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Jensen’s Inequality and a Convexity Adjustment in Bond Pricing

Article Quant Q&A · Author: Betty
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Summary

The document illustrates a convexity adjustment by comparing two ways to price cash flows when an interest rate can take one of two values. One calculation discounts the amount using the average of the rates, while the other averages the two prices obtained by discounting separately at each rate. The difference between these prices is the adjustment being asked about.

This setup demonstrates why pricing at an average input need not equal the average of prices: the discounting function is nonlinear in the rate. That gap is the practical effect associated with Jensen’s inequality and convexity. The example gives formulas for the two price calculations but does not show the arithmetic leading to the requested numerical difference, nor does it explain the broader theory or assumptions behind the cash-flow setup. It is therefore a compact illustration rather than a complete treatment of convexity adjustments in fixed-income valuation.

Key ideas

  • Discounting at an average interest rate can differ from averaging prices discounted at separate rates.
  • The difference between the two valuation methods illustrates a convexity adjustment.
  • The adjustment arises because the bond pricing function is nonlinear in interest rates.
  • The example presents the comparison formulas but does not provide a general valuation framework.

Tags

Fixed incomeStatisticsDerivatives pricing
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# Jensen’s inequality in Convexity adjustment premium


# Jensen’s inequality in Convexity adjustment premium












I'm preparing for my FRM II test in May. Could someone help to explain where does the 0.0823 come from? 😥

## Answer by Magic is in the chain (score 3, accepted)

https://quant.stackexchange.com/a/51549

$P_2-P_1$ where:

$P_1=\frac{1000}{\left(1+\frac{0.06+0.04}{2} \right)\left(1+0.05 \right)}$

$P_2=0.5\frac{1000}{\left(1+0.06 \right)\left(1+0.05 \right)}+0.5 \frac{1000}{\left(1+0.04 \right)\left(1+0.05 \right)}$

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.