RE5003 Real Estate Investment
Revision
Financial Mathematics
Sing Tien Foo
Department of Real Estate
National University of Singapore
1
RE 4212 Real Estate Securitization
REVISION
Sing Tien Foo, Dept of Real Estate, NUS
Sing Tien Foo, Dept of Real Estate, NUS 2011
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Financial mathematics




Concept of earning interest on interest!
It implies that interest paid on a loan, or income /dividend received for
an investment is added to the initial principal
Four basic components are used in any compounding problem:
o
PV = Initial Amount / Present Value
o
n = Time
o
i = Interest Rate
o
FV = Future Amount / Future Value
Given any three of the above, we can solve for the fourth
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Sing Tien Foo, Dept of Real Estate, NUS
RE 4212 Real Estate Securitization
The value of $1 today
Which option will you choose?
B
A
Spend $1 today
Why do you choose “A” over “B”?
Lend out $1 today
and receive $1.1 at
the end of 1 year
Why do you choose “B” over “A”?
Sing Tien Foo, Dept of Real Estate, NUS
Sing Tien Foo, Dept of Real Estate, NUS 2011
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RE 4212 Real Estate Securitization
Compounding Effects


What is the value of $10,000 that earns an interest rate of 10% at
the end of 3 years?
In a simple interest case (no compounding effects):
o
o

FV = PV (1+i)n
o
o
o
o

FV3 = $10,000 + ($10,000 x 10% x 3)= $13,000
Note: Interest expenses: $10,000 x 10% x 3 = $3,000
Year 1: FV1 = $10,000 x (1.1) = $11,000
Year 2: FV2 = FV1(1+i) = $11,000 x (1.1) = $12,100
Year 3: FV3 = FV2 (1+i) = $12,100 x (1.1) = $13,310
FV3 = PV (1+i) (1+i) (1+i) = PV (1+i)3 = $10,000 x (1.1)3 = $13,310
Difference = $13,310 – $13,000 = $310 is attributed to the
COMPOUNDING effects!
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Sing Tien Foo, Dept of Real Estate, NUS
RE 4212 Real Estate Securitization
What is “interest on interest”?

How much will a lender expect to receive if a
borrower of $1 defers the payment till the end of
year 2?
Interest
$0.1
Interest
$0.1
$1.1 + $0.1
= $1.20
$1.0
Why?
Principal
$1
today
Year 1
$1.0 x (1.10)
= $1.1
Year 2
Sing Tien Foo, Dept of Real Estate, NUS
Sing Tien Foo, Dept of Real Estate, NUS 2011
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RE 4212 Real Estate Securitization
Assumption: discount rate is constant

He will reinvest the “interest” received at the end
of year 1 and earn the same rate of return
Interest
$0.1
$0.1 x 10%
= $0.11
$1.10
$1.0
X(1.10)
Principal
$1.0
today
Year 1
Sing Tien Foo, Dept of Real Estate, NUS
$1.21
$1.0 x 10%
= $1.1
Year 2
Sing Tien Foo, Dept of Real Estate, NUS
RE 4212 Real Estate Securitization
Future value formula
FVi ,n  PV (1  i ) n

(1+i)n is known as the Future Value Interest Factor

Notation: FVIFi,n

When you borrow money from a bank, you are expected to repay the
loan with interest.

FV=$10,000(1+.10)1=$11,000

Interest over 1 year = ($10,000) x 10% = $1,000

FV = PV + Interest = $10,000 + $1,000 = $11,000
Sing Tien Foo, Dept of Real Estate, NUS
Sing Tien Foo, Dept of Real Estate, NUS 2011
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RE 4212 Real Estate Securitization
Be a millionaire!
Growth in money over time
Future Value (FV(PV=10,000, i=10%,n))
1200000
At n = 49, FV = 1,067,190.57
1000000
800000
600000
400000
At n =20,
FV = $67,275.00
200000
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Sing Tien Foo, Dept of Real Estate, NUS
RE 4212 Real Estate Securitization
Interest rate sensitivity
$1,000,000
$900,000
FV ($10,000, i, n)
$800,000
$700,000
$600,000
$500,000
$400,000
$300,000
$200,000
$100,000
$0
1
6
11
16
21
26
>25
5%
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Sing Tien Foo, Dept of Real Estate, NUS 2011
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>32
10%
41
46
51
56
61
66
71
76
81
>48
86
91 Year
>94
15%
20%
Sing Tien Foo, Dept of Real Estate, NUS
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Present Value of a Lump Sum

Discounting is an important concept of time value of
money





What is the value today of a future payoff? If you expect to
receive $13,310 from an investment three year from today
How much are you willing to accept today in exchange for the
payoff in 3 year time?
What are risks involved in waiting for the deferred payment?
What is the adequate discount rate to compensates you for the
delayed payment?
If you are willing to accept $10,000 today in return for
“giving up” your payoff of $13,310 at the end of 3 year, if
your discount rate is 10%. Why?
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Sing Tien Foo, Dept of Real Estate, NUS
RE 4212 Real Estate Securitization
Present Value formula
FV

From the FV formula:

Re-arranging the formula:

i ,n
PV
 PV ( 1  i ) n
i,n

FV
(1  i )
Computing the PV using the above formula:
n
PV10%,3 
13,310
 $10,000
(1  10%)3
Present value=?
Today, t = 0
3 years
10% p.a.
DISCOUNTING
Sing Tien Foo, Dept of Real Estate, NUS
Sing Tien Foo, Dept of Real Estate, NUS 2011
$13,310
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Present Value Factor

PVIF is the reciprocal of FVIF:
PVIF
i ,n
PVIF10%,3 

1
FVIF


i ,n
1
(1  i ) n
1
1

 0.7513
FVIF10%,3 1.331
PV = FV x PVIF10%,3 = $13,310 x 0.7513 = $10,000
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Sing Tien Foo, Dept of Real Estate, NUS
RE 4212 Real Estate Securitization
Annuities



Instead of a single lump sum payment, we receive a
series of payments made at equal intervals
The series of payments is known as Annuity
There are two types of annuity
o
o
Annuity Due = payment at the beginning of period
Regular Annuity= payment at the end of period
Sing Tien Foo, Dept of Real Estate, NUS
Sing Tien Foo, Dept of Real Estate, NUS 2011
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Future Value of Annuity Due (FVAD)
If Mr X borrows $10,000 per year at the beginning of each year for 3
years, how much will he owe at the end of 3 years?
Assume three separate loans of $10,000 each. First loan is for 3
years, second loan is for 2 years and the third loan is for 1 year.
Work out the FV for each loan:



$10,000
$10,000
now
1 year
$10,000
2 years
3 years
Future value?
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Sing Tien Foo, Dept of Real Estate, NUS
RE 4212 Real Estate Securitization
FVAD computation





Loan 1: FV = $10,000 (FVF10%,3)=$13,310
Loan 2: FV = $10,000 (FVF10%,2)=$12,100
Loan 3: FV = $10,000 (FVF10%,1)=$11,000
Total FV = $13,310+12,100+11,000=$36,410.
So the future value of annuity due (FVDA) is
FVAD i ,n  ANN (1  i ) n  ANN (1  i ) n 1  ANN (1  i ) n  2  ...  ANN (1  i )1
 ANN
n
 (1  i )
t
t 1

where ANN=amount of annuity
Sing Tien Foo, Dept of Real Estate, NUS
Sing Tien Foo, Dept of Real Estate, NUS 2011
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FVAD simplified

The FVAD formula:
(1  i ) n 1  (1  i )
i
FVAD i , n  ANN

Example:
FVAD10%,3  ANN
(1  0.1) 4  (1  0.1)
 $36,410
0 .1
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Sing Tien Foo, Dept of Real Estate, NUS
RE 4212 Real Estate Securitization
Future Value of Regular Annuity (FVRA)


The difference in FVRA is that the annuity payment
occurs at the end of the period.
If you save $10,000 at the end of each year for a down
payment of property, and if the savings could earn you
an interest rate of 10%, how much will you accumulate
after 3 years?
$10,000
now
1 year
$10,00
0
2 years
$10,000
3 years
Future value?
Sing Tien Foo, Dept of Real Estate, NUS
Sing Tien Foo, Dept of Real Estate, NUS 2011
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Estimation FVRD





Loan 1: FV = $10,000 (FVF10%,2)=$ 12,100
Loan 2: FV = $10,000 (FVF10%,1)=$ 11,000
Loan 3: FV = $10,000 (FVF10%,0)=$ 10,000
Total FV = $12,100+11,000+10,000=$33,100
The FVRD (or in short Future value annuity, FVA)
formula
FVA i , n  ANN (1  i ) n  1  ANN (1  i ) n  2  ...  ANN (1  i ) 0
 ANN
n 1
 (1  i )
t
 ANN
t 1

Why is the last term raised to the power of 0?
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Sing Tien Foo, Dept of Real Estate, NUS
RE 4212 Real Estate Securitization
FVA formula

The FVA formula is defined as:
FVA i , n  ANN FVAF i , n 
FVA i , n
(1  i ) n  1
 ANN
i
FVAF i , n 

(1  i ) n  1
i
FVA is more commonly used, as annuities are assumed to be regular,
unless annuity due is otherwise stated.
Sing Tien Foo, Dept of Real Estate, NUS
Sing Tien Foo, Dept of Real Estate, NUS 2011
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Sinking Fund Factor (SSF)

Suppose you wish to accumulate $33,100 by the
end of 3 years? How much must you set aside if
i=10%?
ANN?
now
ANN?
1 year
ANN?
2 years
3 years
Future value
= $33,100
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Sing Tien Foo, Dept of Real Estate, NUS
RE 4212 Real Estate Securitization
SFF formula
ANN i , n  FVA
i
FVA

n
(1  i )  1 FVAF i , n
ANN10%,3  33,100
SFF i , n 
1
 $10,000
FVAF10%,3
1
FVAF i , n
Sing Tien Foo, Dept of Real Estate, NUS
Sing Tien Foo, Dept of Real Estate, NUS 2011
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Present Value of Annuity (PFA)


Now we want to find the present value of an annuity
rather than the future value.
Suppose you can pay $10,000 for 3 years for a loan
which you take now. What is the loan amount if interest
rate=10% compounded annually?
now
2 years
$10,000
1 year
$10,000
3 years
$10,000
Present value?
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Sing Tien Foo, Dept of Real Estate, NUS
RE 4212 Real Estate Securitization
Present Value Annuity formula

We can now define PVAF as
PVA i , n

1  1 /(1  i ) n
 ANN
i

1  1 /(1  i ) n
PVAFi ,n 
i



Since PVAF10%,3=2.4869,
PVA 10 %, 3  10 ,000 ( 2 . 4869 )  $ 24 ,869
Sing Tien Foo, Dept of Real Estate, NUS
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Mortgage constant


Mortgage constant is used to find the amount of
annuity necessary to amortise a given amount
Suppose you borrow $100,000 mortgage loan at 12%
per annum for 20 years. What is the yearly payment?
now
1 year
ANN?
20 years
2 years…..
ANN?
ANN?....
Present value =1000
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Sing Tien Foo, Dept of Real Estate, NUS
RE 4212 Real Estate Securitization
Mortgage Constant (MC) formula
PVA

i ,n
ANN
MC i ,n 
ANN
1  1 /( 1  i ) n
 ANN
i
 PVA
i ,n
i
1  1 /( 1  i ) n

1
i

PVAFi ,n 1  1 /(1  i ) n

 $ 100000
Sing Tien Foo, Dept of Real Estate, NUS
Sing Tien Foo, Dept of Real Estate, NUS 2011



. 12
 $ 13387
1  1 /( 1  . 12 ) 20


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Single period and multi-period time value factors
with m compounding per annum
Factor
Formula
Future value of lump sum factor
FVIF
i ,n ,m
i 

 1 

m

mn
(1  i / m) mn  1
i/m
i/m
SFF i , n , m 
1  i / m mn  1
1
PVIF i ,n . m 
(1  i / m ) mn
Future value annuity factor
FVAFi ,n,m 
Sinking fund factor
Present value of lump sum factor
Present value annuity factor
PVAFi ,n,m 
Mortgage constant
MC
i ,n .m


1  1 / 1  i / m
i/m
mn

i/m
1  1 /( 1  i / m ) mn


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Sing Tien Foo, Dept of Real Estate, NUS
RE 4212 Real Estate Securitization
SUMMARY
1) Lump Sum
Present
Future
PVIF
FVIF
PVAF
FVAF
MC
SFF
2) Annuity
a)Given annuity
b) Given lump sum
•

Sinking Fund Factor is to determine the periodic
amount to cumulate to a desired future value.
Mortgage Constant is to determine the periodic
amount to repay a present value.
Sing Tien Foo, Dept of Real Estate, NUS
Sing Tien Foo, Dept of Real Estate, NUS 2011
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RE 4212 Real Estate Securitization
Reference

Brueggeman and Fisher, chapters 2-6
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Sing Tien Foo, Dept of Real Estate, NUS 2011
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