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)
Time
compounding
discounting
Value
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)
10%
£
5%
0
1
2
YEARS
3
Discounting...
• Discounting is the opposite of compounding and is a basic concept
underpinning valuation
£
£5,000
£ ??
time
now
1 year
Single sum payments
(the return is assumed to accrue in arrears in the following formulae)
• Future value, FV:
FV  1  r   PV
• Present value, PV:
PV 
n
FV
 FV (1  r )  n
n
1  r 
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
Multi-sum payments
•
•
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
Level annuities (in arrears) where PMT is the regular payment
n

1 r  1
FV  PMT 
r
PMT  FV 
r
1  r n  1
1  1  r 
PV  PMT 
r
n
also known as a sinking fund
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
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
£5,000
immediately
£5,000
after 1 year
£5,000
after 2 years
£5,000
after 3 years
£5,000
after 4 years
Total discounted value
Discounted Value @ 10%
£5,000
£4,545
£4,132
£3,757
£3,415
£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:
P
1
y
or
MR
P
y
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:
y
•
•
•
•
MR
P
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
Yields and valuation
•
Formula can be rearranged to help value property investments
V
MR
y
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%...?
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?