There are two major techniques that are necessary for investment decisions: compounding and discounting. In this chapter, we will discuss compounding; and in Chapter 17, we talk about discounting.
1. SIMPLE INTEREST
1.1 What Is Interest?
The time value of money is a concept that a dollar received now is more valuable than a dollar received at any future dates. This is due to opportunity costs. The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner.
Interest may exist whether there is money or not. For example, David borrows from Peter a tool and returns it with a piece of bread as interest on the borrowing.
1.2 What Is Simple Interest?
Definition
Simple interest is interest which is earned in equal amounts every year (or month) and which is a given proportion of the original investment (the principal).
If a sum of money is invested for a period of time, then the amount of simple interest which accrues is equal to the Number of periods x Interest rate x Amount invested.
Key Point
The formula for simple interest is as follows:
FV = PV + nrPV
Note:
• PV: The original sum invested
• n: The number of periods (normally years)
• r: The interest rate (expressed as a proportion, so 10% = 0.1)
• FV: The sum invested after n periods, consisting of the original capital (PV) plus interest earned.
In diagram form, if you deposit $100 in an account earning 6% interest, how much would you have in the account after one year?
PV = -$100
1
FV = 106
0
Figure 1 Calculation of simple interest
Example 1
How much will an investor have after 10 years if he invests $5,000 at 8% simple interest per annum?
Using the formula FV = PV + nrPV, where PV = $5,000, n = 10 and r = 8%,
FV = $5,000 + (10 x 0.08 x $5,000) = $9,000
1.3 Investment Periods
If, for example, the sum of money is invested for four months and the interest rate is a rate per annum, then n = 4/12 =1/3. If the investment period is 160 days and the rate is an annual rate, then n = 160/365.
2. COMPOUND INTEREST
2.1 Compounding
Interest is normally calculated by means of compounding.
Definition
Compounding means that, as interest is earned, it is added to the original investment and starts to earn interest itself.
If a sum of money, the principal, is invested at a fixed rate of interest such that the interest is added to the principal and no withdrawals are made, then the amount invested will grow by an increasing number of pounds in each successive time period, because interest earned in earlier periods will itself earn interest in later periods.
Example 2
Suppose that $1,000 is invested at 10% interest per annum. After one year, the original principal plus interest will amount to $1,100.
$
Original investment 1,000
Interest in the first year (10%) 100
Total investment at the end of one year 1,100
(a) After two years, the total investment will be $1,210.
$
Investment at the end of one year 1,100
Interest in the second year (10%) 110
Total investment at the end of two years 1,210
The second-year interest of $110 represents 10% of the original investment,
and 10% of the interest earned in the first year.
(b) Similarly, after three years, the total investment will be $1,331.
$
Investment at the end of two years 1,210
Interest in the third year (10%) 121
Total investment at the end of three years 1,331
Instead of performing the calculations shown above, we could have used the following formula.
Key Point
The future value of an investment if compounded annually at a rate of i for n years will be:
FVn = PV (l + i)n
OR
FVn = PV (FVIFi,n)
Note:
• FVn: The future value of the investment at the end of n years
• n: The number of years during which the compounding occurs
• i: The annual interest (or discount) rate
• PV: The present value or original amount invested at the beginning of the first period
• FVIF: Future value interest factor
Example 3
Referring back to Example 2, using the formula for compound interest,
FV = PV(1 + r)n, where PV = $1,000, r = 10% = 0.1, and n = 3,
FV3 = $1,000 x 1.103 = $1,000 x 1.331 = $1,331
By table, FV3 = PV (FVIFi,n) = $1,000 (FVIF0.1,3) = $1,000(1.331) = $1,331
The interest earned over three years is $331, which is the same answer that was calculated in Example 2.
2.2 Changes In The Rate Of Interest
If the rate of interest changes during the period of an investment, the compounding formula must be amended slightly.
Key Point
The formula for compound interest when there are changes in the rate of
interest is as follows.
FV = PV(1 + r1 )y (1 + r2) n-y
Note:
• r1: The initial rate of interest
• y: The number of years in which the interest rate r1 applies
• r2: The next rate of interest
• n - y: The (balancing) number of years in which the interest rate r2 applies.
Example 4
(a) If $10,000 is invested now to earn 10% interest for three years and 8% thereafter, what would be the value of the total investment at the end of five years?
(b) An investor puts $10,000 into an investment for 10 years. The annual rate of interest earned is 10% for the first four years, 12% for the next four years and 8% for the last two years. How much will the investment be worth at the end of 10 years?
(a) $10,000 x 1.103 x 1.082 = $15,524.784
(b) $10,000 x 1.104 x 1.124 x 1.082 = $27,370.99
3. EQUIVALENT RATES OF INTEREST
3.1 Non-annual Compounding
In the previous examples, interest has been calculated annually, but this isn't always the case. Interest may be compounded daily, weekly, monthly or quarterly.
Example 5
If $5,000 is invested for 10 years at an interest rate of 2% per month, what will be the future value?
$5,000 x (1 + 0.02)120 = $53,825.815
Notice that n relates to the number of periods (10 years x 12 months) that r is compounded.
3.2 Effective Annual Rate Of Interest
Definition
An effective annual rate of interest is the corresponding annual rate when interest is compounded at intervals shorter than a year.
The non-annual compounding interest rate can be converted into an effective annual rate of interest. This is also known as the APR (annual percentage rate) which lenders such as banks and credit companies are required to disclose.
Key Point
Effective annual rate of interest: (1 + R) = (1 + r)n
Note:
• R: The effective annual rate
• r: The period rate
• n: The number of periods in a year
Example 6
Calculate the effective annual rate of interest (to two decimal places) of:
(a) 1.2% per month, compound
(b) 4% per quarter, compound
(a) 1 + R = (1 + r)n
1 + R = (1 + 0.012)12
R = 1.15389 - 1 = 0.15389 = 15.389%
(b) 1 + R = (1 + 0.04)4
R = 1.16986 - 1 = 0.16986 = 16.986%
3.3 Nominal Rates Of Interest And Annual Percentage Rate
Definition
A nominal rate of interest is an interest rate expressed as a per annum figure although the interest is compounded over a period of less than one year.
The corresponding effective rate of interest is the annual percentage rate (APR) (sometimes called the compound annual rate, or CAR).
Most interest rates are expressed as per annum figures even when the interest is compounded over periods of less than one year. In such cases, the given interest rate is called a nominal rate. We can, however, also work out the effective rate (APR or CAR).
Example 7
A bank offers investors 5% per annum interest payable half-yearly. If the 5% is a nominal rate of interest, the bank would in fact pay 2.5% every six months, compounded so that the effective annual rate of interest would be
[(1.025)2 -1] = 0.050625 = 5.0625% per annum
Similarly, if a bank offers depositors a nominal 15% per annum, with interest payable quarterly, the effective rate of interest would be 3.75% compound every three months, which is
[(1.0375)4 - 1] = 0.15865 = 15.8655% per annum
4. REGULAR SAVINGS AND SINKING FUNDS
4.1 Final Value Or Terminal Value
An investor may decide to add to his investment from time to time, and you may be asked to calculate the final value (or terminal value) of an investment to which equal annual amounts will be added. An example might be an individual or a company making annual payments into a pension fund: We may wish to know the value of the fund after n years.
Example 8
A person invests $500 now, and then a further $500 each year for three more years. How much would the total investment be worth after four years, if interest is earned at the rate of 8% per annum?
In problems such as this, we call the present "Year 0"; the time one year from now, "Year 1" and so on. It is also a good idea to draw a time line in order to establish exactly when payments are made.
$
(Year 0) The first year's investment will grow to $500 (1.08)4 680.240
(Year 1) The second year's investment will grow to $500 (1.08)3 629.856
(Year 2) The third year's investment will grow to $500 (1.08)2 583.200
(Year 3) The fourth year's investment will grow to $500 (1.08) 540.000
2,433.296
Figure 1 Time line of payments
By table: FV = A (FVIFAi,n)(1 + r)
= $500 (FVIFA0.08,4) (1 + 0.08)
= $500 (4.506) (1 + 0.08)
= $2,433.296
4.2 Sinking Funds
Definition
A sinking fund is an investment into which equal annual instalments are paid in order to earn interest, so that by the end of a given number of years, the investment is large enough to pay off a known commitment at that time. Commitments include the replacement of an asset and the repayment of a mortgage.
With mortgages, the total of the constant annual payments (which are usually paid in equal monthly instalments) plus the interest they earn over the term of the mortgage must be sufficient to pay off the initial loan plus accrued interest. We shall be looking at mortgages later on in this chapter.
When replacing an asset at the end of its life, a company might decide to invest cash in a sinking fund during the course of the life of the existing asset to ensure that the money is available to buy a replacement.
Example 9
A firm has just bought a machine with a life of four years. At the end of four years, it will be replaced at a cost of $15,000, and the company has decided to provide for this amount by setting up a sinking fund into which equal annual investments will be made, starting at year 1 (one year from now). The fund will earn interest at 10% per annum.
Calculate the annual investment.
Let us start by drawing a time line where $A = Equal annual investments.
Figure 2
(Year 0) No payment
(Year 1) The first year's investment will grow to $A x (1.10)3
(Year 2) The second year's investment will grow to $A x (1.10)2
(Year 3) The third year's investment will grow to $A x (1.10)
(Year 4) The fourth year's investment will remain at $A.
The value of the fund at the end of four years is as follows:
A + A (1.10) + A (1.10)2 + A (1.10)3
By table: FV = A (FVIFAi,n) = A (FVIFA 0.1,4) = A (4.641)
These annuities should be able to replace the machine at $15,000
$15,000 = A (4.641)
$15,000 / 4.641 = A
A = 3,232.06
5. LOANS AND MORTGAGES
5.1 Loans
Most people will be familiar with the repayment of loans. The repayment of loans is best illustrated by means of an example.
Example 10
James Wong borrows $10,000 now at an interest rate of 5% per annum. The loan has to be repaid through five equal instalments after each of the next five years. What is the annual repayment?
Let us start by calculating the future value of the loan (at the end of Year 5).
Using the formula FV = PV (1 + r) 5 where PV = $10,000,
r = 5% = 0.05,
n = 5,
FV = The sum invested after 5 years
FV = $10,000 (1 + 0.05)5
= $12,762.82
The value of the initial loan after five years ($12,762.82) must equal the sum of the repayments. A time line will clarify when each of the repayments are made.
Let $A = The annual repayments which start a year from now, i.e. at Year 1.
(Year 0) No payment
(Year 1) The first year's investment will grow to $A x (1.05)4
(Year 2) The second year's investment will grow to $A x (1.05)3
(Year 3) The third year's investment will grow to $A x (1.05)2
(Year 4) The fourth year's investment will grow to $A x (1.05)
(Year 5) The fourth year's investment remains at $A.
The value of the repayments at the end of five years is as follows:
A + (A x 1.05) + (A x 1.05)2 + (A x 1.05)3 + (A x 1.05)4
By table: $12,762.82 = A (FVIFAr,n)
= A (FVIFA0.05,5) = 5.526A
A = $2,309.59
5.2 Mortgages
The final value of a loan/mortgage can be likened to a sinking fund also, since the final value must equate to the sum of the periodic repayments (compare this with a sinking fund where the sum of the regular savings must equal the fund required at some point in the future).
Example 11
Andy Fung has taken out a $500,000 mortgage over 20 years. Interest
is to be charged at 6%. Calculate the monthly repayment.
Final value of mortgage = $500,000 x (1.06)20
= $1,603,567.736
where A = Annual repayment,
r = 0.06, and
n = 20.
Sum of repayments:
$1,603,567.74 = A (FVIFA0.06, 20)
= 36.79A
A = $43,587.05 per annum
If annual repayment = $43,587.05,
monthly repayment = $43,587.05 / 12
= $3,632.25
Chapter Review
• Interest is the opportunity cost for consumption forgone.
• If the rate of interest changes during the period of an investment, the compounding formula must be amended to FV = PV(1+r1)Y(1 + r2 )n-y.
• An effective annual rate of interest is the corresponding annual rate when interest is compounded at intervals shorter than a year.
What You Need To Know
• Simple interest: Interest earned in equal amounts every year (or period). The simple interest formula is FV = PV + nrPV.
• Compounding: As interest is earned, it is added to the original investment and starts to earn interest itself. Formula is FVn = PV (1 + r) n.
• Nominal rate of interest: An interest rate expressed as a per annum figure although the interest is compounded, over a period of less than one year.
• Sinking fund: Equal annual instalments are paid in order to earn interest, so that by the end of a given number of years, the investment is large enough to pay off a known commitment at that time.
Work Them Out
1. At what annual rate of compound interest will $3,000 grow to $4,081.47 after four years?
A 7%
B 8%
C 9%
D 10%
2. A bank adds interest monthly to investors' accounts even though interest rates are expressed in annual terms. The current rate of interest is 9%. Dennis deposits $5,000 on 1 July. How much interest will have been earned by 31 December (to the nearest $)?
A $123.00
B $60.00
C $240.00
D $229.00
3. Interest rate would only exist in an economy where there is money.
A True, because interest rate is expressed in money
B False, it is impossible to have interest without money
C True, because interest is the cost of money
D False, because interest reflects the opportunity cost of consumption
4. The greater the future value of a future amount, the higher the rate of interest.
A True, the higher the rate of interest, the greater the expected yield
B False, the higher the rate of interest, the lower the expected yield
C True, the future value of $1 in the future increases, as the interest rate rises
D False, the future value of $1 in the future decreases, as the interest rate rises
5. What is the future value of depositing $1,000 in a savings account for two years at an interest rate of 6% per annum paid every six months?
A $1,123.6
B $1,125.5
C $1,060.9
D $1,262.5
6. What is the future value of depositing $2,000 in a savings account for three years at an interest rate of 8% per annum for the first two years and 10% per annum for the third year?
A $2,519.4
B $2,662
C $2,566.1
D $2,556.1
7. A newly issued bond costs $500. The holder will get back $700 in seven years' time. What is the interest rate per annum?
A 3%
B 4%
C 5%
D 6%
8. The interest rate
A is the price of earlier availability of money
B is the price of later availability of goods
C is the price of money
D should always be zero since money is barren
9. ___________ is a series of regular and equal instalment which will earn interest so that by the end of a given period of time, the investment is large enough to pay off a commitment.
A mutual fund
B sinking fund
C provident fund
D exchange fund
10. An interest rate expressed as a per annum figure although the interest is
compounded over a period of less than one year. This is referred to as
A real interest rate
B nominal rate of interest
C equivalent interest rate
D notional interest rate
SHORT QUESTIONS
A farm has just bought a truck which has a life of 10 years. At the end of 10 years, a replacement will cost $200,000 and the farm would like to provide for this future amount by setting up a sinking fund into which equal annual investments will be made, starting from now. The fund will earn interest at 15% per annum. What is the farm's annual investment?
Ferrix Lau borrows $80,000 now at an interest rate of 8% per annum. The loan has to be repaid through five equal instalments after each of the next five years. What is the annual repayment?
ESSAY QUESTION
1. (a) David has taken out a $25,000 mortgage over 20 years. Interest is to be charged at 15%. Calculate the monthly repayment.
(b) After eight years, the interest rate changes to 12%. What is the new monthly repayment?
