## 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.Â

- 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

- Â

- As for the 2y zero rate:

- 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

- Similarly for 3y zero rate:

Mathematically 2y zero rate can be calculated as: 100 =

_{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

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:

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

- 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.