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MORE ABOUT INTEREST
X-Kit Textbook
Chapter 5
CONTENT
Depreciation
Inflation
Nominal &
Effective
Interest
Rates
CONTENT- QUESTIONS
How our belongings decrease in value
over time?
How much our money buys us over
time?
How much is our money worth?
EXAMPLE
For his construction business, Jim’s father
bought a car 5 years ago for R50 000.
Now, he wants to sell the vehicle. But the
car is now only worth R20 000.
We call this DEPRECIATION.
TERMINOLOGY
Term Explanation
Assets Things we own that are valuable or useful, like a car or
laptop.
Book Value The original cost of an asset less the depreciation.
Scrap Value The Book Value at the end of an asset’s life.
Straight line
Depreciation
Subtract equal amounts every year from the purchase
price of an asset.
Spread the depreciation evenly over the life of an
asset.
Reducing
Balance
Depreciation
Subtract the greatest amount from the present value
for the first year and smaller amounts for each year
afterwards.
DEPRECIATION
• Straight Line Depreciation:
𝑭𝑽 = 𝑷𝑽 𝟏 − 𝒊 × 𝒏
• Reducing Balance Depreciation:
𝑭𝑽 = 𝑷𝑽 𝟏 − 𝒊 𝒏
𝐹𝑉 = Future Value / Scrap Value
𝑃𝑉 = Present Value / Original Cost of an asset
𝑖 = Depreciation Rate per year
𝑛 = Number of years
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EXAMPLE
Jim buys a new mountain bike for R5 000.
The straight line depreciation rate is 21%
per year. What will his mountain bike be
worth in 4 years’ time?
EXAMPLE
Jim’s older brother buys a new 4x4 for
R185 000. But, after 6 years, the 4x4 will
only be worth R42 000. What is the annual
rate of depreciation using straight line
depreciation?
EXAMPLE
It is Jim’s job to mow the lawn at home.
Jim’s parents buy a new lawn mower for
R1 800. The straight line depreciation
rate is 20%. How long will it take for the
new lawn mower to be worth R0? In
other words, find out when the lawn
mower’s scrap value is zero?
EXAMPLE
Mamphela buys a new car for R80 000.
Find the scrap value of the car after 5 years
using reducing balance depreciation at
18% per annum?
EXAMPLE
A bottling machine in a soft-drink factory
costs R1 200 000. The factory owner
expects that the machine will last for 8
years. After 8 years, its scrap value will be
R100 000. Find the annual rate of
depreciation using reducing balance
depreciation.
EXAMPLE
A tourism company offers flights in a hot-
air balloon. A new hot-air balloon costs
R570 000. Depreciation is at 25% p.a. on
a reducing balance. After how many
years will the hot-air balloon have a value
of R50 000?
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INFLATION
• Some things are worth less over time = Depreciation.
• The price of things goes up over time = Inflation.
• Purchasing power depends on the rate of inflation over the
year (for example, invest R1 000 at an 10% interest rate
for 1 year, but inflation rate is 10%, you gained nothing).
• The effects of inflation can eat up the interest on our
investments.
• The nominal interest rate is the rate at which an
investment increases.
• The real interest rate is the rate at which the purchasing
power of an investment increases.
WORK WITHINFLATION
Suppose the inflation rate over the
year is 6%. In other words, the
prices of goods have gone up, on
average, by 6% over the year. So
every rand that you spend next
year will buy you 6% less.
EXAMPLE
Find the real future value at an inflation rate of 6%, of an
investment of R1 000, invested for 1 year at an 10% annual
interest rate.
𝑅𝑒𝑎𝑙 𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 =
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
1 + 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒 𝑛
• Your R1 100 investment is only worth R1 037,74 in
purchasing power.
• The nominal payoff on the investment is R1 100.
• The Real payoff is R1 037,74
EXAMPLE
Cindy and Londi both inherit R500 000. Cindy buys
a house for R440 000 and a second-hand car for R60
000. Londi buys a house for R280 000 and a new car
for R220 000. The cars depreciate (straight line
depreciation) at 10% p.a. The houses appreciate
(value increases) at 5% p.a. Whose investment is
worth more after 2 years?
COMPARISONOF FUTUREVALUES
Inheritance of R500 000 Value of assets after 2 years
House Car House Car Total
Cindy R440 000 R60 000 R485 100 R48 000 R533 100
Londi R280 000 R220 000 R308 700 R176 000 R484 700
EXAMPLE
In April 2003 a bank’s cheque account statement
included the following as part of the bank’s advertising:
“Did you know that university fees (class fees only)
will be a staggering R26 000 a year by 2011?”
In 2003 Rhodes University’s academic fees were on
average R14 500 per year. What annual rate of inflation
was the bank using?
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NOMINAL& EFFECTIVEINTERESTRATES
• The nominal interest rate is the percentage rate
that the bank will charge for a loan or investment.
• The effective interest rate is the rate the
customer will pay or receive. How much will
depend on how often the bank adds the interest
amount to the loan. (Different compounding
periods – annual, quarterly, monthly or daily)
Jim borrowedR100for 1 year @ 15% p.a.
Period Interval
per year
Formula Future
Value
Interest
Payable
Effective
Annual Rate
Annual 1
100 1 +
0.15
1
1 R115 R15 15%
Half-
yearly
2
100 1 +
0.15
2
2 R115.56 R15.56 15.56%
Monthly 12
100 1 +
0.15
12
12 R116.08 R16.08 16.08%
Daily 365
100 1 +
0.15
365
365 R116.18 R16.18 16.18%
NOMINAL& EFFECTIVEINTERESTRATES
The more often interest is
compounded, the higher the effective
rate of interest.
NOMINAL& EFFECTIVEINTERESTRATES
𝒊 𝒆𝒇𝒇 = 𝟏 +
𝒊
𝒎
𝒎
− 𝟏
• 𝑖 = the nominal (annual) interest rate
• 𝑚 = the number of times interest is
compounded in a year
• 𝑖 𝑒𝑓𝑓 = the effective interest rate
EXAMPLE
If you save money in an investment
account at a nominal rate of 12%
compounded monthly, what is the
effective rate of interest?
EXAMPLE
The owners of a fast-food pizza business buy a new delivery
vehicle for R80 000. After 5 years, they will have to replace
the old bakkie with a new one because old vehicles are
expensive to keep going. The business writes off the
depreciating value of the vehicle every year at a rate of 12%
on a reducing balance. The owners of the business want to
know how much they will be able to sell the bakkie for after
5 years. They also want to know how much it will cost to
buy the same kind of delivery vehicle in 5 years’ time.
Inflation is 14% p.a. Lastly, the owners wants to know how
much extra money they will need to buy a new vehicle after
selling the old one at its scrap value.
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STRUCTURETHE QUESTION
1. We want to know the future value (book
value/scrap value) of the first vehicle with
12% depreciation.
2. We also want to know the future value of
the new vehicle at 14% inflation.
3. Lastly, we want to know the difference
between the cost of a new vehicle and the
scrap value of the old one.
