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Bootstrapping Spot Discount Factors from Deposit and Swap Rates

Article Quant Q&A · Author: Hooimin Ooi
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Summary

The document sketches how to construct spot discount factors from short-term deposit rates and a one-year swap rate. For deposits, it uses simple interest with the stated day-count fractions to convert each quoted rate into a discount factor. It then takes the discount factor at the six-month point as known and uses the swap’s two payment periods to solve for the one-year discount factor.

This is a compact example of bootstrapping: earlier curve points are used to derive a later discount factor from a par swap quote. The answer gives formulas and numerical inputs, but does not provide a broader derivation or discuss conventions such as day-count rules, payment calendars, compounding, or market-specific settlement details. The calculations should therefore be read as an illustration under the assumptions stated, not as a universal curve-building recipe.

Key ideas

  • Deposit quotes are converted to discount factors using simple interest in the example.
  • A previously calculated six-month discount factor is used to solve for the one-year factor.
  • The swap calculation relies on the swap’s two semiannual cash-flow periods.
  • Day-count and market conventions can change the appropriate curve-building formulas.

Tags

Fixed incomeStatistics
Full text
# How do I calculate the spot rate?


# How do I calculate the spot rate?












How can I calculate the rates to construct the curve? I was thinking to use the formula converting par yield to spot rate, but I am not confident about it. Please give some hints or working on how to solve this. Thank you so much.

## Answer by Magic is in the chain (score 0)

https://quant.stackexchange.com/a/42052

For the two deposit rates, use:

Discount factor (90)=1/(1+90/360*0.035)

Discount Factor (180)=1/(1+180/360*0.04)

This is because deposit rates use simple interest rate formula.

For swap rate, first compute the 6 months (180 days) discount rate , d180, as above, and then use the following formula to compute the 1 year ( 360 days) discount factor, d360:

d360=(1-0.5*0.045*d180)/(1+0.5*0.045)

Where 0.045 is the 1 year swap rate.

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.