Unit 9 – Amortisation
To determine the payments to be made given the term, interest rates and
Present Value
(The reverse operation of Annuities PV)
Terminology
• Amortisation – Calculation to determine the equal monthly, or term,
loan payments necessary to provide the lender with a specified
interest return and to repay a loan capital over a specific period.
This factor or multiplier is used to determine the instalment to
amortize the amount of a loan or mortgage bond.
• A payment on a Present Value.
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In Practice
• Example – You would like to purchase a house via a bond/loan with a
Financial Services Provider (purchase price = PV). Calculate the equal
monthly loan payments (P=?) given a specified interest (i) and a
specific period (n).
Note, the number of multiple compounding periods, m, will always be
the same for payments and interest (in this course)
• NOTE: Compare the formula to the Present value of an Ordinary
Annuity - This is, in fact, the reverse operation of PVord Ann
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Amortisation
𝐏=
𝒊
𝐏𝐕[
]
𝟏−(𝟏+𝒊)−𝒏
Where
PV = Present Value (The purchase price, today)
P = Payment per period
i = interest rate, as a decimal
n = number of periods, years in this case
*** Note that you will be given this formula and will need to adjust it for multiple compounding periods, m.
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Amortisation, adjusted for multiple compounding periods
𝒊
𝒎
𝐏 = 𝐏𝐕[
]
𝒊 (−𝒏𝐱𝐦)
𝟏 − (𝟏 + )
𝒎
Where:
PV = Present Value (The purchase price, today)
P = Payment per period or instalment
i = interest rate, as a decimal
n = number of periods, years in this case
m = multiple compounding periods (e.g. monthly m=12; annually m=1, quarterly m=4 etc)
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Practical Examples
𝒊
𝐏 = 𝐏𝐕[
] ***remember you have to adjust the formula
−𝒏
𝟏−(𝟏+𝒊)
PV = R 1 000 000 (the price of the house you wish to buy, today)
P = ? (bond repayments)
i = 7% or 0.07 compounded monthly (i.e. m=12)
n = 20 yrs (Standard home loan repayment period)
𝒊
𝐏 = 𝐏𝐕[
−𝒏 ]
𝟏−(𝟏+𝒊)
P = 1 000
0.07/12
000[
]
1−(1+0.07/12)(−20𝑥12)
= 1 000 000 x (0.007752989)
= R 7 752.99 (i.e. Bond repayments)
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Practical Examples
𝒊
𝐏 = 𝐏𝐕[
]
−𝒏
𝟏−(𝟏+𝒊)
PV = R 704 000 (the price today of a red Audi Q5)
P=?
i = 10% or 0.1
n = 5 yrs (payback period for a vehicle)
m = 12
𝐏 = 𝐏𝐕[
P = 704
𝒊
]
𝟏−(𝟏+𝒊)−𝒏
0.1/12
000[
]
1−(1+0.1/12)(−5𝑥12)
= 704 000 x (0.021247045)
= R 14 957.92(i.e. Car instalments)
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Practical Examples
𝒊
𝐏 = 𝐏𝐕[
]
−𝒏
𝟏−(𝟏+𝒊)
PV = R 233 100 (the price today of a Toyota Yaris 1.5Xi)
P=?
i = 0.135
n = 5 yrs (general payback period for a vehicle)
m = 12
𝐏 = 𝐏𝐕[
𝒊
]
𝟏−(𝟏+𝒊)−𝒏
P = 233
0.135/12
100[
]
1−(1+0.135/12)(−5𝑥12)
= 233 100 x (0.023009846)
= R 5 363.60 (i.e. Car instalments)
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*** End ***
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