VALUATION

MATHEMATICS

11 April 2018
 - VALUATION TABLE -
Understanding The Construction Of Single Rate
Table
– Amount of $1 (A)
– Present value of $1 (PV)
– The amount of $1 per annum
– Annual sinking fund (S)
– Present value of $1 per annum or year’s purchase
– Year’s purchase in perpetuity
– Year’s purchase of a reversion to perpetuity.
Dual Rate Table
– Present value of $1 per annum or year’s purchases.
– The effect of tax on the sinking fund element of the dual
rate year’s purchase.
Mortgage Installment Table

11 April 2018
 SIMPLE INTEREST FORMULA
Simple interest is the interest calculated
on the original principal for the entire
period it is invested. It is the product of the
principal multiplied by the rate and time.
This may be stated as the formula:
I=Pxrxt
Where I = simple interest
P = principal
r = rate of simple interest
t = time or term in year

11 April 2018
 SIMPLE AMOUNT FORMULA
The simple amount is the sum of the
original principal and the interest earned.
The simple amount formula is given as:

S = Original principal + Interest earned

S = P + Prt
S = P(1 + rt)
Where S = simple amount

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 COMPOUND INTEREST FORMULA

Interest earned after 1 interest period = Pi

Amount after 1 interest period = P + Pi
= P(1 + i)
Interest earned after 2 interest periods = P(1 + i)i
Amount after 2 interest periods
= P(1 + i) + P(1 + i)i = P(1 + i)²

Amount after 3 interest periods = P(1 + i)³

11 April 2018
 Continuing the steps above, we
obtain the future value, S after n
interest periods as:
S = P(1 + i)n

The factor (1 + i)n is called the

future value of $1 at the rate i per
interest period for n interest
periods

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SIMPLE INTEREST
i = 10%
$0.10 $0.10 $0.10

0 1 2 3 years
$1 $1 $1 $1
Amount = $1+0.1+0.1+0.1
= $1.30

COMPOUND INTEREST
i = 10%
$0.10 $0.11 $0.121

0 1 2 3 years
$1 $1.10 $1.21 $1.331
Amount = $1.331

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Amount of $1 (A)
– The amount accumulated if $1 is invested now (at i for n
years) at compounded interest
– Formula : A = (1+i)n

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EXAMPLES

AMOUNT OF $1

Mr. Richard has deposited $5,000 in fixed

deposit for 5 years.
What is the amount he expects to accumulate
after 5 years if the interest rate is at 8% per
annum?
$5,000 8% ?

0 5 years

Amount deposited = $5,000

A of $1 for 5 yrs @ 8% = 1.4693
$7,346.50

11 April 2018
Amount deposited = $5,000
A of $1 for 5 yrs @ 8% = 1.4693
$7,346.50

A = (1 + i)n
= (1 + 0.08)5
= 1.469

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Amount of $1 Per Annum
– The amount
accumulated if $1 is
invested at the end of
each year for n years at
i compounded interest.

– Formula:
Amount of $1
per annum = A – 1
---------
i
A = Amount of $1

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Amount Of $1 Per Annum
– Mr. Ahmad wishes to save for his children’s
education. He decided to save $1,200 per year
for the next 10 years. Interest accumulates at
10% per annum. How much would he have
saved at the end of 10 years?
$1,200 p.a. @ 10%

0 10 yrs
Amount saved in 1 year = 1,200
A of $1 p.a in
10 years at 10% = 15.94
--------
Amount accumulated = $19,128
11 April 2018
 Amount of $1 per annum = 1{(1 + i)n – 1}
(1 + i) – 1
= (1 + i)n – 1
i
or = A - 1
i
Where A = Amount of $1

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Present Value Of $1
– Mr. Richard needs $7,346.50 in 5 years time to buy
furniture for his new home. He decides to save now by
putting a lump sum in fixed deposit for the next 5 years
earning 8% per annum.
What is the amount he has to put in the fixed deposit now
to earn $7,346.50 in 5 years time?
? 8% $10,000

0 5 years

Amount to receive in 5 years = $7,346.50

Present Value (PV) of $1
in 5 yrs @ 8% = 0.6806
------------
Amount to save now = $ 5,000

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 - VALUATION TABLES -
Present Value Of $1
(PV)
– The value now of a
right to receive $1 in
n years time, allowing
for i compound
interest

– Formula :

PV = 1 OR 1
------- -----
(1+i)n A

A = Amount of $1

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Annual Sinking Fund
(ASF)
– The amount to be
invested at the end of
each year at i
compound interest to
accumulate to $1 in n
years

– Formula :

i
--------
A–1

A = Amount of $1

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Annual Sinking Fund

– Mr. Ahmad needs to have $19,128 in 10 years time for
his children’s education. He decided to put aside a
certain amount each year @ 10% p.a. How much should
he put aside yearly?

$ ? @ 10 % $19,128

0 10 years

Amount to redeem = $19,128

ASF for 5 years @ 5% = 0.0627454
-----------
$1,200

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Years Purchase In Perpetuity
– The value now of a right to receive $1 p.a. at the end of
each year at i compound interest in perpetuity.
– Formula : 1
i

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Years Purchase In Perpetuity
– What is the Market Value of Ahmad’s house if it is held
under a freehold title and the current market rent is
$10,000 per annum. The rate of interest is 10%.

now $10,000 p.a. In Perpetuity

? @ 10%

Ahmad’s income = $ 10,000

YP in perpetuity @ 10% = 10
-------------
Market Value = $100,000

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Years Purchase (Single Rate) (YP)
– The value now of a right to retrieve $1 p.a. at the end of
each year for n years at i compound interest.
– Formula : 1 – P
-------
i

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Years Purchase (Single Rate)
– Mr. Lim owns a leasehold interest with an unexpired term
of 60 years.
He receives a rent of $6,000 p.a. from his tenant. What is
the value of his interest if the rate of return is 10% p.a.?
now $6,000 p.a. 60 years

? @ 10%

Annual Income = $ 6,000

YP for 60 years @ 10% = 9.967
-----------
Value of Mr. Lim’s interest $59,802

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Years Purchase (YP) (Dual Rate)

– The value now of a right to receive $1 per annum at the
end of each year for n years at i (remunerative rate) and
a lower rate s (accumulative rate) so that the value now
can be recovered in n years

– Formula : 1
----------
i+S

PV = Present Value Of $1

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Years’ Purchase (Dual Rate)
– Miss Fatimah owns a leasehold interest with an
unexpired term of 20 years. She receives a
rent of $4,000 p.a. from her tenant. What is the
value of her interest if the rate of return is 10%
and the Sinking Fund rate is 3%.

now $4,000 p.a. 20 years

? @ 10%

Annual Income = $ 4,000

YP for 20 years @ 10% + 3% = 7.288
-----------
Value of Miss F’s interest = $29,152

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Years Purchase Of A Reversion To Perpetuity

– The value now of a right to receive, after n years, $1 per
annum at the end of each year at i compound interest in
perpetuity

– Formula :

= PV (n years) of $1 x 1
i
= 1 x 1
( 1 + i )n i

= 1
i(1 + i)n
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11 April 2018
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Years Purchase Of A Reversion To A Perpetuity
– Miss Kaur owns a freehold interest in a property. It is
currently still under construction and will be ready for
occupation in 2 years time.
The current market rent is $5,000 per annum. If the rate
of return is 10%, what is the market value of her property.
now 2 years In Perpetuity

under
? construction

Current Market Rent = $ 5,000

YP of rev. to perp.
After 2 years @ 10% = 8.264
-----------
Market Value = $41,320

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Income Tax
– Only the sinking fund is taxed.
Eg: The value of Miss Fatimah’s interest is:

Annual Rent = $ 4,000 p.a.

YP 20 yrs @ 10% + 3%
(30% tax) = 6.529
-----------
Market Value = $26,116

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Mortgage (Instalment Table)

– The amount of equal monthly installments to recover $100 that is
borrowed for n years at compound interest i calculated based on
the amount of balance annually.

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Mortgage Instalment Table

– Mr. Sim wants to take out a loan to buy a house. He
wants a loan of $100,000 for 15 years @ 10%. What is
his monthly instalment?

Loan = $100,000
Annuity $1 wp 15 yrs @ 10% = 0.13147
------------
Payment per annum $13,147

Payment per month $1,096

11 April 2018