# Present Values, Future Values, Annuities, and Series of Unequal Cashflows

## Future Values

### The Future Value (FV) of a Single Sum of Cash Flow

The Future Value (FV) of a single sum of money is the amount that money invested today at a given interest rate (r) for a specified period will translate into in future. 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 periods from today.
$$r$$ = Rate of interest per period.
$$N$$ = Number of periods (Years).

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

The factor (1 + r)N is called a future value factor.

• For a given interest rate, the higher the number of periods “N”, the greater the future value.
• For a given number of periods “N”, the higher the interest rate, the greater the future value.

Suppose you deposited 5,000 in a savings account that earns an annual compound interest of 7%; what would be the value of the money in the savings account after ten years? #### Solution From the question: PV = 5,000. FVN = ? r = 7%. N =10. $$\Rightarrow \text {FV} _ {N} = \text {PV} \left (1+r\right) ^ {N} = 5,000 \left (1+0.07\right) ^ {10} = 9, 835.7568$$ \begin {aligned} & \textbf {BA II Plus™ Financial Calculator Steps} \\&\begin {array} {l|l|l} \textbf {Steps} & \textbf {Explanation} & \textbf { Display } \\ \hline \text { [2nd] [QUIT] } & \text { Return to standard calc mode } & 0 \\ \hline\left[2^{\text {nd}} \right] [\text {CLRTVM}] & \text {Clears TVM Worksheet} & 0 \\ \hline 10[\mathrm {~N}] & \text {Years/periods} & \mathrm{N} = 10 \\ \hline 7 [\mathrm {I} / \mathrm{Y}] & \text { Set interest rate } & \mathrm{I} / \mathrm{Y} = 7 \\ \hline-5000[\mathrm {PV}] & \text {Set present value} & \mathrm{PV} = -5,000 \\ \hline 0 [\mathrm {PMT}] & \text {Set payment } & \mathrm{PMT} = 0 \\ \hline [\mathrm {CPT}] [\mathrm {FV}] & \text {Compute future value} & \mathrm {FV} = 9, 835.7568 \\\end {array} \end {aligned} When the interest rate is compounded more than once in a year (as is the case of an investment that pays interest more than once a year), the annual interest rate, which should be converted to monthly (or any other frequency such as semi-annually) 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. #### Example: Calculating Future under Monthly Compounding Imagine that you deposited2,000 in a savings account that earns an annual interest rate of 7% compounded monthly. What would be the value of the money in your account after ten years?

Solution

From the question:

m = 12.

N = 12.

r = 7%.

So,

$$\text {FV} _ {N} = \text {PV} \left(1 + \frac {r _ {s}} {m} \right) ^ {\text {mN}} = 2,000\left (1+\frac {0.07} {12} \right) ^ {12\times 10} = 4,019.32$$

\begin {aligned} & \textbf {BA II Plus™ Financial Calculator Steps} \\ & \begin {array} {l|l|l} \textbf {Steps} & \textbf {Explanation} & \textbf {Display} \\ \hline \text { [2nd] [QUIT] } & \text {Return to standard calc mode} & 0 \\ \hline \text { [2nd] [CLR TVM }] & \text {Clears TVM Worksheet} & 0 \\ \hline 120 [\mathrm {N}] & \text {Years/periods} (12 \times 10 = 120) & \mathrm{N} = 120 \\ \hline 0.583333 [\mathrm{I} / \mathrm {Y}] & \text {Set interest rate} (7 / 12 = 0.583333) & \mathrm{I} / \mathrm {Y} = 0.58333 \\ \hline-2,000 [\mathrm {PV}] & \text {Set present value} & \mathrm {PV} = -2,000 \\ \hline 0 [\mathrm {PMT}] & \text {Set payment} & \mathrm{PMT} = 0 \\ \hline [\mathrm {CPT}] [\mathrm {FV}] & \text {Compute future value} & \mathrm {FV} = 4, 019.32 \\ \end {array} \end {aligned}

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

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

Take the above case as an example. If the annual rate of 7% interest were continuously compounded, then the future value of the deposits would be:

$$\text {FV} _ {N} = \text {PV}e^ {r _ {s}N} = 2,000 \times e^ {0.07\times 10} = 4, 027.51$$

Exam tip: there are no TMV buttons on your calculator. When dealing with continuous compounding, simply remember the formula.

Important points to note regarding compounding:

• The greater the “N” (number of periods), the higher the compounded interest earned, all else equal.
• The higher the interest rate, the higher the compounded interest earned, all else equal.

## The Future Value of a Series of Payments

Series of payments are classified into equal cashflows and unequal cashflows.

### Future Values of Equal Cashflows

Annuities are used to determine the future value of equal cashflows. An annuity is a series of even cashflows. There are two types of annuities: ordinary annuities and annuities due.

### Ordinary Annuity

An ordinary annuity is an annuity where cash flows occur at the end of each period. Such payments are said to be made in arrears (beginning at time t = 1).

The future value of an ordinary annuity is derived as follows:

Consider an annuity amount of $$A$$ paid at the end of each period for $$N$$ periods with the interest rate per period denoted by $$r$$. In this instance, the future value of equal cashflows 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:

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

#### Solution

From the information given in the question:

$$A$$ = 2,000.

$$N$$ = 10.

$$r$$ = 9%.

So that:

$$\text {FV} _ {N} = \text {A} \left [\frac {\left (1 + r\right)^ {N} – 1} {r} \right] = 2,000\left [\frac{\left (1 + 0.09\right)^ {10} -1} {0.09} \right] = 30,385.8594$$

## Annuity Due

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:

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

#### Example: Future Value of an Annuity Due

Refer to our ordinary annuity example. If the payments were instead made at the beginning of each period, then the future value of the payments would be:

$$\text {FV} _ {N} = \text {A} \left [\frac {\left(1 + r\right)^ {N} – 1} {d} \right]=2,000\left [\frac {\left (1.09\right)^ {9} – 1}{\frac {0.09} {109}} \right] =$$

### Future Values of Unequal Series of Payments

There are some instances where cash flow payments are not equal. The saving pattern of self-employed individuals who save depending on their income level at a particular time is a good case in point.

The future value of an unequal stream of payments is calculated by working out the sum of the future values of individual payments.

Consider the following example.

#### Example: Future Value of Unequal Cashflows

A small-scale businessman deposits money into his savings account at the beginning of each year, depending on the business returns. He deposits $1,000 in the first year,$2,000 in the second year, $5,000 in the third year, and$7,000 in the fourth year. The account credits interest at an annual interest rate of 7%. What is the closest value of the money accumulated in the savings account at the beginning of year 4?

Solution

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

$$1,000 \left (1.07\right)^3 + 2,000 \left (1.07 \right)^ 2\: + 5,000 \left (1.07 \right)^ 1 + 7000 \left (1.07 \right)^ 0 = 15,864.48$$

Note: He makes payments at the beginning of each year.

## Present Values

### Present Values of Single Cashflow

The present value (PV) is the current value of a future sum of money (Future value, FV) or series of cashflows given a specified rate of return. Note that the future value of a single sum of money is given by:

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

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.
FVN = Future value of the investment N periods from today.
r = Rate of interest per period.

N = Number of  Years.

$$(1 + r)^{-N}$$ 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 (quarterly, monthly, etc.), the formula is analogously (as illustrated above) defined as follows: $$PV = \text {FV} _ {N} \left (1+\frac {r _ {s}} {m} \right)^ {-mN}$$ Where $$m$$ = Number of compounding periods per year. $$N$$ = Number of years. $$r_s$$ = Annual stated rate of interest. #### Example: Calculating the Present Value of Single Cashflow under Multiple Compounding 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.

$$r_s$$ = 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 of the investment 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 present value of this fund. Solution From the question, FV = 4,000. $$r_s$$ = 10%. N = 10. So, $$PV = \text {FV} _ {N}e^ {-\text {N} \text {r} _ {s}} = 4,000\times e^ {-10\times 0.10} = 1, 471.5178$$ ## Present Value of a Series of Cashflows Many investments offer a series of uneven, relatively even, or unequal payments over a given period. Therefore, different methodologies are employed in the valuation of their present values. #### The Present Value of a Series of Equal Cashflows As clarified earlier, annuities are used to determine the present value of a series of equal cash flows. We shall consider ordinary annuity due. ### Ordinary Annuity Remember that the series of payments do not begin immediately in an ordinary annuity. Instead, payments are made at the end of each period. It is further worth noting that 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, the present value of an ordinary annuity is given by: $$PV = A\left [\frac {1 – (1 + r)^ {-N}} {r} \right]$$ #### Example: Calculating the Present Value of the Ordinary Annuity A financial asset generates returns of10,000 at the end of each year for ten years.  The required rate of return is 7% per year. How much must one pay to buy this asset?

Solution

To find the cost of purchasing the asset, we need to find the sum of the present values of the series of payments from the asset. In this case, this 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:

$$PV = A\left [\frac {1-(1+r)^ {-N}} {r} \right] = 10,000 \left [\frac {1 – (1 + 0.07)^ {- 10}} {0.07} \right] = \ 70, 235.81$$

### Annuity Due

Remember that 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, the 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.

Imagine that you have just retired, and your pensioner agrees to pay you 12,000 per year for the next 20 years, where you receive the first payment today. Assuming an interest rate of 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. Here is the formula to use: $$PV = A + A\left [\frac {1-(1 + r)^ {-(N – 1)}} {r}\right] = 12,000 + 12,000 \left [\frac {1- (1.07)^ {-19}} {0.07} \right] = 136,027.1429$$ Alternatively, we could use the usual annuity due formula: $$PV = A\left [\frac {1- (1 + r)^ {-N}} {d} \right] = 12,000\left [\frac {1-(1.07)^ {-20}} {\frac {0.07}{1.07}} \right] = 136,027.1429$$ Calculator steps to change the mode to “BGN” and then back to “END” mode. \begin{aligned} &\textbf {BA II Plus™ Financial Calculator Steps}\\ & \begin {array} {l|l|l} \textbf {Steps} & \textbf {Explanation} & \textbf {Display} \\ \hline \begin {array} {l} {[\text {[2nd }]} \\ {[\mathrm{SET}]} \end {array} [\mathrm {BGN}] [2 \mathrm {nd}] & \text {Puts in BGN mode} & 0 \\ \hline [2 \mathrm {nd}] [\mathrm {QUIT}] & \text {Return to standard calculator mode} & 0 \\ \hline [2 \mathrm {nd}] [\mathrm {CLRTVM}] & \text {Clears TVM Worksheet} & 0 \\ \hline 20 [\mathrm {~N}] & \text {years/periods} & \mathrm {N} = 20 \\ \hline 7 [\mathrm {I} / \mathrm {Y}] & \text {Set interest rate} & \mathrm{I} / \mathrm{Y} = 7 \\ \hline 0[\mathrm {FV}] & \text {Set future value} & \mathrm {FV} = 0 \\ \hline 12000 [\mathrm {PMT}] & {\text {Set payment}} & \begin {array} {l} \mathrm{PMT} \\ 12,000 \end{array} \\ \hline [\mathrm {CPT}] [\mathrm {PV}] & {\text {Compute present value }} & -136,027.1429 \\ \end{array}\end{aligned} Note: Always remember to switch back to “END” mode after solving the problem. To switch back to “END” mode, follow the steps outlined below. • [2nd] [PMT]. You should see BGN on the screen. • Now, press 2nd ENTER to change that to END and then press 2nd CPT to exit setting the calculation mode. • In the “END” mode, the screen’s upper-right corner will be blank. We can also compute the PV of an annuity due by calculating the PV of an ordinary annuity and multiplying that PV by [1 + periodic compounding rate (r)]. That is, $$\text {PV (Annuity due) = PV (Ordinary annuity) × (1 + r)}$$ ## Present Value of a Perpetuity and Present Values Indexed at Times Other Than t = 0 ### Perpetuity A perpetuity is an infinite series of regular cashflows. Consider an ordinary annuity that is paid infinitely. That is, if we take the limit as 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, in a case where a stock offers a constant dividend. #### Example: Calculating the Present Value of a Perpetuity A stock pays a constant dividend of8 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:

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

### Present Values Indexed at Times Other Than t = 0

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, i.e., perpetuity and the present value of a single sum of cash flow. Consider the example given below.

A stock pays a constant dividend of $10, starting at the beginning of year 6 (t = 6). What is the perpetuity’s present value if the required return rate is 20%? Solution First, we need to find the PV of the perpetuity at the 5th time (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: $$\text {PV} _ {0} = 50 \left (1.2\right)^ {-5} = \20.10$$ #### Present Value of a Series of Unequal Cashflows Just like calculating future values, the present value of a series of unequal cash flows is calculated by summing individual present values of cash flows. In finance, the present value of a series of many unequal cash flows is calculated using software such as a spreadsheet. Consider the example below. #### Example: Calculating the Present Values of Unequal Series of Payments A small-scale businessman receives income from his business at the end of each year. He earns$1,000 in the first year, $3,000 in the second year,$5,000 in the third, and $7,000 in the fourth year. An annual interest rate of 7% is applied. Calculate the present value of the cash inflows today. Solution The future value of the unequal payments is the sum of individual accumulations: $$V = 1,000 (1.07)^ {-1} + 3,000 (1.07)^ {-2} + 5,000 (1.07)^ {-3} + 7,000 (1.07)^ {-4} = 12,976.65$$ ## Question Maxwell buys an annuity that makes a series of regular payments, each amounting to$200 per year for a period of 15 years. In addition, he is to receive level payments at the beginning of every year. What premium should he be willing to pay for this annuity, assuming a 13.5% effective interest rate?

A. $1,715. B.$1,430.

C.1,200. The correct answer is B. Solution From the question, it is clear that this is an annuity due since payments are made at the beginning of the year. The premium payable should be the present value of the annuity, and it is determined using the following steps. Step I: Note down the applicable formula. $$\text{PV of ‘1’} = \cfrac{(1-V^n)}{d}$$ Step II: Determine the values for the variables in the formula, i.e., n = 15 years, r = 0.135. $$d = \frac {0.135} {1.135} = 0.11894$$. $$V= \left(1+ 0.135\right)^ {-1} = 0.8810$$. Step III: Determine the present value. \begin {align*} PV & = \cfrac { (1- 0.88106^ {15}) } {0.11894} \\ & = 7.149 \\ \end {align*} Note that this is the premium payable for an annuity of just1 per year.

Therefore, for \$200, the present value will be as follows:

$$\text{PV} = 200 \times 7.15 = \1,430$$

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