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The concept of Present Value

Economists are always evaluating the value of the costs and benefits of a particular project, often over a long time scale. When you do this you don’t just add up the total sum of an expected future income stream, you have to discount it as you look into the future. This is the idea of working out the ‘present value’ of some form of return in the future.

The discount rate is basically the opportunity cost of directing resources elsewhere, so for private investment projects a good benchmark to use is the market interest rate. This is logical because it represents an alternative use of the investment funds – the firm could have just lent it to someone else (or invested it in a bank to lend it on their behalf) and earned the market rate. So if the investment is being used to finance an investment project which is forecast to earn income streams in the future, it is only worthwhile if those income streams will end up being better than what the firm could have earned by simply investing the money at the market rate of interest and taking the returns from that.

It will be easier to see with an example.

Lets say a firm is considering making an investment in a new piece of machinery which will cost £10000 now. The projections are that this piece of machinery will last for six years, and will increase the firm’s profits by £3000 in each of the first two years, by £2000 in each of the next two years, and by £1000 in the final two years before it needs to be replaced. That means in total it will bring in £12000 over 6 years.

Total value of income stream:3000+3000+2000+2000+1000+1000=12000

So it costs £10000 and brings in £12000 over 6 years…sounds like it is worth it?

Not necessarily. Suppose the market interest rate is 4%.

If you use a discount rate of 4%, then the present value of the income stream is \frac {3000}{1.04}+\frac {3000}{1.04^2}+\frac {2000}{1.04^3}+\frac {2000}{1.04^4}+\frac {1000}{1.04^5}+\frac {1000}{1.04^6} = 10758.13

Now given that the cost of the investment is 10000 now, the net present value of the project is -10000 + 10758.13 = 758.13

So yes, it is worth it. The present value of your forecast income streams is worth more than the cost of the investment, but not by the £2000 that you would get simply by adding up the values.

If the net present value of a project > 0, then it is a worthwhile investment.

When it comes to calculating the present value of a government project, the opportunity cost is the net present value of what the funds would otherwise have been used to produce. In the UK most public sector investments take 3.5% as a benchmark discount rate, this is the social time preference rate. The danger with this is that discounting can effectively disenfranchise future generations, if you are looking at a long term project that will yield gains over many years. We are still today benefiting from the investment that our ancestors made in the railways and canals etc, but if you discount every project at 3.5% then the present value of something in 100 years time is effectively the value divided by 31, and something 200 years in the future gets divided by 972. This is pretty controversial when it comes to evaluating the gains of environmental projects. So the UK government uses a declining discount rate, the 3.5% counts only for the first 30 years, and then gradually declines over different time horizons.

Categories: Geometric series, Maths
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