Return on Investment I
A. Interest
Interest is the return for loaning money. It is earned on federal and provincial bonds,
corporate bonds, guaranteed investment certificates, and term deposits When setting
the interest rate, the institution or organization making the loan takes two major factors
into consideration: (a) possible inflation over the period of the loan,and (b)the risk of
default (the borrower’s credit rating,and whether the loan is secured or unsecured
(collateral),
A term deposit provides the depositor with a guaranteed rate of interest. Unlike demand deposits which
may be accessed at any time, the money deposited may be withdrawn only after the term has ended or if
the depositor gives advanced notice of the intention to withdraw (some reduced interest penalty is usually
involved). If the financial institution fails, term deposits are guaranteed by the Canadian Deposit
Insurance Corporation (CDIC).
Guaranteed Investment Certificates (GICs) are sold by banks and trust companies. They may be
cashable or non-cashable (the latter carry a higher rate of interest. They are not covered by the CDIC are
often bought for retirement plans because they provide a low-risk fixed rate of return. The principal is at
risk only if the bank defaults.
Simple Interest
Simple interest is calculated on the original principal only. Accumulated interest
from prior periods is not used in calculations for the following periods. Simple interest is
normally used for a single period of less than a year, such as 30 or 60 days.
Simple Interest: i = p * r * t
where:
p = principal (original amount borrowed or loaned)
r = interest rate for one period
t = time (number of interest periods
Example 1 – time is less than 1 interest period: You invest $10,000 for 60 days at
5% simple interest per year (per annum) (assume a 365 day year).
i= p * r * t i= 10,000 * .05 * (60/365) = 82.1917
Example 2 – time is equal to 1 interest period: You invest $10,000 for 1 year at 5%
simple interest per year.
i=p*r*t i= 10,000 * .05 * 1 = 500
Example 3 –time is greater than 1 interest period: You invest $10,000 for 3 years at
5% simple annual interest.
i=p*r*t i= 10,000 * .05 * 3 = 1,500
Solve the following simple interest problems.
1. Find the interest when you invest $5,000 for 90 days at 6% simple
interest per annum.
2. Find the interest when you invest $350,000 for 1 year at 5%
simple interest per annum.
3. Find the interest when you invest $9,000 for 4 years at 8% simple
interest per annum.
Compound Interest
Compound interest is calculated each period on the original principal and all interest
accumulated during past periods. Although the interest may be stated as a yearly
rate, the compounding periods can be yearly, semi-annually, quarterly, or even
continuously.
You can think of compound interest as a series of back-to-back simple interest
contracts. The interest earned in each period is added to the principal of the previous
period to become the principal for the next period. For example, you borrow $10,000 for
three years at 5% annual interest compounded annually:
interest year 1: i1 = p * r * t = 10,000 * .05 * 1 = 500
interest year 2: i2= (p2=p1 + i1) * r * t = (10,000 + 500) * .05 * 1 = 525
interest year 3: i3 = (p3 = p2 + r2) * r * t = (10,500 + 525) *.05 * 1 = 551.25
Total interest earned over the three years = 500 + 525 + 551.25 = 1,576.25. Compare
this to 1,500 earned over the same number of years using simple interest. This would
be time consuming way to calculate compound interest. Instead, use the formula
shown below.
A = P(1+i)n
where A is the amount in $
P is the principal invested
i is the interest rate per interest period[expressed as a decimal]
n is the number of interest periods
How much can compounding influence return on investment? Below are the results for
an investment of $10,000 for 30 years using 12% simple interest, and 12% interest
compounded yearly and quarterly.
Type of Interest Principal Plus Interest Earned
Simple 46,000.00
Compounded Yearly 299,599.22
Compounded Quarterly 347,109.87
Example 4 - annual interest period: You invest $5000. at 7% per annum per annum
compounded annually for 3 years. What amount of money will be paid after three
years?
A = P(1 + i)n A = 5000(1 + .07)3* = 5000 x 1.225043 = 6125.22
* Use the yx function key on your calculator.
Example 5 - semi-annual interest period: You invest $5000 at 7%
per annum compounded semi-annually for 3 years. What is the
amount paid after 3 years?
A = P(1 + i)n A = 5000(1 + 0.035)6 = 5000 x 1.22926 = 6146.28
Note: - the interest rate ( i ) for each 6-month interest period
is 7%/2 = 3.5% (0.035)
- the number of interest periods ( n ) is 2 per year X 3
years = 6
Example 6 - monthly interest period: You invest $5000. at 7% per
annum compounded monthly for 3 years. What is the amount paid
after 3 years?
A = P(1 + i)n A = 5000(1+0.07/12)36 = 5000 x 1.61173 = 8058.63
1.005833
Solve the following compound interest problems:
1. Find the amount when $5000. is invested at 5.25% compounded
annually for 10 years.
2. Find the amount when $350 000. is invested at 6.50%
compounded annually for 15 years.
3. Calculate the compound amount in each of the following:
a) $900. is invested at 4.25% per annum compounded semi-annually for 7 years?
b) $12 500. is invested at 6.412% per annum compounded semi-annually for 9
years?
c) $ 29 400. is invested at 7.125% per annum compounded monthly for 3
years?
d) $11 900. is invested at 6.00% per annum compounded quarterly for 60
months?
Present Value
How much money has to be invested today in a fixed interest in order
to ensure that a certain amount of money will be accumulated at the
end of a set investment period?
At Simple Interest use the formula:
PV=A/(1+nr)
PV=present value
A=amount desired in future
n=number of years in the future
r=interest rate
Example: you want to accumulate $100,000. in ten years time. How
much must you invest today at a simple interest rate of 5% per
annum?
PV=A/(1+nr) =100 000/(1+10x0.05) = 100 00/1.5=66 667.
Compound Interest Calculated Annually
Example: as above except the interest is compounded annually
PV=A/(1+r)n
PV=100 000/(1+0.05)10=100 000/1.62889=61 391
Compounded more than once per year (q times)
PV=A/(1+r/q)nq
Example: as above except the interest is compounded monthly
PV=100 000/(1+0.05/12)10x12=100 000/(1.0042)120=100 000/1.6536=60 474.12
Solve the following present value problems:
1. Amount required to accumulate $1200 - invested for 7 years
at 4% simple interest.
2. Amount required to accumulate $25 000 - invested at 6% per
annum compounded annually for 9 years.
3. Amount required to accumulate $50 000. - compounded at 7%
per annum compounded monthly for 9 years.
SIMPLE INTEREST PROBLEMS
1. i = p x r x t = 5000 x .06 x (90/365) = 73.98
.2466
2. i = p x r x t = 350000 x .05 x1 = 17500
3. i = p x r x t = 9000 x .08 x 4 = 2880
COMPOUND INTEREST PROBLEMS
1. A = P(1 + i)n = 5000(1+0.0525)10 = 5000x1.6681 = 8340.50
2. A = P(1 + i)n = 350000(1+0.0650)15 = 350000x2.57184 = 900144
3. a) A = P(1 + i)n =900(1+0.02125)14 = 900x1.3423 = 1208.07
.0425/2
b) A = P(1 + i)n = 12500(1+0.03206)18 = 12500x1.76477 = 22059.67
.06412/2
c) A = P(1 + i)n = 29400(1+0.0059375)36 = 29400x1.23753 = 36383.40
0.07125/12
d) A = P(1 + i)n = 11900(1+0.015)15 = 16027.57
.06/4
Present Value Problems
1. PV=A/(1+nr) = 1200/(1+ 7x.04) = 1200/1.28 = 937.50
2. PV=A/(1+r)n 25000/(1+.06)9 = 25000/1.68949 = 14797.37
3. PV=A/(1+r/q)nq = 50000/(1 + 0.07/12)9X12 = 26678.38