# Financial Maths Question: How to 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 + $$v^4$$7501 = 9081.09.

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

Firstly, there are two types of duration, one is Macaulay Duration, and one is Modified Duration. Since you are new to this topic, I am going to assume that the duration you are referring to is Macaulay Duration.

Your formula for duration is absolutely correct, but, I suggest using this formula instead for calculating the $$D_{Mac}$$ (Macaulay Duration) of a cash flow.

\begin{align} D_{Mac} = -\frac{P'(\delta)}{P(\delta)} = \frac{\sum_{t=0}^n \ t \ \cdot \ v^t \ \cdot \ CF_t}{\sum_{t=0}^n \ v^t \ \cdot \ CF_t} \end{align}

Using this formula, you will be able to plug-in your respective cash flow for both assets and liabilities.

Thus, the duration for the $$\mathbf{liabilities}$$ becomes:

\begin{align} D_{{Mac}_{liabilities}} = \frac{0 \ \cdot \ v^0 \ \cdot \ 2,910 \ + \ 4 \ \cdot \ v^4 \ \cdot \ 7,501 }{v^0 \ \cdot \ 2,910 \ + \ v^4 \ \cdot \ 7,501 } \approx 2.71821 \ldots\ years \end{align}

and the duration for the $$\mathbf{assets}$$ becomes:

\begin{align} D_{{Mac}_{assets}} = \frac{1 \ \cdot \ v^1 \ \cdot \ 5,000 \ + \ 3 \ \cdot \ v^3 \ \cdot \ 5,000 }{v^1 \ \cdot \ 5,000 \ + \ v^3 \ \cdot \ 5,000 } \approx 1.95125 \ldots \ years \end{align}

Hope this helps!