Annuities

3.7 FV of Annuities: Formula Approach

A. Future Value of Ordinary Simple Annuity

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 it is for retirement savings, education funds, or other long-term financial goals. We begin by examining the future value of ordinary annuities, where payments are made at the end of each period.

In an Ordinary Simple Annuity, the payment period and the interest compounding period are the same ([asciimath]P//Y = C//Y[/asciimath]), and the payments are made at the end of the payment period.

Timeline for an ordinary simple annuity, illustrating payments made at each period's end, with payment intervals matching compounding periods.

Figure 3.7.1 A timeline for an ordinary simple annuity

Consider a scenario where you invest $1,000 at the end of every year into a savings account that offers a 10% annual interest rate compounded annually, over five years. To find out the total amount in your account at the end of these five years, you need to calculate the future value of this annuity. Here, the future value represents the total value accumulated from all your annual payments, including the interest earned, by the end of the five-year term.

To calculate the total accumulated value, we can calculate the future value of each payment using the formula for the future value of compound interest, [asciimath]FV=PV(1+i)^N[/asciimath] (Formula 2.4a). The timeline for this calculation is depicted in Figure 3.7.2.

Timeline showing five annual payments of $1,000, beginning at the end of year 1 and continuing through to the end of year 5. Arrows from each payment point to the end of year 5, underscoring the calculation of their future values at that time.

Figure 3.7.2 Future value of Ordinary simple Annuity

In our example, since the interest is compounded annually ([asciimath]C//Y=1[/asciimath]), the periodic interest rate (i) is 10%, and the number of compounding periods (N) for each payment is the number of years from the payment date until the end of the fifth year. The present value (PV) for each payment is $1,000.

 [asciimath]"Total FV"=1000(1+0.1)^0+1000(1+0.1)^1+1000(1+0.1)^2[/asciimath] [asciimath]+1000(1+0.1)^3+1000(1+0.1)^4[/asciimath]

 [asciimath]=1000+1100+1210+1331+1464.10[/asciimath]

[asciimath]=$6105.10[/asciimath]

Therefore, the total accumulated value from investing $1,000 at the end of each year for five years amounts to $6,105.10.

Now, consider a different scenario where you deposit $1,000 monthly for 30 years. This would result in 360 payments, and calculating the future value for each payment, as done in the first example, would be impractical due to its time-consuming nature. For such cases, we need a more straightforward method to compute the future value for annuities. The uniformity and periodic nature of the payments enable us to use a simplified formula for this calculation.

 [asciimath]FV=PMT[((1+i)^N-1)/i][/asciimath]Formula 3.5a

(See how the formula is derived)

where [asciimath]PMT[/asciimath] represents the size of the periodic payment, [asciimath]i[/asciimath] is the periodic interest rate (calculated as  [asciimath]i=(I//Y)/(C//Y)[/asciimath], Formula 2.1a), and [asciimath]N[/asciimath] is the number of payments in the annuity term (computed as [asciimath]N=P//Y*t[/asciimath], Formula 3.1a).

B. Computing Interest Amount

When the future value of an annuity is known or needs to be calculated, particularly in investment contexts, it 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. As previously introduced in Section 3.2, 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

 

Example 3.7.1: Compute FV of Ordinary Simple Annuity

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:

  • 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]
  • Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (7.5%)/12 =0.625%[/asciimath]
  • Periodic payment: [asciimath]PMT = $360[/asciimath]
  • The initial balance of the account is zero, so [asciimath]PV = 0[/asciimath]

a) [asciimath]FV =?[/asciimath]

Substituting the values into Formula 3.5a gives

 [asciimath]FV=PMT[((1+i)^N-1)/i][/asciimath]

 [asciimath]FV=360[((1+0.625%)^144-1)/(0.625%)][/asciimath]

 [asciimath]=83,676.891...[/asciimath]

 [asciimath]~~$83,676.89[/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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C. Future Value of Ordinary General Annuity

In an Ordinary General Annuity, the payment period and the interest compounding period are NOT the same ([asciimath]P//Y != C//Y[/asciimath]) and the payments are made at the end of the payment period.

Timeline of an ordinary general annuity showing payments at the end of each period, with payment intervals not matching the compounding periods.Figure 3.7.3 A timeline for an ordinary general annuity

To calculate the future value of an ordinary general annuity, we can adapt the formula originally developed for the future value of an ordinary simple annuity. However, a minor adjustment is needed before applying this formula. In an ordinary simple annuity, the periodic interest rate corresponds to the interest rate per compounding period, which is the same as the payment period. But for an ordinary general annuity, it’s necessary to determine the interest rate per payment period and then incorporate this rate into the future value formula.

The periodic interest rate per payment period is denoted by [asciimath]i_2[/asciimath] and obtained by

 [asciimath]i_2=(1+i)^((C//Y)/(P//Y))-1[/asciimath]

Then, in the future value formula for an ordinary simple annuity, replace [asciimath]i[/asciimath] with to adjust for an ordinary general annuity.

 [asciimath]FV=PMT[((1+i_2)^N-1)/i_2][/asciimath]       Formula 3.6a

Note that if the number of compounding periods per year ([asciimath]C//Y[/asciimath]) is equal to the number of payment periods per year ([asciimath]P//Y[/asciimath]), then [asciimath]i_2[/asciimath] will be the same as [asciimath]i[/asciimath]. In such a case, Formula 3.6 for an ordinary general annuity will be identical to Formula 3.5a for an ordinary simple annuity.

 

Example 3.7.2: Compute FV of Ordinary General Annuity

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]
  • Interest rate per compounding period: [asciimath]i = (I//Y)/(C//Y) = (6%)/2 =3%[/asciimath]
  • Periodic payment: [asciimath]PMT = $250[/asciimath]
  • The initial balance of the account is zero, so [asciimath]PV = 0[/asciimath]

a) [asciimath]FV = ?[/asciimath]

First, we need to find the interest rate per payment period [asciimath]i_2[/asciimath]:

 [asciimath]i_2=(1+i)^((C//Y)/(P//Y) )-1[/asciimath]

[asciimath]i_2=(1+3%)^(2/4)-1[/asciimath]

 [asciimath]=0.0148891...[/asciimath]

Then, by plugging the values into Formula 3.6a, we get the following calculation:

 [asciimath]FV=PMT[((1+i_2)^N-1)/i_2][/asciimath]

 [asciimath]FV=250[((1+0.0148891... )^60-1)/(0.0148891... )][/asciimath]

 [asciimath]=23,964.797...[/asciimath]

 [asciimath]~~$23,964.80[/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. By 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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D. Future Value of Simple Annuity Due

In a Simple Annuity Due, the payment period and the interest compounding period are the same ([asciimath]P//Y = C//Y[/asciimath]) and the payments are made at the beginning of the payment period.

Timeline for a simple annuity due, illustrating payments made at each period's beginning, with payment intervals matching compounding periods.

Figure 3.7.4 A timeline for a Simple annuity due.

Consider a scenario we used at the start of this section for an ordinary simple annuity. Now consider this time you invest $1,000 at the beginning of every year into a savings account that offers a 10% annual interest rate compounded annually over five years. To find out the total amount in your account at the end of these five years, you need to calculate the future value of this annuity. Similar to the previous scenario, to calculate the total accumulated value, we calculate the future value of each payment using the formula for the future value of compound interest (Formula 2.4a). The timeline for this calculation is depicted in Figure 3.7.5.

Timeline showing five annual payments of $1,000, starting at the beginning of year 1 and continuing through to the beginning of year 5. Arrows from each payment point to the end of year 5, underscoring the calculation of their future values at that time.

Figure 3.7.5 Future Value of Simple Annuity Due

 [asciimath]"FV"=1000(1+0.1)^1+1000(1+0.1)^2+1000(1+0.1)^3[/asciimath] [asciimath]+1000(1+0.1)^4+1000(1+0.1)^5[/asciimath]

If we factor out [asciimath](1+0.1)[/asciimath] from the right-hand side of the equation, we arrive at

 [asciimath]"FV"=(1+0.1)[1000(1+0.1)^0+1000(1+0.1)^1+1000(1+0.1)^2[/asciimath] [asciimath]+1000(1+0.1)^3+1000(1+0.1)^4][/asciimath]

It is important to note that the expression in the square brackets matches the Future Value (FV) calculated for an ordinary simple annuity in our earlier example. Therefore

 [asciimath]FV=(1.1)[$6105.10][/asciimath]

 [asciimath]=$6715.61[/asciimath]

Generally, the future value of a simple annuity due is equal to the future value of an ordinary simple annuity multiplied by a factor of [asciimath](1 + i)[/asciimath]. Therefore, the formula for the future value of an annuity due can be expressed as [asciimath]FV_("due")=FV_("Ordinary")(1+i)[/asciimath] and is given by

 [asciimath]FV=PMT[((1+i)^N-1)/i](1+i)[/asciimath]Formula 3.7a

(See how the formula is derived)

 

Example 3.7.3: Compute FV of Simple Annuity Due

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 monthly 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]
  • Interest rate per compounding period: [asciimath]i = (I//Y)/(C//Y) = (3.24%)/12 =0.27%=0.0027[/asciimath]
  • The periodic payment:  [asciimath]PMT = $500[/asciimath]
  • There is no initial lump sum in the fund at the start of the annuity, so [asciimath]PV = 0[/asciimath]

[asciimath]FV =?[/asciimath]

Plugging the values into Formula 3.7a results in

 [asciimath]FV=PMT[((1+i)^N-1)/i](1+i)[/asciimath]

 [asciimath]FV=500[((1+0.0027 )^120-1)/(0.0027 )](1+0.0027 )[/asciimath]

 [asciimath]=70,939.865...[/asciimath]

 [asciimath]~~$70,939.87[/asciimath]

The future value of the deposits in Kian’s sabbatical fund will be $70,939.87 at the end of the 10-year term.

 

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E. Future Value of General Annuity Due

In a General Annuity Due, the payment period and the interest compounding period are different ([asciimath]P//Y != C//Y[/asciimath]) and the payments are made at the beginning of the payment period.

Timeline of a general annuity due showing payments at the beginning of each period, with payment intervals not matching the compounding periods.

Figure 3.7.6 The timeline and timing of the payments in a general Annuity Due

 

Just as with the future value of ordinary annuities, the formula initially developed for the future value of a simple annuity due can be modified by including the interest rate per payment period, [asciimath]i_2[/asciimath], into the future value calculation. Therefore, in the future value formula for the simple annuity due, substitute [asciimath]i[/asciimath] with [asciimath]i_2[/asciimath] to make it suitable for calculating the future value of a general annuity due.

 [asciimath]FV=PMT[((1+i_2)^N-1)/i_2](1+i_2)[/asciimath]Formula 3.8a

where the periodic interest rate per payment period is obtained by

 [asciimath]i_2=(1+i)^((C//Y)/(P//Y))-1[/asciimath]

 

Example 3.7.4: Compute FV of General Annuity Due

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]
  • Interest rate per compounding period: [asciimath]i = (I//Y)/(C//Y) = (6.12%)/4 =1.53%[/asciimath]
  • Periodic payment:  [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]

First, we need to find the interest rate per payment period [asciimath]i_2[/asciimath]:

 [asciimath]i_2=(1+i)^((C//Y)/(P//Y))-1[/asciimath]

 [asciimath]i_2=(1+1.53%)^(4/12)-1[/asciimath]

 [asciimath]=0.00507420...[/asciimath]

Then substituting the values into Formula 3.8a gives

 [asciimath]FV=PMT[((1+i_2)^N-1)/i_2][/asciimath]

 [asciimath]FV=250[((1+0.00507420... )^216-1)/(0.00507420... )][/asciimath] [asciimath](1+0.00507420... )[/asciimath]

[asciimath]=98,244.204...[/asciimath]

 [asciimath]~~$98,244.20[/asciimath]

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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F. 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.

 

Example 3.7.5: Compute FV of Annuity Combined with Compound Interest

$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 in 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:

  1. First, calculate the future value of the annuity at the end of the first five-year segment.
  2. 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.

Timeline for a 9-year investment divided into two phases. The first 5 years consist of annual payments of $2,650, culminating in a calculated future value (FV1). No payments are made in the subsequent 4 years; instead, FV1 accrues interest, now treated as the present value (PV2) for this compound interest phase. The timeline ends at year 9 with a question mark, indicating the goal to determine the final future value (FV2).

a)
Term 1: First 5 years – Ordinary Simple Annuity
Given Information
  • 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 annuity payment: [asciimath]PMT_1 = $2650[/asciimath]
  • Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (4.7%)/2 =2.35%=0.0235[/asciimath]

[asciimath]FV_1 =?[/asciimath]

Substituting the values into Formula 3.5a gives

 [asciimath]FV_1=PMT_1[((1+i)^(N_1)-1)/i][/asciimath]

 [asciimath]FV_1=2650[((1+0.0235 )^10-1)/(0.0235 )][/asciimath]

 [asciimath]=29,485.420...[/asciimath]  (Do not round)

Term 2: Next 4 years – Compound Interest

Given Information

  • Present value: [asciimath]PV_2 = FV_1=$29,485.420...[/asciimath]
  • Investment Term: [asciimath]t_2 = 4[/asciimath]  years
  • Interest is compounded semi-annually so [asciimath]C//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_2=0[/asciimath]

[asciimath]FV_2=?[/asciimath]

Substituting the values into the future value formula of compound interest (Formula 2.4a) gives

 [asciimath]FV_2=PV_2(1+i)^(N_2)[/asciimath]

 [asciimath]FV_2=29,485.420... (1+0.0235 )^8[/asciimath]

 [asciimath]=35,506.682...[/asciimath]

 [asciimath]~~ $35,506.68[/asciimath]  (Rounded to the nearest cents)

Thus 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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G. 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.

 

Example 3.7.6: Compute FV of an Annuity with Intial Lump Sum Investment (Nonzero PV)

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.

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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 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]
  • Interest rate per compounding period: [asciimath]i = (I//Y)/(C//Y) = (7%)/4 =1.75%=0.0175[/asciimath]
  • Investment Term: [asciimath]t = 37[/asciimath]  years
  • Number of payments in the term: [asciimath]N =P//Y*t=12 (37) = 444[/asciimath]
  • Periodic payment: [asciimath]PMT = $500[/asciimath]
  • The initial value of $15,000 at the start of the annuity is the present value: [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. To calculate this, we use the future value formula for an ordinary general annuity for the periodic payments, and the formula for the future value of compound interest for the initial lump sum.

Timeline depicting contributions to the account with 444 monthly payments (PMTs) ending at the end of year 30, plus an initial account balance at the beginning of the investment (PV). Arrows from the payments and the initial balance direct to year 30, highlighting the requirement to calculate their future values.

1. Calculating the future value of payments:

First, we need to find the interest rate per payment period [asciimath]i_2[/asciimath] :

 [asciimath]i_2=(1+i)^((C//Y)/(P//Y) )-1[/asciimath]

 [asciimath]i_2=(1+0.0175)^(4/12)-1[/asciimath]

 [asciimath]=0.00579963...[/asciimath]

 

Then, by plugging the values into Formula 3.6a, we get the following calculation:

 [asciimath]FV_1=PMT[((1+i_2)^N-1)/i_2][/asciimath]

 [asciimath]FV_1=500[((1+0.00579963... )^444-1)/(0.00579963...)][/asciimath]

 [asciimath]=1,037,521.258...[/asciimath]

2. Calculating the future value of the initial lump sum:

For compound interest, N represents the number of compounding periods in the term. Thus, we need to calculate N before using the compound interest future value formula (Formula 2.4a). Also, for compound interest problems, we use the periodic interest rate per compounding period ([asciimath]i[/asciimath]).

[asciimath]N=C//Y*t=4(37)=148[/asciimath]

 [asciimath]FV_2=PV(1+i)^N[/asciimath]

 [asciimath]=15000(1+0.0175)^148[/asciimath]

 [asciimath]=195,517.262...[/asciimath]

The future value of Nancy’s 401(k) when she retires is obtained by summing the above future values:

[asciimath]FV=FV_1+FV_2[/asciimath]

 [asciimath]=1,037,521.258...+ 195,517.262...[/asciimath]

 [asciimath]=1,233,038.520...[/asciimath]

 [asciimath]=$1,233,038.52[/asciimath]

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 * 500) - 15000[/asciimath]

[asciimath]=1,233,038.52-222,000-15,000[/asciimath]

 [asciimath]=$996,038.52[/asciimath]

 

Try an Example

 

H. Summary of Formulas

If we use the formula for the future value of an ordinary simple annuity as the base, the future value formulas for other types of annuities can be derived with slight modifications to this primary formula as follows:

 

The FV of Ordinary Simple Annuity:

 [asciimath]FV=PMT[((1+i)^N-1)/i][/asciimath]

For FV of Ordinary General Annuity, replace [asciimath]i[/asciimath] with [asciimath]i_2[/asciimath]:

 [asciimath]FV=PMT[((1+i_2)^N-1)/i_2][/asciimath]

For FV of Simple Annuity Due, multiply by [asciimath](i+1)[/asciimath]:

 [asciimath]FV=PMT[((1+i)^N-1)/i](i+1)[/asciimath]

For FV of General Annuity Due, multiply by [asciimath](i+1)[/asciimath] and replace [asciimath]i[/asciimath] with [asciimath]i_2[/asciimath]:

 [asciimath]FV=PMT[((1+i_2)^N-1)/i_2](i_2+1)[/asciimath]

 

Where

[asciimath]N=P//Y*t[/asciimath]

[asciimath]i=(I//Y)/(C//Y)[/asciimath]

[asciimath]i_2=(1+i)^((C//Y)/(P//Y) )-1[/asciimath]

Section 3.7 Exercises

  1. 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?
    Show/Hide Answer

     

    a) FV = $67,015.84

    b) I = $10,615.84

  2. $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.
    Show/Hide Answer

     

    a)  FV = $221,936.95

    b) I = $49,136.95

  3. 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?
    Show/Hide Answer

     

    a) FV = $10,153.73

    b) Total deposit = $8,971.48

    c) I = $1,182.25

  4. 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

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