Page 1
Contents
Recap from previous grades ............................................................................................................................................. 1
Converting from a nominal to effective rate: ................................................................................................................... 1
Future Value and Present Value Formulae: ...................................................................................................................... 2
Paying at the beginning of the month: ............................................................................................................................. 4
Sinking funds ..................................................................................................................................................................... 4
Deferred payments: .......................................................................................................................................................... 5
Balance Outstanding ......................................................................................................................................................... 6
Final Payment.................................................................................................................................................................... 8
Recap from previous grades
𝐴 = 𝑃(1 + 𝑖. 𝑛) 𝐴 = 𝑃(1 + 𝑖)𝑛
𝐴 = 𝑃(1 − 𝑖. 𝑛) 𝐴 = 𝑃(1 − 𝑖)𝑛
𝐴 = Accumulated amount (original plus interest or original minus depreciation)
i.e. the Future Value of the investment/asset
𝑃 = Principal amount (original amount)
i.e. the Present Value of the investment/asset
𝑛 = number of investment periods
𝑖 = interest rate per period
Converting from a nominal to effective rate:
Example:
1. Convert 14% p.a. compounded quarterly to an effective rate.
0,14 4
(1 + 𝑖) = ൬1 + ൰
4
𝑖 = 1 − 1,14652 …
𝑖 = 0,14752 …
𝑟𝑒𝑓𝑓 = 14,75%
, Page 2
Future Value and Present Value Formulae:
Both formulae are based on the following assumptions:
• The first payment is made one time period from the present.
• The final payment is the 𝑛𝑡ℎ payment.
• The regularity of compounding the interest is the same time period as the regularity
of the payments made.
𝑥[(1 + 𝑖)𝑛 − 1] 𝑥[1 − (1 + 𝑖)−𝑛 ]
𝐹= 𝑃=
𝑖 𝑖
Often used for annuities, savings or sinking Often used for loans (get the money now).
funds (investing money and find its value in
the future).
𝑥 = regular payment
𝑛 = number of payments
𝑖 = interest rate per period
Examples:
2. David applies for a home loan. The bank charges 11,5% p.a. interest, compounded
monthly. David can afford to pay R5 000 per month. The bank offers a loan over 20
years. The first payment will be made one month after the loan is granted. Calculate
the amount that David can afford to the nearest rand.
0,115
𝑥 = 𝑅 5000 𝑖= 𝑛 = 20 × 12 = 240
12
𝑥[1 − (1 + 𝑖)−𝑛 ] Note that 𝑖 is,
𝑃= •
11,5
11,5% = 100 = 0.115
𝑖
0.115
• Then 𝑖 = , because 11,5 is the
0,115 −240 12
5000 ቈ1 − ቀ1 +
12 ቁ rate per year so to determine the
𝑃= monthly rate you divide by 12.
0,115
12
= 468 854,19
= 𝑅 468 854
Contents
Recap from previous grades ............................................................................................................................................. 1
Converting from a nominal to effective rate: ................................................................................................................... 1
Future Value and Present Value Formulae: ...................................................................................................................... 2
Paying at the beginning of the month: ............................................................................................................................. 4
Sinking funds ..................................................................................................................................................................... 4
Deferred payments: .......................................................................................................................................................... 5
Balance Outstanding ......................................................................................................................................................... 6
Final Payment.................................................................................................................................................................... 8
Recap from previous grades
𝐴 = 𝑃(1 + 𝑖. 𝑛) 𝐴 = 𝑃(1 + 𝑖)𝑛
𝐴 = 𝑃(1 − 𝑖. 𝑛) 𝐴 = 𝑃(1 − 𝑖)𝑛
𝐴 = Accumulated amount (original plus interest or original minus depreciation)
i.e. the Future Value of the investment/asset
𝑃 = Principal amount (original amount)
i.e. the Present Value of the investment/asset
𝑛 = number of investment periods
𝑖 = interest rate per period
Converting from a nominal to effective rate:
Example:
1. Convert 14% p.a. compounded quarterly to an effective rate.
0,14 4
(1 + 𝑖) = ൬1 + ൰
4
𝑖 = 1 − 1,14652 …
𝑖 = 0,14752 …
𝑟𝑒𝑓𝑓 = 14,75%
, Page 2
Future Value and Present Value Formulae:
Both formulae are based on the following assumptions:
• The first payment is made one time period from the present.
• The final payment is the 𝑛𝑡ℎ payment.
• The regularity of compounding the interest is the same time period as the regularity
of the payments made.
𝑥[(1 + 𝑖)𝑛 − 1] 𝑥[1 − (1 + 𝑖)−𝑛 ]
𝐹= 𝑃=
𝑖 𝑖
Often used for annuities, savings or sinking Often used for loans (get the money now).
funds (investing money and find its value in
the future).
𝑥 = regular payment
𝑛 = number of payments
𝑖 = interest rate per period
Examples:
2. David applies for a home loan. The bank charges 11,5% p.a. interest, compounded
monthly. David can afford to pay R5 000 per month. The bank offers a loan over 20
years. The first payment will be made one month after the loan is granted. Calculate
the amount that David can afford to the nearest rand.
0,115
𝑥 = 𝑅 5000 𝑖= 𝑛 = 20 × 12 = 240
12
𝑥[1 − (1 + 𝑖)−𝑛 ] Note that 𝑖 is,
𝑃= •
11,5
11,5% = 100 = 0.115
𝑖
0.115
• Then 𝑖 = , because 11,5 is the
0,115 −240 12
5000 ቈ1 − ቀ1 +
12 ቁ rate per year so to determine the
𝑃= monthly rate you divide by 12.
0,115
12
= 468 854,19
= 𝑅 468 854
