What is the formula in finding the present value of an ordinary annuity identify each variable Brainly?

Though your retirement is probably still a long way off, the earlier you start investing the more you can take advantage of the power of compounding interest to generate your savings.

Did it startle you to learn in Section 10.5 that your annual gross retirement income might need to be $160,000 when you turn 65 years old? How long will you live? The average Canadian life expectancy is around 80 years old; if you live to be that age, you will need 15 years of retirement income. Using a conservative interest rate of 5% compounded annually along with 3% annual inflation, this works out to saving up approximately $2 million for when you retire. A daunting goal, isn’t it? You might well wonder, "If I start saving $300 per month today, will I have enough?"

Clearly it is important to know how much your annuities are worth in the future. This matters not only for investments but also for debts, since most businesses and individuals pay off their debts through annuity structures. After you have made several annuity payments, can you tell at any time how much you or your company still owes on an outstanding debt?

In the previous section you learned to recognize the fundamental characteristics of annuities, so now you can start to solve any annuity for any unknown variable. There are four annuity formulas. This section covers the first two, which calculate future values for both ordinary annuities and annuities due. These formulas accommodate both simple and general annuities.

Ordinary Annuities

The future value of any annuity equals the sum of all the future values for all of the annuity payments when they are moved to the end of the last payment interval. For example, assume you will make $1,000 contributions at the end of every year for the next three years to an investment earning 10% compounded annually. This is an ordinary simple annuity since payments are at the end of the intervals, and the compounding and payment frequencies are the same. If you wanted to know how much money you have in your investment after the three years, the figure below illustrates how you would apply the fundamental concept of the time value of money to move each payment amount to the future date (the focal date) and sum the values to arrive at the future value.

While you could use this technique to solve all annuity situations, the computations become increasingly cumbersome as the number of payments increases. In the above example, what if the person instead made quarterly contributions of $250? That is 12 payments over three years, resulting in 11 separate future value calculations. Or if they made monthly payments, the 36 payments over three years would result in 35 separate future value calculations! Clearly, solving this would be tedious and time consuming—not to mention prone to error. There must be a better way!

The Formula

The formula for the future value of an ordinary annuity is indeed easier and faster than performing a series of future value calculations for each of the payments. At first glance, though, the formula is pretty complex, so the various parts of the formula are first explored in some detail before we put them all together.

The annuity formula is a more complex version of the rate, portion, and base formula introduced in Chapter 2. Relating Formula 2.2 and the first payment from the figure above gives the following:

\[\begin{array}{l}{\text { Portion }=\text { Base } \times \text { Rate }} \\ {\$ 1,210=\$ 1,000 \times(1+0.1)^{2}}\end{array} \nonumber \]

The portion equals the future value and the base equals the annuity payment amount. The rate is expressed as a formula and written as \((1 + 0.1)^2\). However, notice that each payment in the figure has a different exponent to reflect the compounding required. This requires a mathematical adaptation of \((1 + i)^N\), which permits the determination of an equivalent rate representing all payments in a single calculation.

This equivalent rate turns out to be a pretty complex expression, which is examined in three parts: the percent change overall, the percent change with each payment, and their quotient.

1. Numerator - The Percent Change Overall:\(\left[(1+i)^{\frac{CY}{PY}}\right]^{N}-1\). This part of the formula determines the percent change from the start of the annuity to the end of the annuity. It has three critical elements:
   1. Interest Rate Conversion \((1+i)^{\frac{CY}{PY}}\). The rate of interest that occurs with each payment must be known. All annuity calculations require the compounding period to equal the payment interval. If this is not already the case, then you must convert the expressed interest rate into an equivalent interest rate.
      • For simple annuities, no conversion is necessary since the frequencies are the same: \(CY = PY\). The exponent of \(\dfrac{CY}{PY}\) always equals 1 and has no effect.
      • For general annuities, recall Formula 9.4 for calculating equivalent interest rates. Here the old compounding frequency forms the numerator (\(CY_{Old}\)) and the new compounding frequency (which matches the payment frequency) forms the denominator (\(CY_{New}) = PY\). Thus \(\dfrac{CY_{OLD}}{PY_{New}}\) becomes \(\dfrac{CY}{PY}\).
   2. The Compounding \(N\). The exponent \(N\) compounds the periodic interest rate (which matches the payment interval) in accordance with the number of annuity payments made. For example, assume there are two end-of-year payments at 10% compounded semi-annually. If the interest rate is left semi-annually, there are four compounds over the two years, which does not match the payments. The interest rate is converted within the brackets from 10% compounded semi-annually to its equivalent 10.25% compounded annually rate. The end result is that interest will now compound twice over the two years, matching the number of payments.
   3. Removing the Starting Point (−1). Since you added 1 to perform the compounding, mathematically you now need to remove the 1. The end result is that you now know (in decimal format) how much larger the future value is relative to its starting value.
2. Denominator - The Percent Change with Each Payment: \((1+i)^{\frac{CY}{PY}}-1\). The denominator in the formula shows the percent change in the account with each payment made. It too ensures that the denominator has a periodic rate matching the payment interval.
3. The Quotient: By taking the numerator and dividing by the denominator, the percent change overall is divided by the percent change with each payment. This establishes a ratio between what is happening overall in the annuity versus what is happening with each transaction. Thus, this computation determines how much bigger the final value is relative to what happens with each payment. This ratio (the rate) is then multiplied against the annuity payment (the base) to get the final balance (the portion). You are now in a position to see the whole formula by which you calculate the future value of any ordinary annuity. This single formula applies regardless of the number of payments that are made. Formula 11.2 summarizes the variables and calculations required.

Formula 11.2

How It Works

There is a five-step process for calculating the future value of any ordinary annuity:

Step 1: Identify the annuity type. Draw a timeline to visualize the question.

Step 2: Identify the known variables, including \(PV, IY, CY, PMT, PY\), and Years.

Step 3: Use Formula 9.1 to calculate \(i\).

Step 4: If \(PV\) = $0, proceed to step 5. If there is a nonzero value for \(PV\), treat it like a single payment. Apply Formula 9.2 to determine \(N\) since this is not an annuity calculation. Move the present value to the end of the time segment using Formula 9.3.

Step 5: Use Formula 11.1 to calculate N for the annuity. Apply Formula 11.2 to calculate the future value. If you calculated a future value in step 4, combine the future values from steps 4 and 5 to arrive at the total future value.

Revisiting the RRSP scenario from the beginning of this section, assume you are 20 years old and invest $300 at the end of every month for the next 45 years. You have not started an RRSP previously and have no opening balance. A fixed interest rate of 9% compounded monthly on the RRSP is possible.

Step 1: This is a simple ordinary annuity since the frequencies match and payments are at the end of the payment interval.

Step 2: The known variables are \(PV\) = $0, \(IY\) = 9%, \(CY\) = 12, \(PMT\) = $300, \(PY\) = 12, and Years = 45.

Step 3: The periodic interest rate is \(i\) = 9% ÷ 12 = 0.75%.

Step 4: Since \(PV\) = $0, skip this step.

Step 5: The number of payments is \(N\) = 12 × 45 = 540. Applying Formula 11.2 gives the following:

\[FV_{ORD}=\$ 300\left[\dfrac{\left[(1+0.0075)^{\frac{12}{12}}\right]^{540}-1}{(1+0.0075)^{\frac{12}{12}}-1}\right]=\$ 2,221,463.54 \nonumber \]

Hence, 540 payments of $300 at 9% compounded monthly results in a total saving of $2,221,463.54 by the age of retirement.

Important Notes

Calculating the Interest Amount. For investment annuities, if you are interested in knowing how much of the future value is principal and how much is interest, you can adapt Formula 8.3, where \(I = S – P = FV – PV\).

• The \(FV\) is the solution to Formula 11.2.
• The \(PV\) is the principal (total contributions) calculated by taking \(N × PMT + PV\). In the figure above, \(N × PMT + PV\) = 540 payments × $300 + $0 = $162,000 in principal. Therefore, \(I\) =$2,221,463.54 − $162,000 = $2,059,463.54, which is the interest earned.

Your BAII+ Calculator. Adapting your calculator skills to suit annuities requires three important changes:

1. You now have a value for PMT. Be sure to enter it with the correct cash flow sign convention. When you invest, the payment has the same sign as the \(PV\). When you borrow, the sign of the payment is opposite that of \(PV\).
2. The P/Y is no longer automatically set to the same value as C/Y. Enter your values for P/Y and C/Y separately. If the values are the same, as in the case of simple annuities, then taking advantage of the “Copy” feature on your calculator let’s you avoid having to key the value in twice.
3. If a present value (\(PV\)) is involved, by formula you need to do two calculations using Formula 9.3 and Formula 11.2. If you input values for both \(PV\) and \(PMT\), the calculator does these calculations simultaneously, requiring only one sequence to solve.

Things To Watch Out For

In many annuity situations there might appear to be more than one unknown variable. Usually the extra unknown variables are "unstated" variables that can reasonably be assumed. For example, in the RRSP illustration above, the statement "you have not started an RRSP previously and have no opening balance" could be omitted. If something were saved already, the number would need to be stated. As another example, it is normal to finish a loan with a zero balance. Therefore, in a loan situation you can safely assume that the future value is zero unless otherwise stated.

Paths To Success

The ability to recognize a simple annuity can allow you to simplify Formula 11.2. Since CY = PY, these two variables form a quotient of 1 for the exponent. For a simple annuity, you can simplify any appearances of the following algebraic expressions in any annuity formula (not just Formula 11.2) as follows:

\((1+i)^{\frac{CY}{PY}}\) in the numerator can be simplified to \(1+i\)

and

\((1+i)^{\frac{CY}{PY}}-1\) in the denominator can be simplified to just \(i\)

Thus, for simple annuities only, you simplify Formula 11.2 as \[FV_{ORD}=PMT\left[\dfrac{(1+i)^{N}-1}{i}\right] \nonumber \]

Exercise \(\PageIndex{1}\): Give It Some Thought

Assume you had planned to make 10 annuity payments to an investment. However, before you started paying in to the investment, you changed your mind, doubling your original payment amount while still making 10 payments. What happens to the maturity value of your new investment compared to that of your original plan? Will your new balance be exactly double, more than double, or less than double? Explain and justify your answer.

Your new balance will be exactly double. Mathematically, you have taken PMT in Formula 11.2 and multiplied it by 2. That is the only difference between your original plan and your new plan. Therefore, the future value will double as well.

Example \(\PageIndex{1}\): Future Value of an Investment Account

A financial adviser is reviewing one of her client's accounts. The client has been investing $1,000 at the end of every quarter for the past 11 years in a fund that has averaged 7.3% compounded quarterly. How much money does the client have today in his account?

Solution

Step 1:

The payments are at the end of the payment intervals, and both the compounding period and the payment intervals are the same. This is an ordinary simple annuity. Calculate its value at the end, which is its future value, or \(FV_{ORD}\).

What You Already Know

Step 1 (continued):

The timeline shows the client's account.

Step 2:

\(PV\) = $0, \(IY\) = 7.3%, \(CY\) = 4, \(PMT\) = $1,000, \(PY\) = 4, Years = 11

How You Will Get There

Step 3:

Apply Formula 9.1.

Step 4:

Skip this step since \(PV\) = $0.

Step 5:

Apply Formula 11.1 and Formula 11.2.

Perform

Step 3:

\(i=7.3 \% \div 4=1.825 \%\)

Step 5:

\(N=4 \times 11=44\) payments

\[FV_{ORD}=\$ 1000\left[\dfrac{\left[(1+0.01825)^{\frac{4}{4}}\right]^{44}-1}{(1+0.01825)^{\frac{4}{4}}-1}\right]=\$ 66,637.03 \nonumber \]

Calculator Instructions

The figure shows how much principal and interest make up the final balance. After 11 years of $1,000 quarterly contributions, the client has $66,637.03 in the account.

Example \(\PageIndex{2}\): Future Value of a Savings Annuity

A savings annuity already contains $10,000. If an additional $250 is invested at the end of every month at 9% compounded semi-annually for a term of 20 years, what will be the maturity value of the investment?

Solution

Step 1:

The payments are at the end of the payment intervals, and the compounding period and payment intervals are different. This is an ordinary general annuity. Calculate its value at the end, which is its future value, or \(FV_{ORD}\).

What You Already Know

Step 1 (continued):

The timeline appears below.

Step 2:

\(PV\) = $10,000, \(IY\) = 9%, \(CY\) = 2, \(PMT\) = $250, \(PY\) = 12, Years = 20

How You Will Get There

Step 3:

Apply Formula 9.1.

Step 4:

Apply Formula 9.2 and Formula 9.3.

Step 5:

Apply Formula 11.1 and Formula 11.2 The final future value is the sum of the answers to step 4 (\(FV\)) and step 5 (\(FV_{ORD}\)).

Step 3:

\(i=9 \% \div 2=4.5 \%\)

Step 4:

\(N=2 \times 20=40\) compounds

\[FV=\$ 10,000(1+0.045)^{40}=\$ 58,163.64538 \nonumber \]

Step 5:

\(N=12 \times 20=240\) payments

\[FV_{ORD }=\$ 250\left[\dfrac{\left[(1+0.045)^{\frac{2}{12}}\right]^{240}-1}{(1+0.045)^{\frac{2}{12}}-1}\right]=\$ 163,529.9492 \nonumber \]

\[\text {Total }FV=\$ 58,163.64538+\$ 163,529.9492=\$ 221,693.59 \nonumber \]

Calculator Instructions

The figure shows how much principal and interest make up the final balance. The savings annuity will have a balance of $221,693.59 after the 20 years.

Important Notes

If any of the variables, including \(IY, CY, PMT\), or \(PY\), change between the start and end point of the annuity, or if any additional single payment deposit or withdrawal is made, a new time segment is created and must be treated separately. There will then be multiple time segments that require you to work left to right by repeating steps 3 through 5 in the procedure. The future value at the end of one time segment becomes the present value in the next time segment. Example \(\PageIndex{3}\) illustrates this concept.

Things To Watch Out For

Pay extra attention when the variable that changes between time segments is the payment frequency (\(PY\)). When inputted into a BAII+ calculator, the \(PY\) automatically copies across to the compounding frequency (\(CY\)). Unless your \(CY\) also changed to the same frequency, this means that you must scroll down to the CY window and re-enter the correct value for this variable, even if it didn't change.

Paths To Success

When working with multiple time segments, it is important that you always start your computations on the side opposite the unknown variable. For future value calculations, this means you start on the left-hand side of your timeline; for present value calculations, start on the right-hand side.

Example \(\PageIndex{3}\): Saving Up for a Vacation

Genevieve has decided to start saving up for a vacation in two years, when she graduates from university. She already has $1,000 saved today. For the first year, she plans on making end-of-month contributions of $300 and then switching to end-of-quarter contributions of $1,000 in the second year. If the account can earn 5% compounded semi-annually in the first year and 6% compounded quarterly in the second year, how much money will she have saved when she graduates?

Solution

Step 1:

There is a change of variables after one year. As a result, you need a Year 1 time segment and a Year 2 time segment. In both segments, payments are at the end of the period. In Year 1, the compounding period and payment intervals are different. In Year 2, the compounding period and payment intervals are the same. This is an ordinary general annuity followed by an ordinary simple annuity. You aim to calculate the future value, or \(FV_{ORD}\).

What You Already Know

Step 1 (continued):

The timeline for her vacation savings appears below.

Step 2:

Time segment 1: \(PV\) = $1,000, \(IY\) = 5%, \(CY\) = 2, \(PMT\) = $300, \(PY\) = 12, Years = 1

Time segment 2: \(PV = FV_1\) of time segment 1, \(IY\) = 6%, \(CY\) = 4, \(PMT\) = $1,000, \(PY\) = 4, Years = 1

How You Will Get There

For each time segment repeat the following steps:

Step 3:

Apply Formula 9.1.

Step 4:

Apply Formula 9.2 and Formula 9.3.

Step 5:

Apply Formula 11.1 and Formula 11.2. The total future value in any time segment is the sum of the answers to step 4 (\(FV\)) and step 5 (\(FV_{ORD}\)).

Perform

For the first time segment:

Step 3:

\(i=5 \% \div 2=2.5 \%\)

Step 4:

\(N=2 \times 1=2\) compounds

\[FV_{1}=\$ 1,000(1+0.025)^{2}=\$ 1,050.625 \nonumber \]

Step 5:

\(N=12 \times 1=12\) payments

\[FV_{ORD1}=\$ 300\left[\dfrac{\left[(1+0.025)^{\frac{2}{12}}\right]^{12}-1}{(1+0.025)^{\frac{2}{12}}-1}\right]=\$ 3,682.786451 \nonumber \]

\[\text {Total } FV_{1}=\$ 1,050.625+\$ 3,682.786451=\$ 4,733.411451 \nonumber \]

For the second time segment:

Step 3:

\(i=6 \% \div 4=1.5 \%\)

Step 4:

\(N=4 \times 1=4\) compounds

\[FV_{2}=\$ 4,733.411451(1+0.015)^{4}=\$ 5,023.870384 \nonumber \]

Step 5:

\(N=4 \times 1=4\) payments

\[FV_{ORD2}=\$ 1,000\left[\dfrac{\left((1+0.015)^{\frac{4}{4}}\right]^{4}-1}{(1+0.015)^{\frac{4}{4}}-1}\right]=\$ 4,090.903375 \nonumber \]

\[\text { Total } FV_{2}=\$ 5,023.870384+\$ 4,090.903375=\$ 9,114.77 \nonumber \]

Calculator Instructions

The figure shows how much principal and interest make up the final balance. When Genevieve graduates she will have saved $9,114.77 toward her vacation.

Annuities Due

An annuity due occurs when payments are made at the beginning of the payment interval. To understand the difference this makes to the future value, let's recalculate the RRSP example from earlier in this section, but treat it as an annuity due. You want to know the future value of making $1,000 annual contributions at the beginning of every payment interval for the next three years to an investment earning 10% compounded annually. This forms a simple annuity due. The figure below illustrates how you apply the fundamental concept of the time value of money to move each payment amount to the future date (the focal date) and sum the values to arrive at the future value.

Note that you do not end up with the same balance of $3,310 achieved under the ordinary annuity. Instead, you have a larger balance of $3,641. Why is this? Placing the two types of annuities next to each other in the next figure demonstrates the key difference between the two examples.

Both annuities have an identical sequence of three $1,000 payments. However, in the ordinary annuity no interest is earned during the first payment interval since the principal is zero and the payment does not occur until the end of the interval. On the other hand, in the annuity due an extra compound of interest is earned in the last payment interval because of the existing principal at the end of the second year. If you take the ordinary annuity's final balance of $3,310 and give it one extra compound you have \(FV\) = $3,310(1 + 0.1) = $3,641. In summary, the key difference between the two types of annuities is that an annuity due always receives one extra compound of interest.

The Formula

Adapting the ordinary annuity future value formula to suit the extra compound creates Formula 11.3. Note that all the variables in the formula remain the same; however, the subscript on the FV symbol is changed to recognize the difference in the calculation required.

Formula 11.3

How It Works

The steps required to solve for the future value of an annuity due are almost identical to those you use for the ordinary annuity. The only difference lies in step 5, where you use Formula 11.3 instead of Formula 11.2. Example \(\PageIndex{4}\) and Example \(\PageIndex{5}\) illustrate the adaptation.

Important Notes

To adapt your calculator to an annuity due, you must toggle the payment timing setting from END to BGN. The calculator default is END, which is the ordinary annuity. The payment timing setting is found on the second shelf above the PMT key (because it is related to the PMT!). To toggle the setting, complete the following sequence:

1. 2nd BGN (the current payment timing of END or BGN is displayed)
2. 2nd SET (it toggles to the other setting)
3. 2nd Quit (to get out of the window)

When the calculator is in annuity due mode, a tiny BGN appears in the upper right-hand corner of your calculator. To return the calculator to ordinary mode, repeat the above

Exercise \(\PageIndex{2}\): Give It Some Thought

Under equal conditions,

1. For any investment, which will always have the higher future value: an ordinary annuity or an annuity due? Explain.
2. For any debt, which will always have a higher future value: an ordinary annuity or an annuity due? Explain.

1. The annuity due will have the higher future value, since it always has one extra compound compared to an ordinary annuity.
2. The ordinary annuity will have the higher future value, since the principal in the first payment interval is higher and therefore more interest accrues than in the annuity due.

Example \(\PageIndex{4}\): Lottery Winnings

The Set for Life instant scratch n' win ticket offers players a chance to win $1,000 per week for the next 25 years starting immediately upon validation. If a winner was to invest all of his money into an account earning 5% compounded annually, how much money would he have at the end of his 25-year term? Assume each year has exactly 52 weeks.

Solution

Step 1:

The payments start immediately, and the compounding period and payment intervals are different. Therefore, this a general annuity due. Calculate its value at the end, which is its future value, or \(FV_{DUE}\).

What You Already Know

Step 1 (continued):

The timeline for the lottery savings is below.

Step 2:

\(PV\) = $0, \(IY\) = 5%, \(CY\) = 1, \(PMT\) = $1,000, \(PY\) = 52, Years = 25

How You Will Get There

Step 3:

Apply Formula 9.1.

Step 4:

Skip this step since there is no \(PV\).

Step 5:

Apply Formula 11.1 and Formula 11.3.

Perform

Step 3:

\(i=5 \% \div 1=5 \% \)

Step 5:

\(N=52 \times 25=1,300\) payments

\[\begin {aligned}FV_{DUE}&=\$ 1,000\left[\dfrac{\left[(1+0.05)^{\frac{1}{52}}\right]^{1300}-1}{(1+0.05)^{\frac{1}{52}}-1} \times(1+0.05)^{\frac{1}{52}}\right]\\&=\$2,544,543.22 \end{aligned} \nonumber \]

Calculator Instructions

The figure shows how much principal and interest make up the final balance. If the winner was to invest all of his lottery prize money, he would have $2,544,543.22 after 25 years.

Example \(\PageIndex{5}\): Saving into a Trust Fund with a Variable Change

When Roberto's son was born, Roberto started making payments of $1,000 at the beginning of every six months to a trust fund earning 5.75% compounded monthly. After five years, he changed his contributions and started depositing $500 at the beginning of every quarter. How much money will be in his son's trust fund when his son turns 18?

Solution

Step 1:

There is a change of variables after five years. As a result, you need two time segments. In both segments, payments are at the beginning of the period and the compounding periods and payment intervals are different. Therefore, Roberto has two consecutive general annuities due. Combined, calculate the future value, or \(FV_{DUE}\).

What You Already Know

Step 1 (continued):

The timeline for the trust fund is below.

Step 2:

Time segment 1: \(PV\) = $0, \(IY\) = 5.75%, \(CY\) = 12, \(PMT\) = $1,000, \(PY\) = 2, Years = 5

Time segment 2: \(PV = FV_1\) of time segment 1, \(IY\) = 5.75%, \(CY\) = 12, \(PMT\) = $500, \(PY\) = 4, Years = 13

How You Will Get There

For each time segment, repeat the following steps:

Step 3:

Apply Formula 9.1.

Step 4:

If there is a \(PV\), apply Formula 9.2 and Formula 9.3.

Step 5:

Apply Formula 11.1 and Formula 11.3. The total future value in any time segment is the sum of the answers to step 4 (\(FV\)) and step 5 (\(FV_{ORD}\)).

Perform

Step 3:

\(i=5.75 \% \div 12=0.4719 \overline{6} \%\)

Step 4:

No \(PV\), so skip this step.

Step 5:

\(N=2 \times 5=10\) payments

\[FV_{DUE1}=\$ 1,000\left[\dfrac{\left[(1+0.004719 \overline{6})^{\frac{12}{2}}\right]^{10}-1}{(1+0.004719 \overline{6})^{\frac{12}{2}}-1} \times(1+0.004719 \overline{6})^{\frac{12}{2}}\right]=\$ 11,748.47466=FV_1 \nonumber \]

\[\text { Total } FV_{2}=\$ 24,765.17+\$ 38,907.21529=\$ 63,672.39 \nonumber \]

Calculator Instructions

The figure shows how much principal and interest make up the final balance. When Roberto’s son turns 18, the trust fund will have a balance of $63,672.39.