Annuities
3.2 FV of Annuities: Calculator Approach
A. Future Value of Annuities and Interest Amount
After covering the basics and types of annuities, we now focus on understanding and calculating the future value of annuities. This skill is crucial for financial planning, whether for retirement savings, education funds, or other long-term financial goals. Consider, for instance, a scenario where you’re investing $1,000 monthly into a retirement plan with a 10% annual interest rate compounded monthly over 30 years. To determine the amount you will have at retirement, you need to calculate the future value of this annuity. The future value here refers to the total accumulated value of all payments at the end of the annuity’s term, including interest.
For solving annuity problems, just like with compound interest, we use the time-value-of-money (TVM) worksheet in financial calculators (refer to Figure 2.2.1 in Section 2.2, Chapter 2). It’s also important to adhere to the cash flow sign convention (see Table 2.2.1 in Section 2.2) when inputting monetary values into the calculator. This ensures consistency and accuracy in our calculations.
It’s important to first verify the payment timing setting on your calculator before solving annuity problems. The default setting is usually at ‘END’ (end of the payment period), which is appropriate for ordinary annuities. However, for annuities due, where payments are made at the beginning of each period, this setting needs to be adjusted to ‘BGN’ (beginning of the payment period).
To change the payment timing, use the ‘2ND’ key followed by the ‘PMT’ key. This action accesses the secondary function ‘BGN’. To switch between ‘BGN’ and ‘END’, press the “2ND” and “ENTER” keys. Figure 3.2.1 illustrates the process for adjusting the payment timing setting on your calculator.
Figure 3.2.1 The process for adjusting the payment timing setting on a financial calculator
When the calculator is set to ‘BGN’ (annuity due), a small ‘BGN’ indicator will display in the top-right corner of the calculator screen, and it will remain until the setting is switched back to ‘END’. Remember, this setting adjustment is crucial for accurate calculations in your annuity problems.
Computing Interest Amount
The future value of an annuity includes both the total payments made and interest earned over the term of the annuity. To calculate the total interest earned, you should deduct the sum of all payments (calculated as [asciimath]PMT * N[/asciimath]) from the future value. The amount of interest earned can be calculated using
[asciimath]I=FV-(N*PMT)[/asciimath] Formula 3.2
If there is an initial investment at the start of the annuity (a nonzero Present Value, PV), you need to modify the calculation. In this scenario, the future value (FV) includes the initial investment, all subsequent payments, and interest earned over the term of the annuity. Therefore, the formula to calculate the interest earned is adjusted to
[asciimath]I = FV - (N*PMT)-PV[/asciimath] Formula 3.3)
In this formula, PMT is the regular payment amount, and N is the total number of payments, calculated as [asciimath]N = P//Y * t[/asciimath] (Formula 3.1a).
How to Compute the Future Value of Annuities Using a Financial Calculator
1. Clear the TVM Worksheet: Before starting, clear your TVM worksheet to remove any previous values by pressing ‘2ND’ followed by the ‘FV’ key.
2. Identify the type of annuity: Determine whether the annuity is an ordinary annuity (payments at the end of each period) or an annuity due (payments at the beginning of each period). Adjust your calculator’s setting to ‘END’ for an ordinary annuity or to ‘BGN’ for an annuity due.
3. Enter the values: Input the given values for the number of periods (N), the nominal interest rate in percent (I/Y), and the periodic payment amount (PMT). Remember to follow the cash flow sign convention, typically entering the payment (PMT) as a negative value (outflow).
4. Determine present value (PV):
- If you are starting the annuity without an initial lump sum investment, set PV to zero.
- If there’s an initial lump sum investment, input this value as PV, which is entered as a negative value if it is an outflow.
5. Compute future value (FV): With all other variables correctly inputted, proceed to calculate the FV. The calculator will display the future value of the annuity, which is the total value accumulated at the end of the term, including the compounded interest.
6. Compute the amount of interest (I): When required, apply Formula 3.2 to calculate the total amount of interest earned over the entire term of the annuity. However, if there is an initial investment at the start of the annuity (a nonzero Present Value, PV), you need to modify the calculation. In this scenario, the future value (FV) includes both the initial investment and all subsequent payments. Therefore, the formula to calculate the interest earned is adjusted to Formula 3.3.
It’s important to note that when working with algebraic formulas, such as Formulas 3.2 or 3.3 for calculating the amount of interest, all values are treated as positive values. This approach differs from the cash flow sign convention used in financial calculators.
B. Future Value of Ordinary Annuity
We begin by examining the future value of ordinary annuities, where payments are made at the end of each period. It’s important to first verify the payment timing setting on your calculator before solving annuity problems.
Lauren deposits $360 at the end of each month for 12 years in her Registered Retirement Savings Plan (RRSP) account. The interest rate on the RRSP account is 7.5% compounded monthly. a) How much will be the accumulated value of her investment? b) How much will she have contributed to the account by the end of the term? c) How much interest will be earned on the account?
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Given information:
- Payments are made at the end of each month, so it is an ordinary Annuity: END
- Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
- Payments are made at the end of each month so [asciimath]P//Y = 12[/asciimath]
[asciimath]C//Y = P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary Simple Annuity
- Investment Term: [asciimath]t = 12[/asciimath] years
- Number of payments in the term: [asciimath]N =P//Y*t=12 (12) = 144[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 7.5%[/asciimath]
- In investments, the periodic payment is cash outflow, so [asciimath]PMT = -$360[/asciimath]
- The initial balance of the account is zero, so [asciimath]PV = 0[/asciimath]
a) [asciimath]FV =?[/asciimath]
Thus the accumulated value of her investment will be $83,676.89.
b) Lauren’s contribution to the account is through periodic payments. She will have contributed N payments of size PMT:
Contribution through payments [asciimath]=N*PMT[/asciimath]
[asciimath]=144(360)[/asciimath]
[asciimath]=$51,840[/asciimath]
c) [asciimath]I=?[/asciimath]
The difference between FV and total payment (i.e., Lauren’s contribution) is the amount of interest earned in the account. By Formula 3.2, we have
[asciimath]I=FV-(N*PMT)[/asciimath]
[asciimath]=83,676.89-51,840[/asciimath]
[asciimath]=$31,836.89[/asciimath]
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Consider a scenario where you deposit $250 at the end of every three months for 15 years into an account that offers a 6% interest rate compounded semi-annually. a) What will be the maturity value of these deposits at the end of the 15-year period? b) How much interest will be earned during the 15-year term?
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Given information:
- Interest is compounded semi-annually so [asciimath]C//Y = 2[/asciimath]
- Deposits are made at the end of every three months so [asciimath]P//Y = 4[/asciimath]
[asciimath]C//Y != P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary General Annuity
- Investment Term:[asciimath]t = 15[/asciimath] years
- Number of payments in the term: [asciimath]N =P//Y*t=4 (15) = 60[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 6%[/asciimath]
- The periodic payment is cash outflow, so [asciimath]PMT = -$250[/asciimath]
- The initial balance of the account is zero, so [asciimath]PV = 0[/asciimath]
a) [asciimath]FV =?[/asciimath]
Thus the maturity value will be $23,964.80.
b) [asciimath]I=?[/asciimath]
The difference between FV and total deposits is the amount of interest earned in the account. Using Formula 3.2, we have
[asciimath]I=FV-N.PMT[/asciimath]
[asciimath]I=23,964.80-60(250)[/asciimath]
[asciimath]=23,964.80 -15,000[/asciimath]
[asciimath]=$8,964.80[/asciimath]
The below figure displays a donut chart with the Future Value (FV) of $23,964.80 at its center, breaking down into components of interest ([asciimath]I[/asciimath]) and total payments ([asciimath]N*PMT[/asciimath]) around the ring, illustrating how each contributes to the [asciimath]FV[/asciimath].
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C. Future Value of Annuity Due
In the upcoming examples, we will focus on calculating the future value of annuities due. In these annuities, since payments are made at the start of each payment period, it is necessary to change the payment timing setting on your calculator to ‘BGN’ (beginning of the payment period).
Kian plans to start saving for a sabbatical trip he intends to take in 10 years. To finance his adventure, he decides to make monthly deposits of $500 at the beginning of each month into a high-yield savings account that offers a 3.24% annual interest rate compounded monthly. Determine the future value of Kian’s sabbatical fund at the end of 10 years if he starts to make the deposit at the beginning of the next month.
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Given information:
- Interest is compounded semi-annually so [asciimath]C//Y = 12[/asciimath]
- Deposits are made at the beginning of every month so [asciimath]P//Y = 12[/asciimath]
[asciimath]C//Y = P//Y[/asciimath] [asciimath]=>[/asciimath] Simple Annuity Due
- Investment term: [asciimath]t = 10[/asciimath] years
- Number of payments in the term: [asciimath]N =P//Y*t=12 (10) =120[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 3.24%[/asciimath]
- For investments PMT is a cash outflow, so [asciimath]PMT = -$500[/asciimath]
- There is no initial lump sum in the fund at the start of the annuity, so [asciimath]PV = 0[/asciimath]
a) [asciimath]FV = ?[/asciimath]
Ensure to first change the payment timing to BGN for an annuity due before starting to enter values.
The future value of the deposits in Kian’s sabbatical fund will be $70,8939.87 at the end of the 10-year term.
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Adriel made deposits of $250 in a fund at the beginning of every month for 18 years. The fund was earning interest of 6.12% compounded quarterly. a) What was the total amount Adriel deposited in the fund? b) What was the accumulated value of the investment? c) How much interest was earned in the fund during the term?
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Given information:
- Interest is compounded quarterly so [asciimath]C//Y = 4[/asciimath]
- Deposits are made at the beginning of every month so [asciimath]P//Y = 12[/asciimath]
[asciimath]C//Y != P//Y[/asciimath] [asciimath]=>[/asciimath] General Annuity Due
- Investment Term: [asciimath]t = 18[/asciimath] years
- Number of payments in the term: [asciimath]N =P//Y*t=12 (18) =216[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 6.12%[/asciimath]
- For investments PMT is a cash outflow, so [asciimath]PMT = -$250[/asciimath]
- There is no initial lump sum in the fund at the start of the annuity, so [asciimath]PV = 0[/asciimath]
a) Total amount Adriel deposited in the fund is N deposits of size PMT:
Total deposits [asciimath]=N*PMT[/asciimath]
[asciimath]=216(250)[/asciimath]
[asciimath]=$54,000[/asciimath]
b) [asciimath]FV=?[/asciimath]
Ensure to first change the payment timing to BGN for an annuity due before starting to enter values.
The accumulated value of the deposits in the fund will be $98,244.20 at the end of the 18-year term.
c) [asciimath]I=?[/asciimath]
The difference between FV and total deposits is the amount of interest earned in the account. Using Formula 3.2, we have
[asciimath]I=FV-(N*PMT)[/asciimath]
[asciimath]I=98,244.20-216(250)[/asciimath]
[asciimath]=98,244.20 -54,000[/asciimath]
[asciimath]=$44,244.20[/asciimath]
The below figure displays a donut chart with the Future Value (FV) of $98,244.20 at its center, breaking down into components of interest ([asciimath]I[/asciimath]) and total payments ([asciimath]N*PMT[/asciimath]) around the ring, illustrating how each contributes to the [asciimath]FV[/asciimath].
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D. Adjusting for Variable Changes Within the Term
It is important to note that if any key financial variables alter during the term, like the payment amount, the nominal interest rate, or the frequency of compounding, the term must be divided into separate time intervals at the point of each change. The upcoming example demonstrates the procedure for dealing with such adjustments.
$2,650 was deposited at the end of every six months for 5 years into a fund earning 4.7% compounded semi-annually. After this period, no further deposit was made but the accumulated money was left in the account for another 4 years at the same interest rate.
a) Calculate the accumulated amount at the end of the 9-year period.
b) Calculate the total amount of interest earned during the 9-year period.
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Given that payments stop midway through the term, we need to divide the investment term into two segments at the point of change, which occurs at the end of year 5. In the initial segment of five years, periodic deposits are made, classifying this period as an annuity term. In the subsequent four years, no additional deposits are made, and the account simply earns interest on the accumulated amount, which is a compound interest term. To approach this calculation:
- First, calculate the future value of the annuity at the end of the first five-year segment.
- The amount accumulated at the end of year 5 will then serve as the present value for the following four-year term, during which compound interest will be applied to this sum to determine the final future value at the end of year 9.
- Interest is compounded semi-annually so [asciimath]C//Y = 2[/asciimath]
- Payments are made at the end of every six months so [asciimath]P//Y = 2[/asciimath]
[asciimath]C//Y = P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary Simple Annuity
- Investment Term: [asciimath]t_1 = 5[/asciimath] years
- Number of payments in the term: [asciimath]N_1 =P//Y*t_1=2 (5) = 10[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 4.7%[/asciimath]
- The payment is cash outflow, so [asciimath]PMT_1 = -$2650[/asciimath]
- The initial balance of the account is zero, so [asciimath]PV_1 = 0[/asciimath]
- [asciimath]FV_1 = ?[/asciimath]
Term 2: Next 4 years – Compound Interest
Given Information
- In investments, the present value is cash outflow, so [asciimath]PV_2 = -FV_1=-$29,485.420395...[/asciimath]
- Investment Term: [asciimath]t_2 = 4[/asciimath] years
- Interest is compounded semi-annually so [asciimath]C//Y = 2[/asciimath]
- For compound interest problems, [asciimath]P//Y=C//P[/asciimath], so [asciimath]P//Y=2[/asciimath]
- Number of compounding periods in the term: [asciimath]N_2 =C//Y.t_2=2 (4) = 8[/asciimath]
- Same interest rate
- No periodic payment, so [asciimath]PMT=0[/asciimath]
- [asciimath]FV_2=?[/asciimath]
So the accumulated amount at the end of the 9-year period is $35,506.68.
b) [asciimath]I=?[/asciimath]
The total interest earned in the account ([asciimath]I[/asciimath]) is the sum of interest earned during the first term ([asciimath]I_1[/asciimath]) and the interest earned during the second term ([asciimath]I_2[/asciimath]). These interests can be computed individually and then added together.
Alternatively, given that the only additions to the account are the deposits of $2,650, the total interest can also be determined by subtracting the total deposit amount from the final accumulated value ([asciimath]FV_2[/asciimath]) of the account.
[asciimath]I=FV_2-N_1*PMT[/asciimath]
[asciimath]I=35,506.68 -10(2650)[/asciimath]
[asciimath]=$9,006.68[/asciimath]
The below figure displays a donut chart with the final Future Value ([asciimath]FV_2[/asciimath]) of $35,506.68 at its center, breaking down into components of interest ([asciimath]I[/asciimath]) and total payments ([asciimath]N*PMT[/asciimath]) around the ring, illustrating how each contributes to the final accumulated value. The interest is further expanded to show interest earned during the first term ([asciimath]I_1[/asciimath]) and interest earned during the second term ([asciimath]I_2[/asciimath]).
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E. Annuities with Initial Lump Sum
In our earlier examples, we assumed that the annuities began without any initial investment, meaning the present value (PV) was zero. However, if an annuity starts with an initial lump sum investment, you must enter this amount as the present value (PV) in your calculations. Remember to input the PV as a negative number as it represents a cash outflow.
When the annuity calculation includes an initial lump sum (PV), the future value will include this initial investment, all the periodic payments made thereafter, and the interest that accrues over time. To calculate the total interest earned over the term of the annuity, you need to use Formula 3.3.
Nancy is diligently preparing for her retirement and has already saved $15,000 in her 401(k) retirement fund. She is currently 30 years old and aims to retire at 67. To supplement her savings, Nancy begins to contribute $500 at the end of each month to her 401(k). The account is expected to earn an average interest rate of 7% per year compounded quarterly. a) Calculate the future value of Nancy’s 401(k) when she retires. b) Calculate the total amount of interest that will have been earned on the account by the time Nancy reaches retirement age.
Show/Hide Solution
Given information:
- Payments are made at the end of each month, so it is an ordinary annuity: END
- Interest is compounded quarterly so [asciimath]C//Y = 4[/asciimath]
- Deposits are made at the end of every month so [asciimath]P//Y = 12[/asciimath]
[asciimath]C//Y != P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary General Annuity
- Nominal interest rate: [asciimath]I//Y = 7%[/asciimath]
- Investment Term: [asciimath]t = 37[/asciimath] years
- Number of payments in the term: [asciimath]N =P//Y*t=12 (37) = 444[/asciimath]
- The periodic payments of $500 are cash outflow, so [asciimath]PMT = -$500[/asciimath]
- The initial value of $15,000 at the start of the annuity is also a cash outflow, so [asciimath]PV = -$15,000[/asciimath]
- The questions ask for FV and I.
a) [asciimath]FV = ?[/asciimath]
The total value accumulated at the end of the term is composed of two parts: the sum of the future values of all periodic payments and the future value of the initial lump sum investment.
Thus the future value of Nancy’s 401(k) when she retires is $1,233,038.52.
b) [asciimath]I=?[/asciimath]
Because the present value (PV) is not zero, to calculate the amount of interest earned over the 37-year term, we use the modified formula for interest amount (Formula 3.3):
[asciimath]I = FV - (N * PMT) - PV[/asciimath]
[asciimath]I = 1,233,038.52 - (444 xx 500) - 15000[/asciimath]
[asciimath]=1,233,038.52-222,000-15,000[/asciimath]
[asciimath]=$996,038.52[/asciimath]
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Section 3.2 Exercises
- Thiago would like to save $470 at the end of every month for the next 10 years in a savings account at 3.44% compounded annually. a) What would be the accumulated value of the investment at the end of the term? b) What would be the amount of interest earned?
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a) FV = $67,015.84
b) I = $10,615.84
- $4,800 was deposited at the end of every three months for 9 years into a fund earning 2.8% compounded quarterly. After this period, the accumulated money was left in the account for another 4.5 years at the same interest rate. a) Calculate the accumulated amount at the end of the 13.5-year term. b) Calculate the total amount of interest earned during the 13.5-year period.
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a) FV = $221,936.95
b) I = $49,136.95
- Erick made deposits of $320.41 in a fund at the beginning of every quarter for 7 years. The fund was earning interest of 3.38% compounded semi-annually. a) What was the accumulated value of the investment? b) What was the total amount Erick deposited in the fund? c) How much interest was earned in the fund during the term?
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a) FV = $10,153.73
b) Total deposit = $8,971.48
c) I = $1,182.25
- Carolyn is preparing for her retirement and has already saved $26,500 in her 401(k) retirement fund. She is currently 30 years old and aims to retire at 61. To supplement her savings, Carolyn begins to contribute $1,520 at the end of every quarter to her 401(k). The account is expected to earn an average interest rate of 5.46% compounded quarterly. a) Calculate the future value of the 401(k) fund when Carolyn retires. b) Calculate the total amount of interest that will have been earned on the account by the time Carolyn reaches retirement age.
Show/Hide Answer
a) FV = $629,167.72
b) I = $414,187.72
