Calculating Cash-Flow Duration When Payments Can Be Negative

Summary

The document explains how to calculate duration for a stream of fixed cash flows that includes both positive and negative payments. It defines present value as the sum of each cash flow discounted at its corresponding yield and time, then differentiates present value with respect to a parallel yield shift. Dividing that derivative by present value gives the duration sensitivity measure.

The derivation applies regardless of whether individual cash flows are positive or negative, so the usual formula remains usable for the example mortgage pool. The explanation assumes fixed cash-flow amounts and a parallel shift in the yield curve. It does not address how prepayment behavior, optionality, or yield-curve shape changes might alter cash flows or sensitivity, so those features require additional modeling.

Key ideas

• Duration follows from differentiating the discounted value of fixed cash flows with respect to yield.
• The derivative weights each cash flow by its time to payment.
• Negative cash flows do not invalidate the duration formula.
• The stated sensitivity assumes a parallel shift in yields and fixed cash flows.

Tags

# Duration calculation with negative cashflows


# Duration calculation with negative cashflows












I have a pool of (mortgage) assets that pay cashflows as below. How could I correctly calculate the duration? Does it have a meaning in the sense of a vanilla/callable bond as the measure of price sensitivity?

Year CF

0 1

1 25

2 25

3 0

4 0

5 -5

6 0

7 -5

8 5

## Answer by Ami44 (score 2, accepted)

https://quant.stackexchange.com/a/28390

You can calculate the duration and use it as sensitivity measure, as you are used to.

That is because the npv of the cashflows is: $NPV = \sum_i [ c_i * \exp(- y_i * t_i ) ]$

With $c_i$ the fixed cashflow amount at time $t_i$.

From that it follows, that the derivative of the NPV with respect to a parallel shift of the yieldcurve is

$d(NPV)/dy = - \sum[ c_i * t_i * \exp(- y_i * t_i ) ]$

If you divide that by the NPV, you get the definition of the duration. You see the formulas are valid regardless of the sign or amount of the $c_i$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.