So we know how to convert a monthly interest rate into an effective annual interest. Similarly, we can convert a quarterly interest, or a semi-annual interest rate or an interest rate of any frequency for that matter into an effective annual interest rate.
For a quarterly interest rate of say 3% per quarter, the interest will be paid four times per year (every three months). We can calculate the effective annual interest rate by solving for
:
where
is the quarterly interest rate.
So
, and so
. This is the effective annual interest rate.
In general, for interest paid at a frequency of
times per annum, the follow equation holds:
where
is the interest rate paid
times per annum.
De-coding the terminology
Market convention however, is not to state the interest rate as say 1% per month, but rather to express this amount as an annual amount which in this example would be paid monthly. This annual amount is called the nominal amount.
The market convention is to quote a nominal interest rate of “12% per annum paid monthly" instead of saying (an effective) 1% per month. We know from a previous example, that a nominal interest rate of 12% per annum paid monthly, equates to an effective annual interest rate of 12,68%, and the difference is due to the effects of interest-on-interest.
So if you are given an interest rate expressed as an annual rate but paid more frequently than annual, we first need to calculate the actual interest paid per period in order to calculate the effective annual interest rate.
For example, the monthly interest rate on 12% interest per annum paid monthly, is:
The same principle applies to other frequencies of payment.
Consider a savings account which pays a nominal interest at 8% per annum, paid quarterly. Calculate (a) the interest amount that is paid each quarter, and (b) the effective annual interest rate.
We are given that a savings account has a nominal interest rate of 8% paid quarterly. We are required to find:
the quarterly interest rate,
the effective annual interest rate,
We know that:
and
where
is 4 because there are 4 payments each year.
The effective annual interest rate (
) is calculated as:
The quarterly interest rate is 2% and the effective annual interest rate is 8,24%, for a nominal interest rate of 8% paid quarterly.
On their saving accounts, Echo Bank offers an interest rate of 18% nominal,
paid monthly. If you save R100 in such an account now, how much would the amounthave accumulated to in 3 years' time?
Interest rate is 18%
nominal paid monthly. There are 12 months in a year. We are working with ayearly time period, so
. The amount we have saved is R100, so
. We need the accumulated value,
.
We know that
for converting from nominal interest rate to effective interest rate, we have
and for calculating accumulated value, we have
There are 12 month in a year, so
and then, we have
The accumulated value is R
. (Remember to round off to the the nearest
cent.)
Nominal and effect interest rates
Calculate the effective rate equivalent to a nominal interest rate of 8,75% p.a. compounded monthly.
Cebela is quoted a nominal interest rate of 9,15% per annum compounded every four months on her investment of R 85 000. Calculate the effective rate per annum.
Formulae sheet
As an easy reference, here are the key formulae that we derived and used during this chapter. While memorising them is nice (there are not many), it is the application that is useful. Financial experts are not paid a salary in order to recite formulae, they are paid a salary to use the right methods to solve financial problems.
Definitions
Principal (the amount of money at the starting point of the calculation)
interest rate, normally the effective rate per annum
period for which the investment is made
the interest rate paid
times per annum, i.e.
Equations
End of chapter exercises
Shrek buys a Mercedes worth R385 000 in 2007. What will the value of the Mercedes be at the end of 2013 if
the car depreciates at 6% p.a. straight-line depreciation
the car depreciates at 12% p.a. reducing-balance depreciation.
Greg enters into a 5-year hire-purchase agreement to buy a computer for R8 900. The interest rate is quoted as 11% per annum based on simple interest. Calculate the required monthly payment for this contract.
A computer is purchased for R16 000. It depreciates at 15% per annum.
Determine the book value of the computer after 3 years if depreciation
is calculated according to the straight-line method.
Find the rate, according to the reducing-balance method, that would
yield the same book value as in
[link] after 3 years.
Maggie invests R12 500,00 for 5 years at 12% per annum compounded monthly for
the first 2 years and 14% per annum compounded semi-annually for the next 3years. How much will Maggie receive in total after 5 years?
Tintin invests R120 000. He is quoted a nominal interest rate of 7,2% per annum compounded monthly.
Calculate the effective rate per annum correct to THREE decimal
places.
Use the effective rate to calculate the value of Tintin's investment if
he invested the money for 3 years.
Suppose Tintin invests his money for a total period of 4 years, but
after 18 months makes a withdrawal of R20 000, how much will hereceive at the end of the 4 years?
Paris opens accounts at a number of clothing stores and spends freely. She gets heself into terrible debt and she cannot pay off her accounts. She owes Hilton Fashion world R5 000 and the shop agrees to let Paris pay the bill at a nominal interest rate of 24% compounded monthly.
How much money will she owe Hilton Fashion World after two years ?
What is the effective rate of interest that Hilton Fashion World is charging her ?
Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?