8 QUESTIONS ON LUMP SUM

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A lump sum is a single payment paid at a certain point in time.

Before we solve the seven questions, let's review some key formulas.

The formula for the future value of a lump sum is:

$$FV=PV\left(1+\frac{i}{m}\right)^{nm}$$

The formula for the present value of a lump sum is:

$$PV=\frac{FV}{\left(1+\frac{i}{m}\right)^{nm}}$$

Question 1 

Find the value of N10,000 in 10 years if the investment earns 8% for four years and then earns 4% for the remaining six years.

Solution:

The formula for a lump sum that has two interest rates is 

$$FV=PV(1+i_1)^{n_1}(1+i_2)^{n_2}$$

Where,

$i_1$  is the first interest rate

$i_2$ is the second interest rate

$n_1$ is the number of years for the first interest rate

$n_2$ is the number of years for the second interest rate

PV is the present value or principal

Therefore

$FV=10,000(1.08)^4(1.06)^6$

$FV=17214.52556$

Question 2

An investor deposits N10,000. Ten years later, it is worth N17,910. What rate of return did the investor earn on the investment?

Solution:

The formula for future value is

$$FV=PV\left(1+\frac{i}{m}\right)^{nm}$$

$17910=10,000(1+i)^{10}$

Dividing both sides by N10,000

$\frac{17910}{10000}=\frac{10000(1+i)^{10}}{10000}$

${1.7910}=(1+i)^{10}$

${1.7910}^{\frac{1}{10}}=(1+i)$

$1.060=1+i$

$i=1.060-1=0.060$

$0.060$ expressed in percentage is 6%.

Hence, the rate of return is 6%.

Question 3

You deposited N10,000 in a savings account for two year. During the first year, the investment earned 20% for the year. During the second year, you earned only 4% for that year. How much is your deposit worth at the end of the two years.

Solution:

$FV=10,000(1+0.2)^1 × (1+0.04)^1$

$FV=10,000(1.2)(1.04)=12480$

Question 4

How much should I deposit today in a bank account paying interest compounded quarterly if you wish to have N10,0000 at the end of 3 months assuming the banks pay 5 per cent annually.

Solution:

The formula for present value is 

$$PV=\frac{FV}{\left(1+\frac{i}{m}\right)^{nm}}$$

n is the case is 0.25 because 3 months is one-fourth of a year. and m is 4 because interest compounds quarterly.

$PV=\frac{10,000}{(1+\frac{0.05}{4})^{0.25(4)}}$

$PV=\frac{10,000}{(1.0125)^{1}}$

$PV=\frac{10,000}{1.0125}=9876.54$

Question 5

At the end of the 24 months, Kenny intends to have 50,000 in his saving accounts. How much should he deposit in his savings account if the bank offer to pay him an 8% annual interest rate assuming interest is compounding monthly.

Solution

Here n=2 because 24 months is equivalent to 2 years. 

$PV=\frac{50,000}{\left(1+\frac{0.08}{12}\right)^{2(12)}}$

$PV=\frac{50,000}{\left(1+0.006666667\right)^{24}}$

$PV=\frac{50,000}{\left(1.006666667\right)^{24}}$

$PV=\frac{50,000}{1.17288864}=42629.8$

Question 6

James wishes to have N6000 at the end of the 12 months. Given that the annual interest rate is 9%, How much should he deposit today in his savings account to earn such an amount?

Solution:

n=1 because 12 same is equivalent to 1 year

$PV=\frac{6,000}{\left(1+\frac{0.09}{12}\right)^{1(12)}}$

$PV=\frac{50,000}{\left(1+0.0075\right)^{12}}$

$PV=\frac{50,000}{\left(1.0075\right)^{12}}$

$PV=\frac{50,000}{1.093806898}=5485.42893$

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Question 7

What rate of return is the bank charging you if you borrow N77650 and must reimburse N80,000 at the end of 2 quarters assuming interest is compounded quarterly.

Solution:

The future value of a lump sum is calculated thus:

$$FV=PV\left(1+\frac{i}{m}\right)^{n \times m}$$

Here, m is 4 because it compounds quarterly, n=0.5 because 2 quarters is equivalent to 0.5 year

$80,000=77650(1+\frac{i}{4})^{0.5\times 4}$

$\frac{80000}{77650}= (1+\frac{i}{4})^{2}$

$1.030264005=(1+\frac{i}{4})^{2}$

$\sqrt{1.030264005}=(1+\frac{i}{4})$

$1.01501921418-1=\frac{i}{4}$

$0.01501921418=\frac{i}{4}$

$0.060=i$

Expressed in percentage, this becomes

$i=6\%$

Question 8

You started saving for a car when you were in middle school. You have done some research and discovered that it will cost you about N9,500 for a used car that would fit your needs. You were encouraged to invest some of your savings to help you get to your financial goal faster. So far you have saved N7,500. If you invest this money in a savings account with a 3.3% interest rate compounded annually, how long will it take to earn enough money to go purchase the car?

Solution:

Recalled that the formula for an annuity that compound annually is

$FV=PV(1+i)^n$

Here the Future value is N9500 and the present value is N7500.

$9500=7500(1+0.033)^n$

$\frac{9500}{7500}=1.033^n$

$1.266666667=(1.033)^n$

Adding logarithm to both sides

$\log 1.266666667=n\log 1.033$

$n=\frac{\log.1266666667}{\log 1.033}$

$n=7.28085$

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