# HARD QUESTIONS ON MATHEMATICS OF FINANCE

Being the diligent student that you are, your mind would have probably being ingrained with the differences between annuity and lump sum.

We'll utilize the concept of present and future value of a lump sum and annuity to answer some problems in this post.

Forthwith, let's get started

Example 1

Suppose you put N350 in an account that earns 6% annual interest, compounded monthly, at the start of each month. Calculate the total value in this account after 25 months.

Solution:

$FV=350\left[\frac{\left(1+\frac{0.06}{12}\right)^{25}-1}{\frac{0.06}{12}}\right]\left(1+\frac{0.06}{12}\right)$

$FV=350\left[\frac{\left(1+0.005\right)^{25}-1}{0.005}\right]\left(1+0.005\right)$

$FV=350\left[\frac{\left(1.005\right)^{25}-1}{0.005}\right]\left(1.005\right)$

$FV=350\left[\frac{1.1327955751-1}{0.005}\right]\left(1.005\right)$

$FV=350\left[\frac{0.1327955751}{0.005}\right]\left(1.005\right)=9342.17$

Example 2

Dayo is 20 years old. How much does he need to deposit every month in an account paying 6% annually to accumulate N1,000,000 by age 65

Solution:

The formula for the future value of the ordinary annuity is exemplified as

$$FV=C\left[\frac{\left(1+\frac{i}{m}\right)^{n m}-1}{\frac{i}{m}}\right]$$

Here, n=65-20=45 and m=12, FV=1,000,000, I=0.06

$1,000,000=C\left[\frac{\left(1+\frac{0.06}{1}\right)^{12(45)}-1}{\frac{0.06}{12}}\right]$

$1,000,000=C\left[\frac{\left(1+0.005\right)^{540}-1}{0.005}\right]$

To simplify the calculation, we cross-multiply

$1,000,000 × 0.005=C\left[(14.779963)-1\right]$

$5000=C(13.779963)$

$\frac{5000}{13.779963}=C$

$C=362.845676$

Example 3

Kenny believes he will be able to deposit N4000 at the end of each of the next three years in a bank account paying 8% interest. He has N7000 in his account right now. How much money will he have in the account at the end of the three years.

Solution:

The N7000 means we should calculate the future value of the lump sum. However, N4000 payment for the next three years means that the question also involved annuity.

Hence, the future value will be the sum of the lump sum and annuity.

$Fv=7000(1+0.08)^3+\frac{4000[(1+0.08)^3-1]}{0.08}$

$FV=7000(1.259712)+\frac{4000(0.259712)}{0.08}$

$FV=8817.984+12985.6=21803.584$

Therefore, N21803.584 will be in the account at the end of the third years

Example 4

Daniel needs N1000 in one year and N2000 more in two years. What is the present value of the cash flows if he can make 9% on his money?

Solution

The present value of the cash flow is simply the sum of the present value of the two cash flows.

$PV=\frac{2000}{(1+0.09)^2}+\frac{1000}{(1+0.09)^1}$

$PV=1683.3599987+917.4311927=26900.7912$

Example 5

Joshua has been offered an investment that will make three N5000 payment. The first payment is due four years from today, the second in five years, and the third in six years. What is the present value of this investment if he earns 11% interest?

Solution

The present value of the investment is simply the sum of the present value of the six year, fifth year and fourth year

$PV=\frac{5000}{1.11^{6}}+\frac{5000}{1.11^{5}}+\frac{5000}{1.11^{4}}$

$PV=2673.204+2967.257+3293.655=8934.116$

Example 6

I owe N100. To pay off this debt, I've decided to make a monthly payment of N20 at 1.5 % per month. How many months will it take me to pay off this debt completely?

Solution:

This is a present value of an ordinary annuity problem since we already known the present worth of the debt

$1000=20\left[\frac{1-\left(1+0.015\right)^{-n}}{0.015}\right]$

The interest rate is expressed "per month," not "per year." As a result, the interest rate does not need to be divided by m.

We don't need to multiply $-n$ by m because we're just interested in the number of months, not the number of years.

Cross multiplying

$50 \times 0.0015=1-(1.015)^{-n}$

$0.75=1-(1.015)^{-n}$

$(1.015)^{-n}=1-0.75$

$(1.015)^{-n}=0.25$

$-n\log 1.015=\log 0.25$

$-n=\frac{\log 0.25}{\log 1.015}$

$n=93.11$

Example 7

John Bird life insurance is trying to sell you an investment policy that will pay you and your heirs N700 per year forever. What interest rate should be charged if the policy costs N8,500?

Solution:

Forever means it's a perpetuity. Hence

$8500=\frac{700}{i}$

$8500i=700$.

$\frac{i}=\frac{700}{8500}=0.0824$

$0.824$ expressed in percentage is 8.24%

Example 8

I recently turned 20 years old, and a wealthy uncle of mine has set up a trust fund for me that will give me N100,000 when I become 26. How much is this fund worth now, assuming a discount rate of 11%?

Solution:

Number of years is 26-20=6

$PV=\frac{100,000}{(1+0.11)^6}$

$PV=\frac{100,000}{(1.11)^6}=53.464.08361$