David K.H. Begg, Gianluigi Vernasca-Economics-McGraw Hill Higher Education (2011)

11.2 Rentals, interest rates and asset prices
N
PV =L
R
t
t=I (I+ i)t
where R 1 is the revenue in year t, i is the interest rate and 2, is a symbol that means the sum of each year's
discounted earnings R/(l + i)f.
For example, suppose a firm wants to buy today a machine that costs £8000. The machine can give a revenue
of £2000 a year for four years. After four years the machine can be sold as scrap for £3000. Assume that the
interest rate is 10 per cent in all four years. The present value of this stream of future revenues is:
2000 2000 5000
+ +
(1+0.1) (1+0.1) 2 (l+0.1)3 (1+0.1) 4 = £8388. 7
PV = 2000 +
In this case, the present value of the future revenues from the machine is higher than the cost of buying the
machine. The firm should indeed buy the machine in this case.
The difference between the present value of a stream of revenues from a given investment minus the actual
cost of that investment is called the net present value (NPV). In our case, the net present value from buying
the machine is NPV = 8388.7 - 8000 = £388.7.
This provides a rule for investment decisions: you should invest in a particular project if the net present value
of that project is non-negative.
An alternative way to assess whether an investment should be undertaken is given by the calculation of the
required real rate of return. This is discussed in Section 11.5.
When an asset is a perpetuity, earning £K a year for ever, formula ( 1) implies that the present value of this
stream is £K/i.
Real and nominal interest rates: inflation and present values
Thus far we have discussed future payments valued in nominal terms. The first column of Table 11.5 shows
rental receipts in actual pounds. The interest rate of 10 per cent tells us how many actual pounds we earn
by lending £1 for a year.
At a nominal interest rate of 10 per cent, £100 lent today accumulates to £110 by
next year. But we want to know how many goods that £110 will then buy. This is
what really matters for the lender.
The nominal interest rate
tells us how many actual
pounds are earned by lending
£ 1 for a year.
Suppose the nominal interest rate is 10 per cent and inflation is 6 per cent when
the lender receives back the money lent. Lending £1 for a year gives £1.10. Since
inflation is 6 per cent, it costs £1.06 to buy goods we could have bought for £1 today. With £1.10 to spend
next year, our purchasing power rises by only 4 per cent. The real interest rate is 4 per cent. Thus
Real interest rate = nominal interest rate - inflation rate
Consider another example: nominal interest rates are 17 per cent and inflation
is 20 per cent. Lending £100 for a year, you get £117. But it will cost you £120 to
buy goods you could have bought today for £100. You are worse off by 3 per cent
by delaying purchases for a year and lending your money at the apparently
(2)
The real interest rate on
a loan is the extra quantity of
goods that can be purchased.
high rate of 17 per cent. Real interest rates are negative. The real interest rate is -3 per cent. In real terms,
it costs you to be a lender. The nominal interest rate does not compensate for higher prices of goods
you ultimately wish to buy. Notice that the nominal interest rate cannot be negative whereas the real interest
rate can.
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