# Present Value and Discounting Tweet

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1. Present Value and Discounting
Hi,

Normally when you wish to calculate the present value to evaluate whether or not you wish to purchase a piece of machinery, you compare the present cost to the present value of all it's future benefits. Usually, you would discount the profit stream by 1 + real interest rate. (nominal interest = real interest rate + inflation)

However, I have been given a question where they use the nominal interest rate and they try to explain it but I am not quite sure why.

Here it is; (included is the lecturer's answer, ne is pi which is the rate of inflation, r the real interest rate)

2. A firm has the opportunity to buy a new machine which it calculates would add £100 to its profits in a year’s time, £200 to its profits in two year’s time, and £150 to its profits in three year’s time. The machine would then become worthless. It costs £x and the firm would have to take out a loan of that amount to buy it. The nominal interest rate charged on the loan would be 15% and the firm expects the inflation rate to be 10% in each of the next three years.

(a) What is the value of £x that would make the firm just willing to buy the machine?
(b) What would the value of £x be if the firm’s expected an inflation rate of only 5% over the next three years whilst all the values of all the other variables given in the question remained as before?
(a) The standard present value calculation suggests that the present real value of the future stream of profits the firm anticipates will be:

Notice this calculation uses the nominal interest rate as the discount factor. This is the correct interest rate to use here. To see why: If we the take the current period- i.e. the period when the machine is to be purchased – as the base year then the £100, £200 and £150 have first to be converted into expected real values by discounting them by (1+πe), (1+πe)2 and (1+πe)3 respectively where πe is the expected inflation rate. Then these terms have to be further discounted by (1+r), (1+r)2 and (1+r)3 where r is the real interest rate. This gives us the expected present value of the future profit streams measured in the same prices as the present cost of the machine. This in effect means that the discount factors are (1+r)(1+ πe), ((1+r)(1+ πe))2, and ((1+r)(1+ πe))3. But the nominal interest rate is strictly defined such that as (1+i) = (1+r)(1+ πe). Hence since the future streams of profits are expressed in nominal terms it is correct to discount them by the nominal interest rate.

(b) It would be the same. The previous argument explains why. If the nominal interest rate is still 15% and the expected inflation rate now only 5% then the real interest rate is 10%. So (1+r)(1+ πe) would be the same as before. The calculation would therefore result in the same answer.

No what I THINK has happened here, is that the answer implies you discount future streams of profits if they are in nominal terms with the nominal interest rate, and if they are in real terms with the real interest rate. However I do not understand his argument.

For example; (1+i) = (1+r)(1+ πe) This identity does not hold up. What does he mean by it? And why can't you just discount it by 1 + the real interest rate? It makes sense logically to do it that way. Can anyone shed light on this?
2. Present Value and Discounting
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