Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
14.1 Future Value of an Annuity Find the future value of:
an annuity using the simple interest formula
an ordinary annuity using a $1.00 ordinary annuity future value table
an annuity due using the simple interest formula
an annuity due using a $1.00 ordinary annuity future value table
an annuity using a formula
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
14.1.1 Future Value of an Annuity
Calculate the value of a growing account subject to periodic investments of payments.
Some examples include:
Retirement funds
College education
Vacation
Company’s future investment in capital expenses
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Key Terms
Annuity payment: a payment made to an investment fund each period at a fixed interest rate.
Sinking fund payment: a payment made to an investment fund each period at a fixed interest rate to yield a predetermined future value.
Annuity certain: an annuity paid over a guaranteed number of periods.
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Contingent annuity: an annuity paid over an uncertain number of periods.
Ordinary annuity: an annuity for which payments are made at the end of each period.
Annuity due: an annuity for which payments are made at the beginning of each period.
Key Terms
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Future value of an annuity using the simple interest formula
1. Find the end-of-period principal. First end-of-period principal = annuity payment
2. For each remaining period in turn:End-of-period principal = previous end-of-period principal x (1 + period interest rate) + annuity payment.
3. Identify the last end-of-period principal as the future value.Future value = last end-of-period principal
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Look at this example What is the FV of an annual ordinary annuity of
$1,000 for 3 years at 4% annual interest?
End-of-year 1 = $1,000 (no interest earned Y1)
End-of-year 2 = $1,000 + $1,000 (1.04) = $2,040
End of year 3 = $1,000 + $ 2,040 (1.04) = $3,121.60
The future value is $3,121.60.
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Figure 14-2
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try this example
Find the future value of an annual ordinary annuity of $1,500 for four years at 3% annual interest.
$6,270
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
14.1.2 Find the FV Using a $1.00 Ordinary Annuity FV
Table
Using Table 14-1 in your text:
1. Select the periods row corresponding to the number of interest periods.
2. Select the rate per month column corresponding to the period interest rate.
3. Locate the value in the cell where the periods row intersects with the rate-per-period column.
4. Multiply the annuity payment by the table from step 3.
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
FV = annuity payment x table value
Using Table 14-1 to find the FV of a semiannual ordinary annuity of $6,000 for five years at 6% annual interest, compounded semiannually.
5 years x 2 periods per year = 10 periods 6% annual interest rate = 3% period interest rate
2 periods per year See Table 14-1 for 10 periods at 3% = 11.464 FV = $6,000 x 11.464 = $68,784 The future value of this annuity is $68,784.
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try this example
Find the future value of a semiannual ordinary annuity of $ 5,000 for 10 years at 4% annual interest compounded semiannually.
$121,485
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
14.1.3 Find the FV of Annuity Due Using the Simple Interest Formula
1. Find the first end-of-month period principal: multiply the annuity payment by the sum of 1 and the period interest rate.
2. For each remaining period in turn, find the next end-of-period principal = previous end of period principal = annuity payment x 1 + period interest rate
3. Identify the last end-of-period principal as the future value.
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Look at this example
Find the total interest earned on the annuity of $6,000 we looked at on Slide 11.
Total invested =$6,000 x 10 (number of payments) = $60,000
Total interest = $68,784 - $60,000 = $8,784.
The total interest earned on this annuity is $8,784
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Ordinary annuity versus annuity due
The difference between an ordinary annuity and an annuity due is whether you made the first payment immediately or at the end of the first period.
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Find the FV of this annuity due Find the FV of annuity due of $1,000 for three
years at 4% annual interest. Find the total investment and total interest earned.
End-of-Y 1 value = $1,000 x 1.04 = $1,040.
End-of-Y 2 value = $2,040 x 1.04 = $2,121.60
End-of-Y 3 value = $3,121.60 x 1.04 = $3,246.46
The future value of this annuity is $3,246.46
The interest earned = $246.46
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try this example
Find the future value of an annual annuity due of $5,000 for three years at 4%. Find the total investment amount and the total interest earned.
Total investment = $15,824.32 Total interest earned = $824.32
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
14.1.4 Find the FV of an Annuity Due Using a $1.00 Ordinary Annuity FV Table
Using Table 14-1:
Select the periods row corresponding to the number of interest periods.
Select the rate-per-period column corresponding to the period interest rate.
Locate the value in the cell where the periods row intersects the rate-per-period column.
(next slide)
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Using a $1.00 ordinary annuity FV table
4. Multiply the annuity payment by the table value from step 3. This is equivalent to an ordinary annuity.
5. Multiply the amount that is equivalent to an ordinary annuity by the sum of 1 and the period interest rate to adjust for the extra interest that is earned on an annuity due.
Future value = annuity payment x table value x (1 = period interest rate)
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Look at this example
Using Table 14-1, find the FV of a quarterly annuity due of $2,800 for four years at 8% annual interest, compounded quarterly.
4 years x 4 periods per year = 16 periods
8% annual interest rate ÷ 4 periods p/year = 2%
Table 14-1 value for 16 periods at 2% = 18.639
FV = $2,800 x 18.639 x 1.02 = $52,232.98
The future value of this annuity is $52,232.98
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try this example
Using Table 14-1, find the FV of a quarterly annuity due of $1,800 for three years at 8% annual interest, compounded quarterly.
$24,624.43
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
14.1.5 Find the FV of an Ordinary Annuity or Annuity Due Using a Formula
Identify the period rate (R) as a decimal equivalent, the number of periods (N), and the amount of the annuity payment (PMT).
Substitute the values from Step 1 into the appropriate formula.
(next slide)
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
14.1.5 Find the FV of an Ordinary Annuity or Annuity Due Using a Formula
(next slide)
.
(1 ) 1N
ord annuity
RFV PMT
R
(1 ) 1(1 )
N
annuitydue
RFV PMT R
R
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try this example
Find the future value of an ordinary annuity of $100 paid monthly at 5.25% for 10 years.
R = .0525/12 = .004375 (Period Int. Rate)
The future value of the ordinary annuity is $15,737.70.
120
.
(1.004375) 1$100
.004375ord annuityFV
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try this example
Find the future value of an annuity due of $50 monthly at 5.75% for 5 years.
R = .0575/12 = .0047916667 (Period Int. Rate)
The future value of the ordinary annuity is $15,737.70.
60(1.0047916667) 1$50 (1.0047916667)
.0047916667annuitydueFV
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
14.2 Sinking Funds and the Present Value of an Annuity
Find the sinking fund payment using a $1.00 sinking fund payment table.
Find the present value of an ordinary annuity using a $1.00 ordinary annuity present value table.
Find the sinking fund payment or the present value of an annuity using a formula.
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
14.2.1 Find the Sinking Fund Payment
1. Select the periods row corresponding to the number of interest periods.
2. Select the rate-per-period column corresponding to the period interest rate.
3. Locate the value in the cell where the periods row intersects the rate-per-period column.
4. Multiply the table value from step 3 by the desired future value
Sinking fund payment = FV x Table 14.2 value
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Look at this example Using Table 14-2, find the annual sinking fund
payment required to accumulate $140,000 in 12 years at 6% annual interest rate.
Table 14-2 indicates that a 12-period value at 6% is equal to 0.0592770
SFP = $140,000 x 0.0592770 = $8,298.78
A sinking fund payment of $8,298.78 is required at the end of each year for 12 years at 6% to yield the desired $140,000.
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try this example
Use Table 14-2 for find the annual sinking fund payment required to accumulate $100,000 in 10 years at 4% annual interest.
Find the number of periods: 10 Find the table value where 10 periods and 4%
intersect: 0.0832909 Multiply the desired FV by the table value The annual sinking fund payment required to
accumulate $100,000 in 10 years is $8,329.09
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
14.2.2 Find the PV of an Ordinary Annuity Using
a $1.00 Ordinary Annuity PV Table
Use Table 14-3 in your text to locate the given number of periods and the given rate per period.
Multiply the table value times the periodic annuity payment.
Present value of an annuity = periodic annuity payment x table value
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Look at this example Use Table 14-3 to find the present value of a semiannual
ordinary annuity of $3,000 for seven years at 6% annual interest, compounded semiannually.
7 years x 2 periods per year = 14 periods
6% annual interest rate ÷ 2 periods p/year = 3% period interest rate
PV annuity = $3,000 x 11.296 (table factor)= $33,888
By investing $33,888 now at 6% interest, compounded semiannually, you can receive an annuity payment of $3,000 twice a year for seven years.
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try this example Roberto Santos wants to know how much he will
have to invest now to receive an annuity payment of $5,000 twice a year for ten years. The money will be invested at 6% annually compounded semiannually.
Number of periods = 20; Interest per period = 3% Table factor = 14.877 5,000 x 14.877 = 74,385 He must invest $74,385 now to receive a $5,000
annuity payment twice a year for 10 years.
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
14.2.3 Find the Sinking Fund Payment or the Present Value of an Annuity
Using a Formula
Identify the period rate (R) as a decimal equivalent, the number of periods (N), and the future value (FV) of the annuity.
Substitute the values from Step 1 into the appropriate formula.
(next slide)
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
14.2.3 Find the Sinking Fund Payment or the Present Value of an Annuity
Using a Formula
(next slide)
. (1 ) 1ord annuity N
RPMT FV
R
.
(1 ) 1
(1 )
N
ord annuity N
RPV PMT
R R
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try this example
Find the monthly contribution to reach $100,000 in 20 years with an annuity fund that earns 5.5% annual interest.
R = .055/12 = .0045833333; N = 240
The payment required each month into the sinking fund is $229.56.
. 240
.0045833333$100,000
(1.0045833333) 1ord annuityPMT
Business Math, Eighth EditionCleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try this example
How much is needed in a fund that pays 5.5% to receive $700 per month for 20 years.
R = .055/12 = .0045833333; N = 240
$85,670.56 is needed in the fund to receive $700 each month for 20 years.
240
. 240
(1.0045833333) 1$700
.0045833333(1.0045833333)ord annuityPV