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Calculating Cash Flow Duration as a Present Value Weighted Average

Article Quant Q&A · Author: DPJDPJ
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Summary

The document derives the duration of a single fixed cash flow under annual effective compounding. Starting from the negative sensitivity of present value to the interest rate, differentiation gives a duration equal to the payment time divided by one plus the rate. A payment at time zero therefore has duration zero, while a later payment has a positive duration.

For a set of cash flows, the document combines individual durations using each cash flow’s present value as its weight. It applies this method to stated asset and liability schedules, although the worked calculation shown is for liabilities: the time-zero payment contributes no duration, and the later payment contributes in proportion to its discounted value. The example uses a fixed rate and cash flow dates; it does not discuss alternative duration definitions, changing rates, or cash flows whose amounts depend on rates.

Key ideas

  • For a single fixed cash flow, duration equals its payment time divided by one plus the annual effective rate.
  • A time-zero cash flow has zero duration under the stated measure.
  • The duration of multiple cash flows is the present value weighted average of their individual durations.
  • The example applies the method to a liability schedule with payments at time zero and a later date.

Tags

Fixed incomeStatisticsRisk management
Full text
# How does one calculate the duration of a cash flow


# How does one calculate the duration of a cash flow












The question reads: A firm has liabilities as follows: £2,910 at time t = 0 and £7,501 at time t = 4 (time is measured in years). On the asset side the firm has two payments, each for £5,000, at time t = 1 and t = 3. The annual effective rate is i = 5% p.a.

Compute the effective duration for both assets and liabilities.

I'm new to this topic and struggle to understand it. I understand duration to be a measure of the volatility of the present value of a cash flow with respect to changes in the interest rate. In order to calculate the duration I suppose I would use this formula:

$v = -1/PV * dPV/di$

I can calculate the present value of, let's say firstly, the liabilities to be:

PV = 2910 + $(\frac{1}{1+i})^4$7501 = 9081.09.

But where do I go from there? How would I use that value to calculate the duration? Thanks in advance.

## Answer by Alex C (score 1, accepted)

https://quant.stackexchange.com/a/47174

Let's consider a single cash flow CF

$PV = (\frac{1}{1+i})^n CF$

As you wrote $v = -\frac{1}{PV} \frac{d PV}{di}$

Taking the derivative of PV with respect to i and plugging it in:

$v= - \frac{(1+i)^n}{CF} n \frac{1}{(1+i)^{n-1}}\frac{-1}{(1+i)^2}CF$

after simplifying we get

$v = \frac{1}{1+i}n$

(which is easy to remember, no need to derive it every time)

So v = (1/1.05)4 = 3.809.

Now consider multiple cash flows. The duration then is the weighted average of the durations, using the PVs as weights.

So on the liability side we have 1 cash flow with duration 0 and PV 2910, and one cash flow with duration 3.809 and PV 6171.09. The combined PV is 9081.09. The combined duration of the liabilities is 0+3.809*6171.09/9081.09 = 2.588

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.