Years purchase - A mathematical method to value real estate/property - UK
The present value of a £1 p.a is usually referred to by real estate appraisers/uk chartered surveyors as 'years purchase'. The Oxford English Dictionary gives a date of 1584 for the use of the prhase 'at so many years purchase' which was used in the stating of the price of land in relation to the annual rent in perpetuity.
The PV £1 p.a increases as the number o years increases, however as it approaches a maximum value as a definite time period is reached. This time period is referred to as infinity, but is often assume to be a 100 years to simply. In real estate appraisal language this referred to as in 'perpetuity'. This is maximum value of the PV of £1 per annum as the present value of £1 p.a in perpetuity, or the years purchase in perpetuity (YP perp.). So this formula will give the present value of the right to receive £1 at the end of each year in perpetuity at i. Real estate investments will often produce perpetual incomes.
In the formula 1 - PV / i what happens to the PV as the time period increase? What changes does this hae on the YP number? Derive the formula for years purcahse in perpetuity and compare with 1 - PV / i
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Years PV @ 10% PV £1 p.a @ 10%
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10 0.38 6.14
20 0.14 8.51
30 0.05 9.42
40 0.02 9.77
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From the table above we can observe 2 things. Firstly as the PV decreases as the n increases, and the second as that, in accordance with the fomula the YP increases with time. It therefore follows that the PV inherent in the YP formula has an important role to play as it reduces the value of the YP as n decreases and thus allows Yp to increase as n increases. As n approaches perpetuity, Pv tends towards 0 (the present value of £1 receivable in an infinite number of years time is infinitely small) and 1 - PV / i tends towards 1 - 0/i. The formula for YP in perpetuity is therefore 1/i.
For example at a interest rate of 10%, the YP in perp is 1 / .10 which equals 10.
Another example
Lets say a property produces a net income of £1000 per annum. If an investor requires a return of 10%, what price should be paid? Since the income is perpetual a YP in perpetuity should be used.
Income £1,000
YP perp @ 10% (100/10) x 10
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Current Value (1000 x 10 = 10,000) so current value = (£10,000
For more information on another variant of the present value of pound is the years purchase of a reversion to a perpetuity. It shows the present value of the right to receive £1 at the end of every year in perpetuity, after a given period of time, at i.