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Estimating Duration and Convexity for Callable, Prepayable Term Loans

Article Quant Q&A · Author: AnonnonA
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Summary

The discussion considers an open term loan that pays monthly interest and returns principal when closed. Because either the lender or borrower can end the loan, its remaining cash flows depend on interest rates and on each party’s decision to call or repay. The answer compares this structure to a mortgage with prepayment: borrowers and lenders may have incentives to close when the contract coupon becomes unfavorable, while transaction costs or other frictions can keep the loan outstanding.

The suggested approach is to model the conditions under which either party terminates the loan, then simulate its value across interest-rate scenarios. Effective duration and convexity can be estimated from the resulting price changes as rates shift. The response emphasizes that the termination behavior must be modeled to value the embedded optionality; it does not specify a particular behavioral model, simulation design, calibration method, or hedge. Hedging a portfolio therefore requires assumptions about borrower and lender behavior in addition to the rate sensitivity of ordinary fixed cash flows.

Key ideas

  • A loan that either party can terminate has cash flows contingent on their exercise decisions.
  • Interest-rate moves can change incentives to call or repay the loan.
  • Transaction costs and other frictions can affect how long loans remain outstanding.
  • A behavioral termination model can support scenario simulation of loan values.
  • Effective duration and convexity can be inferred from simulated price changes under rate shifts.

Tags

Fixed incomeDerivatives pricingRisk management
Full text
# Duration and convexity of an open term loan/bond!


# Duration and convexity of an open term loan/bond!












Imagine an open term loan with monthly interest payments of [x]% and the principle due when the loan is closed. Both the lender can call the loan, and the borrower can return the loan (with no penalty) at any time.

If its helpful, assume the loan asset has liquid secondary market and mature futures market with a known term structure basis.

How would one calculate the duration and convexity of a loan with these attributes? How would one think about hedging these risks when considering a loan book made up of these loans?

## Answer by Si Chen (score 2)

https://quant.stackexchange.com/a/68461

Your loan seems to be one that would be either called by the bank or closed by the borrower as soon as the coupon rate moves below or above the prevailing market rate, so for the loan to be outstanding there needs to be some transaction cost or friction to keep it around. For that reason it is most similar to US mortgages which can be prepaid, but where some borrowers do not for various reasons.

If this sounds like a reasonable explanation then you would need to develop a model of when the bank would call the loan and when the borrower would close the loan, and then use a simulation to model to calculate its values in different scenarios. From there you can derive the effective duration and convexity based on the price change relative to interest rate changes.

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.