Valuation Mathematics - Part One - Compound, Discounting, All Risk Yield and other dastardly computations

No joke, so lets get straight through the hard stuff.

Simple interest
The amount charged for a loan, shown as a percentage of the sum borrowed. Or the amount paid by a bank or building society to a depositor on funds deposited and shown as a percentage.

Simple interest occurs when all interest calculations are based upon an initial sum invested.

P(1+in)
Where P = Principal sum invested
Where i = Interest rate per time period expressed as a decimal number
Where n = Number of time periods over which interest accrues

E.G. £30,000 invested for the last 3 years at 5 per cent per annum
£30,000 x (1 + 0.05 x 3)
£30,000 x (1 + 0.15)
£30,000 x (1.15)
£34,500

Compound Interest
If interest rates are discussed in connection with property and investment, then these interest rates are calculated on compound and not simple basis. Compound interest is calculated at the end of each time period over which interest is based (usually yearly) on the accumulated sum of principal plus interest up to that date. I.E. - Interest earned on interest.

P(1+i)
Where P = Principal sum invested
Where i = Interest rate per time period
Where n = Number of time periods and shown as a small n at the top of the closed bracket

E.G. £30,000 is invested at 5 per cent per annum compound interest for the last three years
£30,000 x (1 + 0.05)3
£30,000 x (1.05 x 1.05 x 1.05)
£30,000 x (1.157625)
£34,728.75

As the calculation finds out how much a sum of money invested for a specific time period will amount to at a stated interest rate, this is known in Parry's Valuation and Conversion Tables as the "Amount of £1"

Concepts of Compounding and Discounting

Compounding is the adding of compound interest to an invested sum, the total amount of money invested increases over time.

Discounting is the inverse of compounding, the present value of an investment is found allowing for compound interest that would be earned on it over future time. The further into the future that a sum of money would be receivable and the higher the interest rate, the greater the discount factor will be and the lower the present value of the investment. A bit complicated but stick with it. In valuation terms, discounting allows for the purchase now of an interest, which will produce income or capital in the future.

Put in simple terms, somebody would pay less than £1 today for the right to receive £1 in three year's time. To work this effect out, we use the Present Value of £1 (PV)

PV = 1 / (1 + i) n
Where PV = Present Value of £1
Where i = Interest rate per time period (usually per annum) expressed as a decimal
Where n = Number of time periods (usually years) which will elapse before the sum of £1 is received.

E.G. An investment will return the sum of £50,000 in six year's time. What would be the present value of this be allowing for discounting at the interest rate of 6 per cent per annum?
PV of £1 in 6 years @ 6% = 1 / (1 + 0.06) 6
= 1 1/1.418519
= 0.70496 x £50,000
= Present Value of £35,248

All Risk Yield (ARYs) and implied risk/growth allowances

In property valuation the All Risk Yield (ARY), or Market Yield, is the standard comparison measure of the rate of return from investment. Lower yields are associated with relatively risk free investments, more uncertain and risky investments will have higher yields. As the capital value increases, the ARY decreases.

Freehold market yields are customarily based on the market rent (MR). Where MR is not receivable or it is a leasehold valuation, the yield will require adjustment. Property yields are influenced by, though not directly determined by, the level of interest rates in the economy. If interest rates are falling, investors are able to bid higher prices for property as borrowing the purchase funds are becoming relatively cheaper. This will result in a decrease in property yields.

The Amount of £1 calculations
This formula calculates for every £1 how much money will accumulate, principal plus compound interest, over a specific period.

A = (1 + i) n
Where A = Amount of £1
Where i = Interest Rate (such as ARY) per time period (per annum) expressed as a decimal.
Where n = number of time periods over which compound interest is to be added to the principal sum

E.G. Where the monthly interest rate of 1% is charged on £1
= (1 + i) n
= (1 + 0.01) 12
= 1.1268

The Amount of £1 per annum calculations
This calculates the amount that will accumulate at a specific rate of interest if £1 is invested at the end of each year for a given number of years. The formula is shown below:

[(1 + i) n - 1] / i
Where i = annual interest divided by 100 and expressed as a decimal number
Where n = number of years over which the annual sums are invested

E.G. £1 per annum is invested at the end of each year for five years at 5% per annum.
= [(1 + 0.05) 5 - 1] / 0.05
= 0.2762816 / 0.05
= £5.525632
= SAY £5.526

If the sum where £2,000 per annum, then it would be...
= £2,000 x 5.525632
= £11,051.26