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What is the Formula for the Future Value of an Annuity Due?Future value is the value of a sum of cash to be paid on a specific date in the future. An annuity due is a series of payments made at the beginning of each period in the series. Therefore, the formula for the future value of an annuity due refers to the value on a specific future date of a series of periodic payments, where each payment is made at the beginning of a period. Such a stream of payments is a common characteristic of payments made to the beneficiary of a pension plan. These calculations are used by financial institutions to determine the cash flows associated with their products. How to Calculate the Future Value of an Annuity DueThe formula for calculating the future value of an annuity due (where a series of equal payments are made at the beginning of each of multiple consecutive periods) is: P = (PMT [((1 + r)n - 1) / r])(1 + r) Where: P
= The future value of the annuity stream to be paid in the future This value is the amount that a stream of future payments will grow to, assuming that a certain amount of compounded interest earnings gradually accrue over the measurement period. The calculation is identical to the one used for the future value of an ordinary annuity, except that we add an extra period to account for payments being made at the beginning of each period, rather than the end. Example of the Future Value of an Annuity DueThe treasurer of ABC Imports expects to invest $50,000 of the firm's funds in a long-term investment vehicle at the beginning of each year for the next five years. He expects that the company will earn 6% interest that will compound annually. The value that these payments should have at the end of the five-year period is calculated as: P = ($50,000 [((1 + .06)5 - 1) / .06])(1 + .06) P = $298,765.90 As another example, what if the interest on the investment compounded monthly instead of annually, and the amount invested were $4,000 at the end of each month? The calculation is: P = ($4,000 [((1 + .005)60 - 1) / .06])(1 + .005) P = $280,475.50 The .005 interest rate used in the last example is 1/12th of the full 6% annual interest rate. 9 Financial Mathematics By the end of this section it is expected that you will be able to:
AnnuitiesA common financial goal is to be comfortable in retirement. During our working lives we contribute to a retirement fund so that upon retirement we receive a financial payment at regular intervals. Financial transactions that involve a series of equal payments at equal intervals are called annuities. Other examples of annuities include payments on a loan, rental payments, and insurance premiums. The term of the annuity is the time from the beginning of the first payment interval to the end of the last payment interval. A payment interval is the time between successive payments. If, for example, a vehicle is purchased with monthly payments on a four-year loan then the term of the loan is 4 years and the payment interval is monthly. In some cases, as with salaries or a senior’s pension, the payments are made at the end of a payment interval. This is referred to as an ordinary annuity. When payments are required at the beginning of a payment interval, as with many loans and mortgages, this is referred to as an annuity due. It is important to note that the term of the annuity does not necessarily coincide with the first and last payment. Consider a one year loan where 12 equal payments are made on the first of each month. This is an example of an annuity due. The term of the annuity is one year and the payment interval is one month. Refer to Figure 1. There are twelve payments, each occurring on the first day of the month. The first payment is made on Jan. 1 and the last payment is made on Dec. 1. Note that the final payment on Dec. 1 does not occur on the last day of the term of the annuity which is Dec. 31. Fig. 1Consider a one year loan where 12 equal payments are made on the last day of each month. This is an example of an ordinary annuity. The term of the annuity is one year and the payment interval is one month. Refer to Figure 2. There are twelve payments, each occurring on the last day of the month. The first payment is made on Jan. 31 and the last payment is made on Dec. 31. Note that the first payment on Jan. 31 does not occur on the first day of the term of the annuity which is Jan. 1. Fig. 2In this section we will only be concerned with ordinary simple annuities. As with all ordinary annuities the payments are made at the end of each payment interval. It is also the case that the compounding interval equals the payment interval. This means that if the payment interval is monthly then interest will also be compounded monthly. Future Value of an Ordinary AnnuityConsider an investment that is in the form of an ordinary simple annuity. This means that a deposit is made at the end of regular intervals and interest is compounded at each of these intervals. The value of the annuity can grow substantially. The final amount of the annuity is called the future value of the annuity. It is the total of all annuity payments and the accumulated compound interest as illustrated in Figure 3. Fig. 3To see how the annuity process works, consider the table in Figure 4 below. This table depicts an ordinary 6-month annuity commencing on Jan. 1 and ending on June 30. The monthly payments are $1000 and the annual interest rate is 6% compounded monthly.
Fig. 4 Since this is an ordinary annuity the payments are made at the end of the month. Interest is calculated as simple interest I = Prt where r = 0.06 and t = 1/12 year and P = the balance at the beginning of the month. The first payment of $1000 is deposited at the end of January and therefore no interest is earned for the month of January. At the beginning of February there is $1000 in the account. Interest for the month of February is I = $1000(0.06)(1/12) = $5. At the end of February there will be a deposit of $1000 so the balance at the end of February is $1000 + $5 + $1000 = $2005. At the beginning of March there is $2005 in the account. Interest for the month of March will be I = $2005(0.06)(1/12) = $10.025 or $10.03 (rounded off). At the end of March $1000 is deposited so the balance at the end of March is $2005 + $$10.03 + $1000 = $3015.03. At the beginning of April there is $3015.03 in the account. Interest for the month of April will be I = $3015.03(0.06)(1/12) = $15.075 or $15.08 (rounded off). At the end of April $1000 is deposited so the balance at the end of April is $3015.03 + $$15.08 + $1000 = $4030.11. The process is continued for the remaining two months. At the end of June, the balance will be $6075.51. This is the future value of the annuity, which is the total of all annuity payments and the accumulated compound interest. Notice that with an ordinary annuity the interest calculation is based on the balance at the beginning of the interval. Since the first payment does not occur until the end of the first payment interval there will not be any interest in the first payment interval. Although the term of the annuity is six months there will only be five intervals where interest is calculated. With an annuity due the payment would be made at the beginning of each payment interval so for a six month term there would be six interest calculations. Since many loans are set up as an annuity due it is advantageous to the lending institution (but not to the loan recipient). The calculation of the future value of an annuity can be very time consuming. Fortunately there is a formula for this. The ordinary annuity formula is:
It is important to note that there are variations on how the ordinary annuity formula is written. This is due to the way in which the variables are defined. The formula that is provided in this section defines r as the annual interest rate, n as the number of compounding periods per year, and t as the time in years (term of the annuity in years). Assume that the formula will be used to calculate the future value of a two year ordinary annuity that offers an annual interest rate of 6%, monthly payments of $1000, and monthly compounding. Define each of the variables but do not calculate the future value. Solution
Note that the term of the annuity is 2 years. The interest calculation involves monthly compounding so n = 12 since there are 12 compounding periods in a year. Assume that the formula will be used to calculate the future value of a 1 year ordinary annuity that offers an annual interest rate of 3%, semiannual payments of $500, and semiannual compounding. Define each of the variables but do not calculate the future value. Show answerAmount (A) = unknown Payment (P) = $500 Annual interest rate (r) = 3% = 0.03 Number of compounding period (n) = 2 Time (t) = 1 year Assume that the formula will be used to calculate the future value of a 6 month ordinary annuity that offers an annual interest rate of 4.8%, weekly payments of $100, and weekly compounding. Define each of the variables but do not calculate the future value. Solution A = amount of annuity (Future Value) = unknown P = periodic payment amount = $100 r = annual interest rate = 4.8% = 0.048 n = number of compounding periods per year = 52 t = time (in years) = 0.5 years Note that although the term of the annuity is 1/2 year, the interest calculation involves weekly compounding so n = 52 since there are 52 compounding periods in a year. Assume that the formula will be used to calculate the future value of a 9 month ordinary annuity that offers an annual interest rate of 5.5%, monthly payments of $200, and monthly compounding. Define each of the variables but do not calculate the future value. Show answerA = unknown P = $200 r = 5.5% n = 12 t = 0.75 years We will now use the formula to calculate the future value of a six month ordinary annuity that offers an annual interest rate of 6%, monthly payments, and monthly compounding. Use the annuity formula to find the annuity amount in 6 months if $1000 is deposited monthly at 6% compounded monthly. Compare this answer to the answer obtained in the table in Figure 4. Solution
The annuity is worth $6075.50. This answer is $0.01 different than in the table in Figure 4 due to rounding off . Use the formula to calculate the future value of a 9 month ordinary annuity at an annual interest rate of 3%, monthly payments of $50, and monthly compounding. Show answerFuture value = $454.53 Use the annuity formula to find the annuity amount in 4 years if $500 is deposited semiannually at 3.6% compounded semiannually. Solution
The annuity is worth $4261.28. Use the formula to calculate the future value of a 5 year ordinary annuity that offers an annual interest rate of 4.8%, semiannual payments of $4000, and semiannual compounding. Show answerFuture value = $44,608.43 The future value or annuity amount includes all payments and compound interest. To determine the total interest we must subtract the total value of all annuity payments from the future value of the annuity. Interest on an Annuity = Future Value of the Annuity – Total Value of the Payments = Future Value of the Annuity – (Payment amount × number of payments per year × number of years) Consider Example 3. For a 6-month annuity where $1000 is deposited monthly the value of the annuity at the end of 6 months is 6075.51. Determine the total interest earned on the annuity. Solution Interest Earned = Future Value of the Annuity – Total Value of the Payments (Deposits) = $6075.51 – ($1000 x 6 payments) = $75.51. Refer to the table in Figure 4 to confirm that the column “interest earned” adds to this identical amount. Refer to Try It 3. For a 9 month ordinary annuity with monthly payments of $50, determine the total interest earned on the annuity. Show answerFuture value = $4.53 a) How much would an annuity be worth in 2 years at 2.4% compounded monthly if the periodic payments are $40 per month? b) Determine the total interest earned on the annuity. Solution: a) The annuity is worth $982.41 after 2 years. b) The total interest earned is: $982.41 – ($40 × 12payments/yr × 2yr) = $982.41 – $960 = $22.41 Consider a ten-year ordinary annuity that offers an annual interest rate of 4.5%, semiannual payments of $1000, and semiannual compounding a) How much would the annuity be worth in 10 years? b) Determine the total interest earned on the annuity. Show answerFuture value = $24911.52 Tish plans to go back to university and opens an account into which she will deposit $300 at the end of every month for 4 years. The account offers an annual interest rate of 4.8% compounded monthly. Assuming a fixed interest rate and no additional deposits or withdrawals, how much will be in the account at the end of 4 years? How much interest will Tish earn in the 4 years? Solution A credit union is offering 6.8% compounded monthly on a savings account. If you deposit $100 at the end of every month for two years (assume no withdrawals) how much will be in the account at the end of two years? How much interest will you earn? Show answerAmount in 2 years = $2563.10 Determining the Annuity PaymentBusinesses and individuals often wish to accumulate a certain amount of money by making regular deposits (payments) into an annuity. Perhaps an individual or business wishes to purchase a larger ticket item such as an appliance or a piece of equipment in one year’s time. Rather than taking out a loan they could choose to deposit a specific amount every month so as to accumulate the required funds by the end of the one year. The amount that needs to be deposited is represented by the payment in the annuity formula. The formula must be solved for the payment (P). The periodic payment formula is:
A home bakery wants to purchase a new oven in one year’s time. The oven is estimated to cost $5000. The baker has found an account that offers 3.2% monthly compounding. a) How much must be deposited at the end of each month to accumulate to the $5000? b) What is the total amount that the baker deposits over the one year? c) How much of the $5000 is interest? Solution
a) The baker must deposit $410.59 per month. b) The baker deposits $410.59/mth × 12 months = $4927.08 in one year. c) Since there is $5000 in the account at the end of the year, the interest component will be: $5000 – $4927.09 = $72.91 Cara is saving to start college in three years and hopes to have saved $12,000 in three years. She opens an account offering 4.8% compounded monthly. a) How much must Cara deposit at the end of each month to accumulate to the $12000? b) What is the total amount that Cara deposits over the three years? c) How much of the $12,000 is interest? Show answera) Monthly deposit must be $310.57 b) Total deposited is $11,180.67 c) $819.33 Sara hopes to accumulate $140,000 in 12 years. She has found an annuity that offers 8% annual compounding and requires that she make a deposit at the end of each year. a) How much must Sara deposit at the end of each year to accumulate to the $140,000? b) What is the total amount that Sara deposits over the twelve years? c) How much of the $140,000 is interest? Solution: a)
Sara must deposit $7,377.30 per year b) 12 years × $7377.30/yr = $88527.60 c) Interest = $140000 – $88527.60 = $51472.40 Zach is saving to go on a trip in one year’s time. He hopes to have $3200 at the end of one year so he makes monthly deposits into an account offering 2.4% compounded monthly. a) How much must Zach deposit at the end of each month to accumulate to the $3200? b) What is the total amount that Zach deposits over the twelve months? c) How much of the $3200 is interest? Show answera) Monthly deposit must be $263.75 b) $3165 c) $35 What monthly payment is necessary for an annuity to be worth $10,000 in 3 years at 7% compounded monthly? Solution
The periodic payment is $250.44 Zach has become more ambitious and is saving to go on world cruise in four years. He anticipates that the cruise will cost $38,000. How much will he need to deposit each month in an account offering 3.6% compounded monthly to accumulate to $38, 000 in four years? Show answer$737.22/month Key Concepts
Interest on an Annuity = Future Value of the Annuity – Total Value of the Payments
Glossaryannuity is a series of payments made at fixed intervals. annuity due is an annuity where the payment is due at the beginning of each payment period. future value of the annuity is the final amount of the annuity. It is the total of all annuity payments and the accumulated compound interest. ordinary annuity is an annuity where the payment is due at the end of each payment period. payment interval is the time between successive annuity payments. term of the annuity is the time from the beginning of the first payment interval to the end of the last payment interval. Exercise Set 9.4
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AttributionSome of the content for this chapter is from “Unit 6: Ordinary annuities” and “Unit 7: Annuity payments” in Financial Mathematics by Paul Grinder, Velma McKay, Kim Moshenko, and Ada Sarsiat, which is under a CC BY 4.0 Licence.. Adapted by Kim Moshenko. See the Copyright page for more information. Which of the following refers to an equal payment made at the end of each period over a period of time?Amortization: Loan payments by equal periodic amounts calculated to pay off the debt at the end of a fixed period, including accrued interest on the outstanding balance.
What is it called when the payment is made at the beginning of each period?Annuity due is an annuity whose payment is due immediately at the beginning of each period. Annuity due can be contrasted with an ordinary annuity where payments are made at the end of each period. A common example of an annuity due payment is rent paid at the beginning of each month.
Which of the following represents equal periodic payments received or paid each period?An annuity is a set of payments made in equal amounts at periodic intervals.
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