate as the discount rate for the second year such that the sum of PV of coupon received at the end of first year and the final principal + coupon received at the end of second year equals par

• Similarly for 3y zero rate:
  • Calculate the PVs of 1y and 2y coupon cash flows using the respective zero rates
  • Now calculate 3y Zero rate as the discount rate for the third year such that the sum of PV of coupons received at the end of first and second year and the final principal + coupon received at the end of third year equals par

Mathematically 2y zero rate can be calculated as:

\[ 100 = \dfrac{6.52}{(1 + 6.12\%)} + \dfrac{(100 + 6.52)}{(1 + R_{2y}^{zc})^{2}} \]

Solving above for the 2y zero coupon rate (denoted as R_2y^zc) = 6.53%… and so on.

This zero rates calculation is popularly referred to as bootstrapping. In fact arriving at breakeven forward rates from zero rates or even par rates is a form of bootstrapping.

Forward rate calculation - zero arbitrage

For zero arbitrage world assuming annual compounding:

\[ (1 + R_{2y})^{2} = (1 + R_{1y}) \times (1 + R_{1y1y}) \]

Where,
R_2y : 2 year rate
R_1y: 1 year rate
R_1y1y: 1 year forward 1y rate

Extending the above for a general notation below:

\[ R_{xy} = \left\{ \dfrac{[1 + R_{x+y}]^{(x+y)}}{[1 + R_{x}]^{x}} \right\}^{\frac{1}{y}} - 1 \]

Where,

R_xy is defined as the annual compounded rate for y years at the end of starting in x years time
R_x+y is the annual compounded rate for tenor (x+y) years

Day count conventions

It’s the number of days for which interest on a fixed income security accrues and is paid. Importantly this impacts the discount factor for calculating the present value of cash flows as different accrual periods would have different discount rates. There are different day count conventions at play in different markets (Actual/365, Semi Bond 30/360, Actual/Actual …) and one has to be cognizant of what applies to which market when we implement pricing or actual cash flow calculations for the same.

Trivia. For instance while calculating bond swap spreads in India one has to consider that bonds trade on a semi 30/360 convention while swaps trade on an Actual/365 convention. Therefore, for the month of February accrual cash flow on 1st March will have a 30 day coupon for a bond but only 28 or 29 days (leap year) for the swap leg.