# How much should I pay to protect that bridge from flood damage?

This is the problem: there is a bridge over a river and it looks like it might be destroyed in a flood.  How much should I pay now to protect the bridge?

This obviously depends on the cost of the bridge but there are two other key factors we need to consider.  One is that we don’t know when a destructive flood with occur, it could be next year or at some distant time in the future.  The other is that the further into the future that the bridge is destroyed, the less we should pay now to protect it.   To work out the present value of the cost of destruction of the bridge, we need to consider both the discount rate, used for present value calculations, and the probability of a destructive flood occurring in any particular year.

Using a standard benefit cost analysis approach, given a discount rate of i, the discount factor for any year t in the future can be calculated as follows.

$\frac{1}{(1+i)^t}$

If the annual exceedance probability of a flood that will destroy the bridge is $p$, then the probably of this flood being exceeded in year t is:

$(1-p)^t \times p$

This is equivalent of t years of no flood damage followed by a damaging flood. We are taking year zero as the present, which is the standard approach for benefit cost analysis.

For any year t, the total discount factor will be the product of these two values

$(1-p)^t \times p \times \frac{1}{(1+i)^t}$

Consider an example using an 8% discount rate and assuming a bridge will be destroyed in a 1% flood. The calculations are shown below.

 Year (t) Discount factor (8% discount rate) Probability of 1% flood Total discount factor (discount factor x probability) 0 1 0.01 0.01 1 0.9259 0.99 x 0.01 0.0099 2 0.8573 0.992 x 0.01 0.009801 … … … … t (1/1.08)t 0.99t  x 0.01 (1/1.08)t x 0.99t  x 0.01

To work out the present value of the bridge damage we need to sum the values in the right hand column.

Using the standard approach for a geometric series, the sum will be:

$\frac{a}{(1-r)}$

Where a is the first term in the series and r is the ratio of two consecutive terms.  In this case:

$a = 0.01$
$r = \frac{1}{1.08} \times 0.99 = 0.9167$

sum = $\frac{0.01}{1-0.9167} = 0.12$

The present value of the cost of bridge destruction will be 0.12 times the current replacement value of the bridge.

The general formula for this factor is:

$\frac{p(1+i)}{i+p}$

So given an estimate of the annual exceedance probability of a flood that will destroy a bridge, and a discount rate, we can calculate the maximum proportion of the bridge replacement cost we should spend on protection.

We can also use this formula to calculate the value of changing the vulnerability of a bridge from being damaged in say a 1% (100-year) flood to say a 0.5% (200-year) flood; calculate both factors and take the difference.