Chapter 3 Time Value of Money

F9 Financial Management

Chapter 3 Time Value of Money

LEARNING OBJECTIVES
1.1Explain the concept of the time value of money.
1.2Calculate the future value of a sum by compounding.
1.3Calculate the present value (PV) of a single sum using formula.
1.4Calculate the PV of a single sum using discount tables.
1.5Calculate the PV of an annuity using formula.
1.6Calculate the PV of an annuity using annuity tables.
1.7Calculate the PV of a perpetuity using formula.
1.8Calculate the annual percentage rate (APR) or effective annual rate (EAR).

1.The Time Value of Money

1.1Why money has time value?

1.1.1Money received today is worth more than the same sum received in the future, i.e. it has a time value.

1.1.2This occurs for three reasons:

(a)potential for earning interest (cost of finance)

(b)impact of inflation

(c)effect of risk.

1.2Compounding

1.2.1 / EXAMPLE 1
An investment of $100 is to be made today. What is the value of the investment after two years if the interest rate is 10%?
Solution:
The formula for calculating the future value (FV) of a sum is:
FV = P × (1 + r)n
FV = $100× (1 + 10%)2 = $121

1.2.2Sometimes financial transactions take place on the basis that interest will be calculated more frequently than once a year.

1.2.3 / EXAMPLE 2
If you put $100 in a bank account earning 12% per annum, then your return after one year is:
FV = $100 × (1 + 12%) = $112
If the interest is compounded semi-annually, then your return after one year is:FV = $100 × (1 + 12%/2)2 = $112.36
If the interest is compounded quarterly, then
FV = $100 × (1 + 12%/4)4 = $112.55
If the interest is compounded daily, then
FV = $100 × (1 + 12%/365)365 = $112.75
Question 1
If $100 is deposited in a bank account that compounds interest quarterly and the nominal return per year is 12%, how much will be in the account after eight years?
Solution:

1.3Continuous compounding

1.3.1If the compounding frequency is taken to the limit we say that there is continuous compounding. When the number of compounding periods approaches infinity the future value is found by

FV = P × ein

Where e is the value of the exponential function. This is set as 2.71828.

1.3.2 / EXAMPLE 3
The future value of $100 deposited in a bank paying 12% nominal compounded continuously after eight years is:
$100 × 2.718280.12x8 = $261.17

1.4Present values

1.4.1Present value (PV) is the cash equivalent now of a sum of money receivable or payable at a stated future date, discounted at a specified rate of return.

1.4.2Discounting starts with the future value, and converts a future value to a present value.

1.4.3 / EXAMPLE 4
If a company expects to earn a (compound) rate of return of 10% on investments, how much would it need to invest now to have the following investments?
(a)$11,000 after 1 year
(b)$12,100 after 2 years
(c)$13,310 after 3 years
Solution:
The discounting formula to calculate the present value of a future sum of money at the end of n time periods is:

(a)
(b)
(c)

1.5Determining the investment period

1.5.1Rearranging the standard equation so that we can find n (the number of years of the investment), we create the following equation:

1.5.2 / EXAMPLE 5
How many years does it take for $10 to grow to $17.62 when the interest rate is 12%?
Solution:

1.6Annuities

1.6.1Quite often there is not just one payment at the end of a certain number of years. There can be a series of identical payments made over a period of years. For example, $100 paid at the end of each of the next years is a 3-year annuity.

1.6.2If payments occur at the end of each period, then we have an ordinary (or deferred) annuity. Payments on mortgages, car loans, and student loans are examples of ordinary annuities.

1.6.3If the payments are made at the beginning of each period, then we have an annuity due. Rental payments for an apartment, life insurance premiums, and lottery payoffs are examples of annuities due.

1.6.4 / EXAMPLE 6
Here are the time lines for a $100, 3-year, 5%, ordinary annuity and for the same annuity on an annuity due basis. With the annuity due, each payment is shifted back to the left by 1 year.

(a)Future value of ordinary annuity
FVA3 = 100 + 100 × (1 + 5%) + 100 × (1 + 5%)2
= 315.25
(b)Future value of annuity due
FVAdue = 100 × (1 + 5%) + 100 × (1 + 5%)2 + 100 × (1 + 5%)3
= 331.01
From the above calculation, we can find that:
FVAdue = FVAordinary× (1 + r)
(c)Present value of ordinary annuity
PVA3 = = 272.32
(d)Present value of annuity due
PVAdue = = 285.94
Similarly, we can find that:
PVAdue = PVAordinary× (1 + r)

1.7Perpetuities

1.7.1Some contracts run indefinitely and there is no end to a series of identical payments. Perpetuities are rare in the private sector, but certain government securities do not have an end date; that is, the amount paid when the bond was purchased by the lender will never be repaid, only interest payments are made.

1.7.2For example, the UK government has issued Consolidated Stocks or War Loans which will never be redeemed.

1.7.3The value of a perpetuity is simply the annual amount received divided by the interest rate when the latter is expressed as a decimal.

PV of perpetuity =

1.7.4 / EXAMPLE 7
If $10 is to be received as an indefinite annual payment then the present value, at a discount rate of 12%, is:
PV = = $83.33

2.Annual Percentage Rate

2.1Sometimes you are presented with a monthly or daily rate of interest and wish to know what that is equivalent to in terms of annual percentage rate (APR) or effective annual rate (EAR).

2.2 / EXAMPLE 8
If m is the monthly interest or discount rate, then over 12 months:
(1 + m)12 = 1 + i, where i is the annual compound rate.
i = (1 + m)12– 1
If a credit card company charges 1.5% per month, the APR is:
i = (1 + 0.015)12– 1 = 19.56%
For daily rate: (1 + d)365 = 1 + i
2.3 / EXAMPLE 9 – Frequency of compounding
Suppose you plan to invest $100 for 5 years at a nominal annual rate of 10%. What will happen to the future value of your investment if interest in compounded more frequently than once a year? Because interest will be earned on interest more often, you might expect the future value to increase as the frequency of compounding increases.
The effect of frequent compounding:
Frequency of Compounding / Nominal Annual Rate / Effective Annual Rate / Future Value of $100 Invested for 5 years
Annual / 10.00% / 10.000% / $161.05
Semiannual / 10.00% / 10.250% / $162.89
Quarterly / 10.00% / 10.381% / $163.86
Monthly / 10.00% / 10.471% / $164.53
Daily / 10.00% / 10.516% / $164.86
APR or EAR = (1 + )m– 1

Exercise

2.What will a $100 investment be worth in three years’ time if the rate of interest is 8%, using (a) simple interest? (b) annual compound interest?

3.You plan to invest $10,000 in the shares of a company.

(a)If the value of the shares increases by 5% a year, what will be the value of the shares in 20 years?

(b)If the value of the shares increases by 15% a year, what will be the value of the shares in 20 years?

4.How long will it take you to double your money if you invest it at: (a) 5%? (b) 15%?

5.A bank lends a customer $5,000. At the end of 10 years he repays this amount plus interest. The amount he repays is $8,950. What is the rate interest charged by the bank?

6.If the flat (nominal annual) rate of interest is 14% and compounding takes place monthly, what is the effective annual rate of interest?

7.What is the present value of $100 to be received in 10 years’ time when the interest rate is 12% and (a) annual discounting is used? (b) semi-annual discounting is used?

8.How much must be invested now to provide an amount of $10,000 in six years’ time assuming interest is compounded quarterly at a nominal annual rate of 8%? What is the effective annual rate?

9.Supersalesman offers you an annuity of $800 per annum for 10 years. The price he asks is $4,800. Assuming you could earn 11% on alternative investments would you buy the annuity?