Introduction to Real Estate
Investment Appraisal
Maths of Finance
Present and Future Values
Pat McAllister
INVESTMENT APPRAISAL:
INTEREST
• Interest is a reward or rent paid to a lender or
investor who has provided an asset (capital,
principal) to another party – the borrower
–
–
–
–
bank lending e.g. a mortgage
a share in a company
government or corporate borrowing, bonds
property investment
• Interest may appear as “income” “profit”
“dividend” or “rent” depending on the asset / deal
• What determines interest rates? The Fisher model:
– Time Preference or Impatience
– Expected or Anticipated Inflation
– The Risk Associated with the Investment
• Other assets: opportunity cost, cost of capital
INVESTMENT APPRAISAL:
SIMPLE INTEREST
• Place £100 into an account paying 8% interest.
After one year it will contain the original £100 plus
£8 interest. 100 * (1 + 0.08) = £108.
• In the second year, it earns a further 8% on the
original sum invested. 100 * (1 + 2 * 0.08) = £116
• In general C (1 + ni) = A where
C = initial capital invested
I = interest rate
n = term
A = final accumulated amount
• e.g invest £100 for 5 years at 8%
• A = 100 (1 + 5 * .08) = 100 * 1.4 = £140
• What would a rational investor do, however?
INVESTMENT APPRAISAL:
COMPOUND INTEREST
• The rational investor withdraws the £108 at the
end of year one and reinvests it at 8%
• At year two, she has £108 (1 + .08) = £116.64
• The extra £0.64 is interest earned on earlier
interest, or COMPOUND interest
Deposit £100 earning 10% compound interest.
At
end Year 1
end Year 2
end Year 3
100 (1.1)
=
100 (1.1) (1.1)
=
or 100 (1.1)2
100 (1.1) (1.1) (1.1) =
or 100 (1.1)3
And so on………….
In general:
An = C(1 + i)n
An is the Future Value of C at i% interest.
110.00
121.00
133.10
INVESTMENT APPRAISAL:
COMPOUND INTEREST
• In Property Valuation, the An with the capital set
at £1 is known as “the amount of one pound”,
written A£1
• There are tables (e.g. Parry’s Tables) which give
values of A£1 – it acts as a multiplier for other
sums invested.
• If you invest £100 in an account bearing 5.5%
compound interest, how much will the account
hold in exactly five years?
• £100 (1 + 0.055)5 = 100(1.3070) = £130.70
INVESTMENT APPRAISAL:
DISCOUNTING
• Reverse the process: you need £1,000 in three years
and can invest at 12% (I wish). How much should
you deposit?
1
n
An C( 1 i)n...so...C An
A(
1
i)
( 1 i)n
• C = 1000 (1.12)-3 = 1000 (0.7118) = £711.80
• £711.80 is the Present Value of £1,000 received in
three years time, discounted at 12%
• £100 received today is more valuable than £100
received in four years. Why?
–
–
–
–
–
Inflation erodes spending power (inflation)
Could invest the £100 and let it grow (opp. Cost)
Will we actually receive the £100 later? (risk)
WE WANT IT NOW NOW NOW (impatience)
Could pay off other debt (cost of capital)
INVESTMENT APPRAISAL:
DISCOUNTING
• Discounting pulls back all future cashflows to a
common base period – today.
• In Property Valuation (1 + i)-n is the Present Value
of £1 - written PV£1
n
• In actuarial notation, this is written
i
v
• This is a useful, universal notation. It is NOT a
formula
In summary:
Future Value (1+i) n
C0 0
n An
Discounting (1+i) -n
INVESTMENT APPRAISAL:
INTEREST RATE CHANGES
• Suppose an investment earns 8% for three years
then 10% for two years (assuming compound
interest). Future value is:
An = C (1.08) (1.08) (1.08) (1.1) (1.1)
= C (1.08)3 (1.1)2 = C (1.2597) (1.21)
= C (1.5243)
• What would the “average” rate be over the whole
period?
An
= C(1 + i*)n So (1 + i*)5 = 1.5243
So (1 + i*) = (1.5243)(1/5)
So i*
= (1.5243) (1/5) -1
= 0.088 or 8.8%
Check: (1.088)5 = 1.5246 (rounding error only).
INVESTMENT APPRAISAL:
INTEREST PAID LESS THAN ANNUALLY
• What is Annual Rate of Interest Charged at 2% per
month?
• (1+.02)12 – 1 = 26.8% per annum NOT 24% NB
• However, what is “10% per annum credited
quarterly”? In fact, this is 2.5% per quarter.
• (1.025)4-1 = 10.38% (and not 10%)
• Distinguish between NOMINAL & EFFECTIVE rates
– the published APR is an Effective Rate
• In general, where i(p) is the nominal rate payable
(“convertible”) p’thly (so p = 12 is monthly, p = 4
quarterly) then the annual effective rate is given
by:
( p)
i [1
i
p
]
p
1
INVESTMENT APPRAISAL:
INTEREST PAID LESS THAN ANNUALLY
• So a loan repayable monthly at a nominal 14.5%
has an effective rate of:
.145 12
[1
] 1 .155 15.5%
12
• Remember that i(p) is not a formula, it is a number
– the p is the number of times that interest is
added in the year.
• For commercial property, in the UK rent is payable
quarterly. In many other countries, it is paid
monthly.
• This means it is important to be able to carry out
calculations that are not annual in nature.
INVESTMENT APPRAISAL:
INFLATION
• Suppose inflation is running at ƒ% per annum.
Then the purchasing power of £X received in one
year is X(1+ ƒ)-1 or X / (1 + ƒ). So £100 received
one year hence with inflation running at 3.5% is
worth 100 / (1.035) = 96.62.
• Now, if an investment pays j% per annum
compound interest, with inflation running at ƒ%
per annum, then the real accumulated values for
years 1,2,n will be, respectively:
(1 j )
(1 f )
(1 j )
(1 f )
2
(1 j )
(1 f )
n
INVESTMENT APPRAISAL:
INFLATION
• There must, thus, be a real interest rate i% such
that:
(1 j )
(1 f )
• Hence:
i
(1 j )
(1 f )
(1 i)
1...
j f
(1 f )
• And NOT simply j-f
• With inflation at 8% and interest at 15%, the real
rate =
i
j f
(1 f )
.15 .08
1.08
.065
• Notice it is NOT 15%-8% = 7%. This becomes
important whenever inflation is high.
Investment Appraisal – Warm Up
• You deposit £200 in an account paying 6%
interest. How much will you have in the account in
five years time?
• An investment pays £5,000 in four years time. If
the appropriate discount rate is 8%, what is the
present value of the investment?
• An investment pays interest at 1.5% per month.
What is the nominal annual rate? What is the
effective annual rate?
Investment Appraisal – Warm Up
• Future value: 200 (1.06)5 = 200(1.3382) = £267.65
• Present value: 5,000 (1.08)-4 = 5,000(0.7350) =
£3,675.15
• Nominal rate is 18%
• Effective rate is (1.015)^12 - 1 = 19.56%
INVESTMENT APPRAISAL:
ANNUITIES
INVESTMENT APPRAISAL
ANNUITIES
•
Annuities are regular payments made (or received).
We may want to know:
– How much a series of deposits earning interest will be
worth in n years (a future value).
– What is the cash equivalent in today’s money of n
payments at £X (a present value).
•
•
There are obvious applications here in investment
and in loans.
Annuities, as the name implies, are traditionally
annual payments. An ordinary annuity is received at
year end, an annuity due is received at the start of
each year. The basic calculations can be applied to
other time periods.
INVESTMENT APPRAISAL
ACCUMULATION: ORDINARY ANNUITIES
• Invest £1 at the end of each of next 10 years, earn
compound interest of 10%. How much will be in
account at end?
0
£1
£1
£1
£1
1
2
8
9
£1
10
• First pound worth £1 (1.1)9; second pound £1(1.1)8
and so on. The last pound earns no interest and is
worth £1:
• In total
£1(1.1)9 + £1(1.1)8 + … £1
• Or £1 [(1.1)9 + (1.1)8 + … (1.1) + 1]
• More generally, with n years and i% interest:
• £1 [(1 +i)n-1 + (1 + i)n-2 + … (1 + i) + 1]
• This can be used as multiplier for any amount invested
INVESTMENT APPRAISAL
ACCUMULATION: ORDINARY ANNUITIES
• The term in the square bracket is the “accumulation”
“accumulated value” or future value. Valuation
terminology calls this the “amount of £1p.a.” written
A£1pa
• Actuarial notation:
n| i %
s
• Now the term is a geometric series and so we can
reduce it to a simple formula. See the handout for the
algebraic derivation.
(1 i ) n 1
s n| i % =
i
• In original question n = 10 and i = 10%
• So accumulated value = [(1.1)10-1] / 0.1 = 15.94
• Note that this is £10 of capital invested and £5.94
accrued interest.
INVESTMENT APPRAISAL
PRESENT VALUES, ORDINARY ANNUITIES
• What is value today of £1000 received at end of each
of the next 8 years if your discount rate is 10%?
£1000
0
1
£1000
£1000
2
6
£1000
£1000
7
8
1
• The first payment is worth £1000
10%
2
• The second payment is worth £1000 10% and so on
• The year eight payment £1000
• In total, we have: 1000
• Which is:
1000[
1
10%
1
10%
8
10%
1000
2
10%
2
10%
...
...1000
8
10%
]
8
10%
INVESTMENT APPRAISAL
PRESENT VALUES, ORDINARY ANNUITIES
• Generalising the square bracket for n years and i%
a n| i %
[
1
i%
2
i%
...
n
i%
]
This is the Present Value of an Ordinary Annuity
or, in UK valuer speak, the Years Purchase
• As before, this is a geometric progression, which
simplifies to the most important valuation formula
you will meet:
n
•
a n| i %
1 vi
i
• In our example, n = 8 and i = 10%
• S0: present value =£1000 a 8| 10%
•
1000 1 1.1
0 .1
8
= 1000 (5.3349) = £5,335
INVESTMENT APPRAISAL
PRESENT VALUES, ORDINARY ANNUITIES
• Note that the total payments received are £8,000 but
the present value to us is only £5,300.
• The lower value reflects the impact of anticipated
inflation, time impatience, alternative investment
opportunities, borrowing costs and the risk
associated with this type of cashflow
• You can acquire a property lease, making an annual
profit of £1000. The lease has six years to run and
alternative investments have returns of 12%. How
much should you pay?
• £1,000
6| 12% = £1,000 (4.1114) = £4,111.41
a
• Valuation:
Profit Rent
£1,000
YP 6 years @ 12% 4.1114
Value
£4,111.41
INVESTMENT APPRAISAL
PRESENT VALUES, ORDINARY ANNUITIES
• We can show that we get our capital back and
receive 12% on the outstanding capital:
YEAR
CAPITAL O/S
AT START
INTEREST
@12%
CAPITAL
RETURNED
1
4111.41
493.37
506.63
2
3604.78
432.57
567.43
3
3037.35
364.48
635.52
4
2401.84
288.22
711.78
5
1690.06
202.81
797.19
6
892.86
107.14
892.86
• Why is this just like a residential mortgage?
INVESTMENT APPRAISAL
PERPETUITIES
• A freehold property gives a (theoretical) income
“forever”; so does owning a share in a company.
How can you value this?
•
a
Well n| i %
1 vin
=
i
n
i% must tend to zero
• If n is infinitely large then
since (1+i)-n = 1/(1+i)n
• So, “in perpetuity” the present value becomes
simply 1
i
• £1,000 received in perpetuity, discounted at 10%
has a present value of 1,000 / 0.10 = £10,000.
• In property valuation, i is the yield, initial yield or
“cap rate” and 1/i is the Years Purchase. In equity
markets, i might be the dividend yield.
INVESTMENT APPRAISAL
REVERSIONS
• You are to receive £1000 per annum from the end
of years 6 to 10 inclusive, but nothing from years 1
to 5. If you discount at 10%, what is this worth?
£1000 £1000 £1000
5
0
6
7
8
£1000
9
£1000
10
• You know how to value the income stream from Yr
6 to 10 – it’s the PV of an annuity £1000
5| 10%
• That is £1000 (3.7908) = £3,791
• But that is a LUMP SUM at the start of Year Six,
which is the end of Year 5
• So we must PV that lump sum back to today:
a
•
£3,791
v
5
10%
= £3,791(0.6209) = £2,354
INVESTMENT APPRAISAL
REVERSIONS - 2
• The same principle applies to perpetuities
• Assume that you are going to receive £1000 in
perpetuity, starting at end Year Six.
• Calculation is essentially the same. The present
value of a perpetuity 1 values the income to start
i
year 6, end year 5. £1,000 / 0.10 = £10,000
• Now we must PV it back to today, five years at
10%: £10,000 (1.1)-5 = £10,000 (.6209) = £6,209
• Reversions are a very common feature of the UK
property market. You will meet “Term and
Reversion” where you receive one income for a set
period and then a higher income thereafter
CALCULATORS AT THE READY …
• If you invest £1,000 in an account paying
compound interest of 4% per annum, how much
will be in the account after 4 years?
• If the appropriate discount rate is 8%, what is the
present value of £1,000 received exactly six years
from now?
• Again assuming a discount rate of 8%, what is the
present value of a cashflow of £1,000 per annum
received at the end of each of the next six years?
• Why do you use different discount rates to
discount different types of project and investment
opportunity?
How Did You Do?
• £1000 (1.04)4 = 1000 (1.1699) = £1,169.86
• £1,000 (1.08)-6 = 1000 (0.6302) = £630.17
• £1,000
a 6| 8%
= 1000(4.6229) = £4,622.88
Since [1-1.08-6]/0.08 = 4.6229
• If they are valued in the same country and at the
same time, it’s differential risk, init? Impatience,
anticipated inflation will be the same for all
projects.
INVESTMENT APPRAISAL
SINKING FUNDS
• You need £2,000 in five years time. You can invest,
from your income and earn 10%. How much should
you invest each year?
• Well X s 5| 10% = 2,000 so X = 2,000 ( 1 / s 5| 10% )
• That is
•
1 / s n| i %
2000
.1
2000(.1638) 327.59
5
(1.1) 1
is the Annual Sinking Fund or ASF in
valuation terminology. It is the amount of annual
investment at i% required to produce £1 after n
years. You can then multiply it by whatever the
required end amount is.
• In UK traditional valuation, this is used extensively
in the leasehold valuation process
INVESTMENT APPRAISAL
ANNUITIES DUE: PAYMENT IN ADVANCE
• Invest £1 at the start of each of the next n years:
£1
£1
0
1
£1
£1
£1
2
n-2
n-1
•
•
•
•
•
•
n
The first £1 will be worth
(1+i)n
The second £1 will be worth (1+i)n-1 and so on
The final pound will be worth (1+i)
That is:
(1+i)n + (1+i)n-1 + … (1+i)2 + (1+i)
Now s n| i % was (1+i)n-1 + (1+i)n-2 +… (1+i) + 1
So each separate term is (1+i) higher than for an
ordinary annuity
• So accumulated (future) value of an annuity due:
s n| i %
= (1+i)
s n| i %
INVESTMENT APPRAISAL
ANNUITIES DUE
• £200 invested at the start of each of the next ten
years earning 8% interest will be worth:
200
10| 8%
s
(1.08)10 1
that is 200(1.08) .08
or 200 (1.08) 14.4866
= £3,129.10
• Exactly the same principle applies to discounting /
calculating present values.
• Each payment is received one year earlier, so it is
discounted one year less.
• Therefore the Present Value of an Annuity Due is:
an| i %
= (1+i)
a n| i %
INVESTMENT APPRAISAL
ANNUITIES DUE
• The present value of £10,000 received at the start of
the next 15 years discounted at 8% is:
15| 8% = 10,000(1.08)a 15| 8%
10,000 a
10,000 (1.08) (8.5595) = £92,442
• Exactly the same principle applies to a perpetuity.
a n |i% = 1
i
 n| i %
so a
1
(1 i )
= (1+i) =
i
i
• An investment that pays £2,000 at the start of each
year in perpetuity, discounted at 8% has a present
value of 2000 (1.08)/.08 = 2000 (13.50) = £27,000
THINGS GET GRIM
QUARTERLY IN ADVANCE
• UK commercial property rents are paid quarterly in
advance even though quoted as annual figures
• We could estimate a quarterly discount rate and
estimate the present value as an annuity due
• However, it is convenient to have a formula which
we can use as a multiplier for the annual rent.
• Effectively, we take the quarterly rent elements
and convert them into an annual equivalent rent
received at year end
• The algebraic transformation to do this is relatively
straightforward:
I Really CAN Do This, You Know …
R
Let the annual rent = R. Then the quarterly payments are
4
The “effective” rental value must take into account the early arrival of income.
Suppose our annual discount rate is i%. Then the quarterly rate is j = (1+i) ¼-1.
Then the effective annual rent is:
1
R4
s 4| j %
s 4| j %
(1+j)s 4| j %
= [(1+j)4 –1] / j
j = (1+i)¼-1 so (1+j) = (1+i)¼
But
Hence (1+j)4 =
[(1+i) ¼]4
s 4| j %
And so
s 4| j %
1
R
4
=
= (1+j)
= (1+i)
= i / [(1+i) ¼ -1]
s 4| j %
= (1+i)¼ {i / [(1+i) ¼ -1]}
This now produces an effective annual in arrears rent, which we can present value using
a n |i%
.
1
R 4 (1+j)
s 4| j % a n | i %
1
= R4
(1+i)¼ {i / [(1+i) ¼ -1]} * [1 – (1+i)-n] / i]
Cross multiplying removes i from denominator1 and numerator.
1 (1 i ) 4 [1 (1 i ) n ]
1
4
R
(1 i ) n 1
Now divide top and bottom by (1+i)¼
1
4
R
1 (1 i )
[1 (1 i )
n
1
4
]
INVESTMENT APPRAISAL
QUARTERLY IN ADVANCE
• So, the Present Value of an Income Stream
received quarterly in advance is:
a
(4)
1 (1 i )
n |i% = R
n
1
4
4[1 (1 i ) ]
• For example, £2,000 received QinA for six years,
discounted at 12% has a present value of:
2000
1 (1.12)
6
1
4
2000(4.4154) 8,830.79
4(1 (1.12) )
• For a perpetuity, the numerator becomes 1
• For other periods, change the 4s to the appropriate
number e.g. monthly, change the 4s to 12s
IS THERE NO END TO THIS SUFFERING?
CONTINUOUS INCOME
• Suppose income is received “continuously”. Then
the present value:
a n| i %
n
i
=
1 v
log e (1 i )
• What is the value of turnover at £10,000p.a.
received continuously for six years, discounting at
10%?
1 (110
. )
10,000
6
loge (110
. )
.4355
= 10,000
.0953
= 10,000 (4.5693) = £45,693
• Loge(1+i) is the “force of interest”, an important
parameter in “serious” investment mathematics
NO, REALLY, IT IS USEFUL ….
INCREASING ANNUITIES
Consider an annuity payable at year end, starting at 1 but increasing by (1+g) per
annum. What is its present value discounting at i%?
Year 1 is worth:
1
(1 i )
(1 g )
Year 2 is worth: (1 i ) 2 and so on until
(1 g ) n 1
Year n: (1 i ) n
Thus
Ia n | i % =
1
(1 i )
(1 g )
(1 g ) n 1
+ (1 i ) 2 + …. + (1 i ) n
This is yet another geometric series. Avoiding the algebra, this reduces to:
Ia n | i %
=
1 g n
1 (
)
1 i
i g
IF ONLY TO BAFFLE YOU ….
INCREASING ANNUITIES
• Now where n tends to infinity (that is, for a
perpetuity), the numerator tends to 1 (assuming
that i > g), leaving:
Ia n | i % =
1
i
g
• In equity markets this is the Gordon Dividend
Growth model: i is the required return, g is the
expected growth in dividends and d = i - g is the
dividend yield.
• The same principle applies to property. The initial
yield k = e - g where k is the initial yield, e is the
equated yield (required return) and g is the
expected rental growth. However, we need to
correct for the rent review period.
INVESTMENT APPRAISAL
INCREASING ANNUITIES
• An investment pays £10,000 per annum in
perpetuity: the income is expected to increase at 3%
per annum and the investor wishes to make a return
of 9% from this type of investment. What should she
pay?
10,000
1
.09 .03
£166,667.
10,000(16.6667 )
• An investment property is sold at an initial yield of
6.5%. Investors require a return of 9%. What rental
growth is required to achieve this?
k = e - g, so e - k = g => 9 – 6.5 = 2.5% per annum
Investment Appraisal – Warm Up
• You deposit £200 in an account paying 6%
interest. How much will you have in the account in
five years time?
• An investment pays £5,000 in four years time. If
the appropriate discount rate is 8%, what is the
present value of the investment?
• Why might the appropriate discount rate be 8%?
• A property investment provides an annual nominal
return of 15%. What is the real return, if inflation
is expected to run at 8% per annum?
Investment Appraisal – Warm Up
• Future value: 200 (1.06)5 = 200(1.3382) = £267.65
• Present value: 5,000 (1.08)-4 = 5,000(0.7350) =
£3,675.15
• Anticipated inflation, risk (of what?), returns on
alternative investments, cost of borrowing, target
returns in investment strategy
• I = (1+j)/(1+f)-1 = (1.15)/(1.08)-1 or (.15-.08)/1.08
= .0648 = 6.48%
and NOT 15%-8% = 7% except as a quick
approximation