Bootstrapping Discount Factors from Deposits and Coupon Bonds
Summary
The document explains how to derive discount factors from a short deposit rate and coupon bond prices when market inputs are sparse. It assumes annual bond coupons and uses the 6-month deposit to establish the first discount factor. The 1.5-year bond price is then written as the discounted value of its coupon and principal payments; with the 6-month factor already known, that equation gives the 1.5-year factor. The longer bond supplies a further equation for its terminal discount factor.
The method is sequential: solve for discount factors in maturity order, using each known factor to isolate the next unknown. The example illustrates the setup rather than providing a full numerical curve. Results depend on conventions that must be checked, including coupon frequency, deposit rate compounding, payment dates, and whether quoted bond prices are clean or adjusted for accrued interest. The response also assumes a unit notional and simplifies cash-flow timing.
Key ideas
- Bootstrap discount factors in increasing maturity order from deposits and bond prices.
- Use the deposit rate and its day-count and compounding convention to determine the earliest discount factor.
- Express each bond price as the sum of its discounted coupon and principal cash flows.
- Use previously determined factors to solve for the next unknown maturity factor.
- Coupon frequency and market conventions affect the equations and must match the instruments.
Tags
Full text
# Get discount factors with limited knowledge?
# Get discount factors with limited knowledge?
I am facing the problem of just having this information:
6% coupon bond with 2.5 years to maturity, traded at a 100% clean price 4% coupon bond with 1.5 years to maturity, traded at a 98% clean price A 6M deposit with 6M rate of 5% (MMY)
And I want to get the discount factors for each maturity.
My problem is, I do not know where to start to get to each discount factor. Is there a straight method for this kind of problem?
## Answer by jensa (score 5)
https://quant.stackexchange.com/a/9540
The general idea is to bootstrap the discount factors in the correct order, based on the data you have given. I'm going to make some assumptions that your bonds are paying annual coupons. The longest maturity is 2.5 years, meaning you need discount factors for 6M, 1.5Y and 2.5Y.
The 6M deposit has a rate of 5%, this tells you that you should use the 5% rate to create the 6M discount factor (you wrote 6M rate = 5%, if the 5% was expressed in yearly terms the rate applicable would have been 2.5% = 5%/2). So know you have the discount factor for 6M, $DF_{6M} = e^{-0.05*0.5}$. To retrieve the remaining discount factors you need to formulate equations, equations relating the given bond prices to an analytical expression involving the sought discount factors. For example, assuming the notional is equal to 1, $0.98 = 0.04 DF_{6M} + (1+0.04) DF_{1.5Y}$. But you already know $DF_{6M}$, so from this equation you can solve for $DF_{1.5Y}$. Now write a similar equation for the longest maturity bond and solve for $DF_{2.5Y}$.
Hope it helps!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.