###### Introduction to Probability: Definitio ...

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The Future Value (FV) of a single sum of money is the future amount of money invested today at a given interest rate (r) for a specified period. Denoted by \(\text{FV}_{N}\), the future value of a single sum of money is given by:

$$\text{FV}_{N}=\text{PV}\left(1+r\right)^{N}$$

Where

PV = present value of the investment

FV* _{N} *= future value of the investment

N=number of periods (Years)

Note that the formula above is based on the time value of money.

Suppose you deposited $5,000 in a savings account that earns an annual compound interest of 7%, what would be the value of money in the savings account after ten years?

From the question:

PV = 5,000

FV* _{N} *=?

N=10

$$\Rightarrow \text{FV}_{N}=\text{PV}\left(1+r\right)^{N}=5000\left(1+0.07\right)^{10}=9,835.7568$$

When the interest rate is compounded more than once in a year (as is the case of an investment paying interest more than once in a year), the annual interest rate, which should be converted to monthly (or any other frequency such as half-yearly) interest rate, is termed as the **stated annual rate of interest **or **quoted interest rate. **It is denoted as \(r_s\).

For example, if the monthly interest rate is 0.65, then the stated interest rate is 0.65×12=7.8.

Under more than one compounding period per year, the future value of a single sum of money is

$$\text{FV}_{N}=\text{PV}\left(1+\frac{r_{s}}{m}\right)^{\text{mN}}$$

Where

*m=*number of compounding periods per year

N=number of years

\(r_s\)=annual stated rate of interest

Suppose you deposited $2,000 in a savings account which earns an annual interest rate of 7% compounded monthly, what would be the value of the money in the account after ten years?

**Solution**

From the question:

*m=*12

N=12

r=7%

So,

$$\text{FV}_{N}=\text{PV}\left(1+\frac{r_{s}}{m}\right)^{\text{mN}}=2000\left(1+\frac{0.07}{12}\right)^{12\times 10}=4,019.32$$

In case the compounding period per year is infinite, that is m→∞, the future value of the single sum of money is expressed as:

$$\text{FV}_{N}=\text{PV}e^{r_{s}N}$$

For example, in our case above, if the annual rate of 7% interest was continuously compounded, then the future value of the deposits would be:

$$\text{FV}_{N}=\text{PV}e^{r_{s}N}=2000 \times e^{0.07\times 10}=4,027.51$$

Under the series payment, they are classified into equal cashflows and unequal cashflows.

Annuities are used to determine the future value of equal cash flows. An annuity is a regular series of payments. An individual submits funds to a financial institution and, in turn, receives a regular series of payments. They are a direct result of the time value of money. Most annuities will require the individual to submit funds at the beginning of the agreement. However, the series of payments could begin immediately or after a specified period of time and could continue throughout the life of the individual. Alternatively, the term of the annuity may be fixed at, say, 20 years. This gives rise to the following types of annuities.

In an ordinary annuity, the series of payments do not begin immediately. Instead, payments are made at the end of each period, usually a month or year. Such payments are said to be made in arrears (beginning at time t=1).

The future value of an ordinary annuity is derived as outlined below.

Consider an annuity amount of A paid at the end of each period for N period with the interest rate per period denoted by *r, *then the future value of equal cash flows is given by:

$$\text{FV}_{N}=\text{A}\left[\left(1+r \right)^{N-1}+\left(1+r \right)^{N-2}+\left(1+r \right)^{N-3}+\dots +\left(1+r \right)^{1}+\left(1+r \right)^{0}\right]$$

This reduces into:

$$\text{FV}_{N}=\text{A}\left[\frac{\left(1+r\right)^{N}-1}{r}\right]$$

The factor \(\frac{\left(1+r\right)^{N}-1}{r}\) is termed as future value annuity factor that gives the future value of an ordinary annuity of $1 per period. Therefore, we multiply any amount by this factor to get the future value of that particular annuity.

Suppose you invest $2000 per year in a stock index fund, which earns 9% per year, for the next ten years, what would be the closest value of the accumulated value of the investment upon payment of the last instalment?

From the information given in the question:

A=2000

N=10

r=9%

So that:

$$\text{FV}_{N}=\text{A}\left[\frac{\left(1+r\right)^{N}-1}{r}\right]=2000\left[\frac{\left(1+0.09\right)^{10}-1}{0.09}\right]=30,385.8594$$

Annuity due is a type of annuity where payments start immediately at the beginning of time, at time t=0. In other words, payments are made at the beginning of each period.

The formula for the future value of an annuity due is derived by:

$$\text{FV}_{N}=\text{A}\left[\left(1+r\right)^{N}+\left(1+r\right)^{N-1}+\left(1+r\right)^{N-2}+…+ \left(1+r\right)^{1}\right]$$

Which reduces to:

$$\text{FV}_{N}=\text{A}\left[\frac{\left(1+r\right)^{N}-1}{d}\right]$$

Where

$$\text{d}=\frac{r}{1+r}$$

If in our ordinary annuity example, the payments were, instead, paid at the beginning of each period, then the future value of the payments would be:

$$\text{FV}_{N}=\text{A}\left[\frac{\left(1+r\right)^{N}-1}{d}\right]=2000\left[\frac{\left(1.09\right)^{9}-1}{\frac{0.09}{109}}\right]=33,120.5868$$

There are some instances where cash flow payments are not equal. A good example is a saving pattern of self-employed individual who saves depending on the income level at a particular time.

The future value of an unequal stream of payments is calculated by summing up the individual future values of the payments.

Consider the following example.

A small-scale businessman deposits money into his savings account at the beginning of each year, depending on the returns of the business. He deposits $1000 in the first year, $2000 in the second year, $5000 in the third, and $7000 in the fourth year. The account credits interest at an annual interest rate of 7%. What is the closest value of the accumulated money in the savings account at the beginning of year 4?

**Solution**

The future value of the unequal payments is the sum of individual accumulations:

$$1000\left(1.07\right)^3+2000\left(1.07\right)^2\:+5000\left(1.07\right)^1+7000\left(1.07\right)^0=16,975.38$$

**Note: **the payments are made at the beginning of each year.

The * present value* (

$$\text{FV}_{N}=\text{PV}\left(1+r\right)^{N}$$

If we make the present value (PV) the subject of the formula, by dividing both sides of the above equation by

$$\frac{\text{FV}_{N}}{\left(1+r\right)^{N}}=\frac{PV\left(1+r\right)^{N}}{\left(1+r\right)^{N}}$$

$$\Rightarrow PV=\text{FV}_{N}\left(1+r\right)^{-N}$$

Where

PV = present value of the investment

FV* _{N} *= future value of the investment

N=number of years

\(\left(1+r\right)^{-N}\) is called the present value factor, which is intuitively the reciprocal of the future value factor.

An investor wishes to save $100,000 in the next eight years. The investor opts for a savings account that pays 6% annual interest. Calculate the closest value of the deposit the investor should make to reach the target.

**Solution**

From the information given:

FV=100,000

r=6%

N=8

PV=?

So,

$$\Rightarrow PV=\text{FV}_{N}\left(1+r\right)^{-N}=100,000(1.06)^{-8}=62,741.2371$$

You could use your financial calculator to save time!

When the frequency of compounding is more than once per year i.e., quarterly, monthly, etc., the formula is analogously (as illustrated above: by making PV the subject of the formula) defined as:

$$PV=\text{FV}_{N}\left(1+\frac{r_{s}}{m}\right)^{-mN}$$

Where

*m=*number of compounding periods per year

N=number of years

\(r_s\)=annual stated rate of interest

An investor wishes to save $100,000 in the next eight years. The investor opts for a savings account that pays 6% annual interest compounded monthly. Calculate the closest value of the deposit the investor should make to reach the target.

**Solution**

From the information given in the question,

FV=100,000

=6%

N=8

m=12

So,

$$PV=100,000\left(1+\frac{0.06}{12}\right)^{-12\times 8}=61,952.3909$$

Similarly, for the continuously compounded interest rate, the present value is given by

$$PV=\text{FV}_{N}e^{-\text{N}\text{r}_{s}}$$

A fund continuously accumulates to $4,000 over ten years at a 10% annual interest rate. Calculate the closest value of the present value of this fund.

**Solution**

From the question,

FV=4000

r_s=10%

N=10

So,

$$PV=\text{FV}_{N}e^{-\text{N}\text{r}_{s}}=4000\times e^{-10\times 0.10}=1,471.5178$$

Many of the investments offer a series of either uneven, relatively even, or unequal payments over a given period of time. Therefore, valuing their present values assumes different methodologies.

The present value of an equal series of cash flows is valued using annuities as defined previously. We shall consider ordinary annuity due.

Note that in an ordinary annuity, the series of payments do not begin immediately. Instead, payments are made at the end of each period. The present value of an annuity is equal to the sum of the current value of each annuity payment:

$$\begin{align}\text{PV} &=A\left(1+r\right)^{-1}+A\left(1+r\right)^{-2}+\dots +A\left(1+r\right)^{-(N-1)}+A\left(1+r\right)^{-N} \\&=A\left[\left(1+r\right)^{-1}+\left(1+r\right)^{-2}+\dots +\left(1+r\right)^{-(N-1)}+\left(1+r\right)^{-N}\right]\\ &=A\left[\frac{1-(1+r)^{-N}}{r}\right]\end{align}$$

So present value of an ordinary annuity is given by:

$$PV=A\left[\frac{1-(1+r)^{-N}}{r}\right]$$

A financial asset generates returns of $10,000 at the end of each year for ten years. The required rate of return is 7% per year. How much must you pay to buy this asset?

**Solution**

To find the cost of purchasing the asset, we need to find the sim of the present values of the series of payments from the asset, which in this case, is the current value of an ordinary annuity.

From the question:

A=10,000

R=7%=0.07

N=10

So the present value is given by:

$$PV=A\left[\frac{1-(1+r)^{-N}}{r}\right]=10000\left[\frac{1-(1+0.07)^{-10}}{0.07}\right]=\$ 70,235.81$$

Recall that the annuity due is a type of annuity where payments start immediately at the beginning of time, that is, at time t=0. The present value of the annuity due is presented as:

$$\begin{align}\text{PV} &=A\left[\left(1+r\right)^{0}+\left(1+r\right)^{-1}+\left(1+r\right)^{-2}\cdots\right]\\ &=A\left[\frac{1-(1+r)^{-N}}{d}\right]\end{align}$$

So present value of an ordinary annuity is given by:

$$PV=A\left[\frac{1-(1+r)^{-N}}{d}\right]$$

Where

$$d=\frac{r}{1+r}$$

It is easy to see that, from

$$\begin{align}\text{PV} &=A\left(1+r\right)^{0}+A\left(1+r\right)^{-1}+A\left(1+r\right)^{-2}+\cdots +A\left(1+r\right)^{-(N-1)}\\ &=A\left[1+\left(1+r\right)^{-1}+\left(1+r\right)^{-2}+\cdots +\left(1+r\right)^{-(N-1)}\right]\\ &=A+A\left[\left(1+r\right)^{-1}+\left(1+r\right)^{-2}+\cdots +\left(1+r\right)^{-(N-1)}\right]\\ &=A+A\left[\frac{1-(1+r)^{-(N-1)}}{r}\right] \end{align}$$

the annuity due is equivalent to a lump sum of A plus the present value of the ordinary annuity for N-1 years.

You have just retired and your pensioner agrees to pay you $12,000 per year for the next 20 years, and you receive the first payment today. Assuming that the interest rate is 7%, calculate the closest value of the present value of your payments.

**Solution**

From the question,

A=12,000

N=20

r=7%

Now, this is an annuity due since the first payment starts today. You should use the formula:

$$PV=A+A\left[\frac{1-(1+r)^{-(N-1)}}{r}\right]=12,000+12,000\left[\frac{1-(1.07)^{-19}}{0.07}\right]=136,027.1429$$

Alternatively, you could use the following usual annuity due formula:

$$PV=A\left[\frac{1-(1+r)^{-N}}{d}\right]=12,000\left[\frac{1-(1.07)^{-20}}{\frac{0.07}{1.07}}\right]=136,027.1429$$

A perpetuity is an infinite series of regular cashflows. Consider an ordinary annuity that is paid infinitely. That is, if we take the limit that is applicable on the formula of an ordinary annuity, we get:

$$PV=\lim_{N \to \infty} A\left[\frac{1-(1+r)^{-N}}{r}\right]=\frac{A}{r}$$

So, the present value of a perpetuity is given by:

$$PV=\frac{A}{r}$$

This formula is mostly applicable in stock valuation, where a stock offers a constant dividend.

A stock pays a constant dividend of $8 at the end of each year for 20 years at a 25% required rate of return. Calculate the present value of the stock dividends.

**Solution**

The constant dividends of the stock are valued as perpetuity. So from the question,

A=8

r=25%

So that:

$$PV=\frac{A}{r}=\frac{8}{0.25}=$32$$

Note that the present value is not always computed at time 0 (t=0) denoted by \(PV_0\). We can also find the present value at any time, such as t=2 or t=3. This concept allows us to combine both present values such as perpetuity and the present value of a single sum of cash flow. Consider the following example.

A stock pays a constant dividend of $10, starting at the beginning of year 6 (t=6). What is the present value of the perpetuity if the required rate of return is 20%?

**Solution**

First, we need to find the PV of the perpetuity at time 5 (because a regular annuity payment occurs at the end of a period) and then discount it to time 0. That is:

$$\text{PV}_{5}=\frac{A}{r}=\frac{10}{0.20}=\$50$$

So the present value today is given by:

$$\text{PV}_{0}=50\left(1.2\right)^{-5}=\$20.10$$

Just like calculating future values, the present value of a series of unequal cash flows is calculated by summing individual present values of the cashflows. In finance, the present value of many unequal series of cash flows is calculated using such a software as a spreadsheet. Consider the following example.

A small-scale businessman receives income at the end of each year from his business. He earns $1000 in the first year, $3000 in the second year, $5000 in the third, and $7000 in the fourth year. The applied annual interest rate is 7%.

Calculate the present value of the cash inflows today.

**Solution**

The future value of the unequal payments is the sum of individual accumulations:

$$V=1000(1.07)^{-1}+3000(1.07)^{-2}+5000(1.07)^{-3}+7000(1.07)^{-4}=12,976.65$$

QuestionMaxwell buys an annuity that makes a regular series of payments, each amounting to $200 per year for a period of 15 years. He is to receive level payments at the beginning of every year. What premium should Maxwell be willing to pay for this annuity, assuming that the effective rate of interest is 13.5%?

A. $1,715

B. $1,430

C.$1,200

The correct answer is B.

SolutionYou should note that this is actually an annuity due since payments are made at the beginning of the year. The premium payable should be the present value of the annuity.

First, you should write down the formula:

PV of ‘1’

=\( \cfrac {(1-V^n)}{d} \)Secondly, establish all the knowns and the unknown. This is why it is important to write down the formula before anything else!

n=15 years, r=0.135,

$$d= \frac{0.135}{1.135}=0.11894$$

$$V= \left(1+ 0.135\right)^{-1} = 0.8810$$

Finally, you can work out the solution.

$$ \begin{align*} PV & =\cfrac { (1- 0.88106^{15}) }{0.11894} \\ & = 7.149 \\ \end{align*} $$

This is the premium payable for an annuity of just $1 per year.

So for $200, \(PV = 200\times 7.15\) = $1,430

Candidates should learn to apply the other formulas in a similar manner.

*Reading 6 LOS 6e:*

*Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, perpetuity (PV only), and a series of unequal cash flows.*