Valuation Maths

TIME VALUE OF MONEY: COMPOUNDING & DISCOUNTING

Introduction• Investment depends on time value of money - rather have money

now than later– Why is £1 today not equivalent to £1 in a year’s time?

• Compounding and discounting measure the value of money over time in opposite directions

• Relationship between capital invested, future cash flows and time forms the basis of investment appraisal

• Two main types of payments to consider: single and multi-sum– E.g. buy a property today and sell it in five years’ time– Pay rent each quarter– Pay back a loan instalment each month

Time value...• A sum of money to be received in the future is not worth its face value

but a sum less than that - the actual amount depending upon the interest rate (or, more specifically, the discount rate)

Compounding...• Calculating the interest that would be earned by the original capital

and adding interest each year• Thus, the size of the future sum is determined by a combination of the

compound rate (i.e. the rate at which the original sum could earn interest) and the deferment period (the delay in payment)

Discounting...• Discounting is the opposite of compounding and is a basic concept

underpinning valuation

£5,000£ ??

now 1 year

£

time

Single sum payments(the return is assumed to accrue in arrears in the following formulae)

• Future value, FV:

• Present value, PV:

where:– n is the number of time periods over which the investment is

held– r is the rate of return– PV is the present value of the investment– FV is the future value if the investment after n periods

PVrFV n 1

n

n rFVr

FVPV

)1(1

Multi-sum payments

Level annuities (in arrears) where PMT is the regular payment

also known as a sinking fund

r

rPMTFV

n 11

11 nr

rFVPMT

r

rPMTPV

n

11 also known as the PV £1 per annum or Years Purchase (YP), i.e. the number of years that will pass before purchase price is recouped

• As well as discounting individual sums, we can discount a series of future sums (an annuity)

• This forms the basis of property valuation since a property bought as an investment will produce a series of rental incomes and the sum paid will reflect the amount and timing of these rents

Present value of real estate annuities

• Your client proposes to purchase a property which will produce a rent of £5,000 pa for the next five years

• Assuming a discount rate of 10% pa, what is the capital value of these rents?

• We can use the PV of an annuity formula to value or ‘discount’ each rent payment in turn. Looking at the latter...

Rent Receivable Discounted Value @ 10%£5,000 immediately £5,000£5,000 after 1 year £4,545£5,000 after 2 years £4,132£5,000 after 3 years £3,757£5,000 after 4 years £3,415Total discounted value £20,850

A note on PVs...• The size of the present value will depend upon the time period

involved and the interest rate (in this case known as the discount rate)• The differences in discounted value of incomes are more pronounced

in the short term than in the long term– E.g. notice in the table that the difference between the present value of year 2 and 3

rents was £375 whereas if the table had been extended to include rents receivable in say 20 years, the difference between the present values of the rents in year 19 and 20 would be only £74 (assuming the rent payable was still £5,000)

Value Discounted Value

Time

• So, in valuation terms, it is the rents receivable in the early years that primarily dictate the overall value of the interest unless for some reason there is a substantial reversionary value expected e.g. when a landlord is able to regain possession after a long lease and perhaps redevelop the property

• The discount rate is crucial too: the five incomes above discounted at 9% pa would have had a capital value of £21,198

• In selecting the appropriate discount rate a valuer must be mindful of all of the following factors:

– type of property– status of the tenant– nature of the lease (particularly the rent review)– yield on alternative investments– anticipated rental growth– when income is first received & frequency

YIELDS

(Present) values and yields• As n gets bigger the PV of a level annuity (also known as the YP or PV

of £1 per annum) simplifies to 1/r• When looking at property investment transactions that have recently

taken place in the market it is possible to substitute r to identify the market rate of return, known as the yield y given a price P. The equation remains the same but the notation changes:

yP

1

y

MRP or for any market rent

(MR) other that £1

Yields• A yield is the relationship between how much money is put in to an

investment (initial cost and subsequent expenditure) relative to how much comes out (income and eventual sale price)

• The yield is a reflection of the attractiveness or risk attached to an investment

• All investors like to receive the highest possible yield compatible with the perceived risk.

• Yields are the main yardstick of property performance• Investing institutions make buying/selling decisions on the basis of

yields rather than cost• The effect of market forces results in popular properties having lower

initial yields than others

Simple investment analysis: yields

• By rearranging the previous equation, we can isolate y:

• Ratio of annual income to capital value• Used to describe the quality of an investment• Used by valuers as a unit of comparison• Known as an initial yield

P

MRy

Yields and valuation• Formula can be rearranged to help value property investments

Example

9,000m2 office block in central Bristol

Market rent is £250/m2

What is the rental value?

If investors require a return (or yield) of 8%

What is the capital value?

What if investors require a yield of

6%, 7%, 9%...?

y

MRV

What do you observe?• An inverse relationship• As yields increase values fall• As yields decrease values rise• Small changes in yield create large changes in value• How does this impact land value?