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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.

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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!

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