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Chapter 13Annuities andSinking Funds13-1 McGraw-Hill/IrwinCopyright 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Compounding Interest (Future Value)Annuity - A series of payments--can be payments goingout or coming in--just a series of payments (in thefinancial community, this is called a stream of payments).Term of the annuity - the time from the beginning of thefirst payment period to the end of the last paymentperiod.13-2
Future Value of an AnnuityThe future dollar amount of a series of paymentsplus interest (You want to know how much youwill end up with.)Example: You know you can afford to save 400each period; you know the interest rate; youwant to know how much you will have at the endof a specific period of time.13-3
Present value of an annuityThe amount of money needed to invest today in order toreceive a stream of payments for a given number ofyears in the future.(You want to know how much to put in the bank so thatyou can start taking out fixed sums of money in thefuture. )Example: How much do your parents put in the bank, ata specific interest rate, on your first day of college sothat you can take out 2,000 each semester over thecourse of four years in college so that it lasts until theend of the fourth year?13-4
Sinking fund: A kind of annuity. You know the amountyou need at a certain date in the future. You are lookingfor the amount of the payment in a stream of payments inorder to end up with that amount.Example: You will need 25,000 five years in the futureto replace your car. You want to know how much to put inthe bank each month in order to end up with 25,000 infive years. You know the interest rate and number ofperiods you have to save the money. The 25,000 thatyou will ultimately have is made up of your stream ofpayments into the sinking fund (in this case, a savingsaccount) and interest payments on the money in thataccount over the five years.13-5
Figure 13.1 Future value of anannuity of 1 at 8% 3.25 3.50 3.00 2.50 2.08 2.00 1.50 1.00 1.00 0.50 0.0012End of period13-63
Classification of AnnuitiesThis is Important!Annuity due - regulardeposits/payments madeat the beginning of theperiodOrdinary annuity regulardeposits/paymentsmade at the end of theperiod13-7Jan. 31MonthlyJan. 1March 30QuarterlyJan. 1June 30SemiannuallyJan. 1Dec. 31AnnuallyJan. 1
Tools for Calculating Compound InterestNumber of periods (N)Number of years timesthe number of times theinterest is compoundedper yearRate for each period (R)Annual interest ratedivided by the number oftimes the interest iscompounded per yearIf you compounded 100 each year for 3 years at 6%annually, semiannually, or quarterly What is N and R?PeriodsAnnually:3x1 3Semiannually: 3 x 2 6Quarterly: 3 x 4 1213-8RateAnnually:6% / 1 6%Semiannually: 6% / 2 3%Quarterly: 6% / 4 1.5%
Calculating Future Value of an Ordinary Annuity ManuallyStep 1. For period 1, no interest calculation is necessary, sincemoney is invested at the end of periodStep 2. For period 2, calculate interest on the balance and addthe interest to the previous balance.Step 3. Add the additional investment at the end of period 2 tothe new balance.Step 4. Repeat steps 2 and 3 until the end of the desiredperiod is reached.13-9
Calculating Future Valueof an Ordinary Annuityby Hand(deposits at ends of years)Find the value of aninvestment after 5years for a 2,000ordinary annuity at 9%13-10Manual Calculation 2,000.00 End of Yr 1180.00 int, yr 22,180.00 new bal.2,000.00 End of Yr 24,180.00 new bal.376.20 int, yr 34,556.20 new bal.2,000.00 End of Yr 36,556.20 new bal.590.06 int, yr 47,146.26 new bal.2,000.00 End of Yr 49,146.26 new bal.823.16 int, yr 59,969.42 new bal.2,000.00 End of Yr 5 11,969.42 new bal.
Calculating Future Value of an OrdinaryAnnuity by Table LookupStep 1. Calculate the number of periods and rate per period.Step 2. Look up the number periods on the appropriate interestrate page in the Math Handbook. When you find the row for thecorrect number of periods, follow across that row to the amountof annuity column. The intersection gives the table factor for thefuture value.Step 3. Multiply the payment for a period by the table factor.This gives the future value of the annuity.Future value of ordinary annuity13-11Annuity pmt.for a periodxOrdinary annuitytable factor
Table 13.1 Ordinary annuity table:Compound sum of an annuity of 1Ordinary annuity table: Compound sum of an annuity of 1 523.275925.129027.152129.360931.7725
Looking Up the Future Value of an Annuity--9%,5 periods---as shown in the Math Handbook9%PeriodCompoundInterest(future value)PresentValueAmount ofAnnuity(future value)PresentValueof 2.36740.422415.19296.41770.065813-13
Future Value of an Ordinary AnnuityFind the value of aninvestment after 5 yearsfor a 2,000 ordinaryannuity at 9%N 5x1 5R 9%/1 9%5.9847 x 2,000 11,969.4013-14
Calculating Future Value of anAnnuity Due ManuallyStep 1. Calculate the interest on the balance for the period andadd it to the previous balanceStep 2. Add additional investment at the beginning of the periodto the new balance.Step 3. Repeat steps 1 and 2 until the end of the desired period isreached.See example on next slide.13-15
YEAR 12,000.00Calculating Future Valueof an Annuity Due manually(deposits at beginning ofperiod)deposit, beginning year 1180.00interest earned during year 12,180.00new balance at end of year 1YEAR 22,000.00deposit, beginning year 24,180.00balance after year 2 deposit376.20interest earned during year 24,556.20new balance at end of year 2YEAR 3Find the value of aninvestment after 5years for a 2,000annuity due at 9%2,000.00deposit, beginning year 36,556.20balance after year 3 deposit590.06interest earned during year 37,146.26new balance at end of year 3YEAR 42,000.00deposit, beginning year 49,146.26balance after year 4 deposit823.16interest earned during year 49,969.42new balance at end of year 4YEAR 52,000.0011,969.4213-16deposit, beginning year 5balance after year 5 deposit1,077.25interest earned during year 513,046.67new balance at end of year 5
Calculating Future Value of an Annuity Due by Table LookupCalculate the number of periods and rate per period. Addone extra period.Look up the number of periods on the appropriateinterest rate page in the Math Handbook.At the intersection of the correct period row and theAmount of Annuity column, you will find the neededtable factor.Multiply the payment each period by the table factor.Subtract 1 payment.13-17
Future Value of an Annuity DueFind the value of aninvestment after 5years for a 2,000annuity due at 9%.N 5x1 5 1 6R 9%/1 9%7.5233 x 2,000 15,046.60 - 2,000 13,046.6013-18
Figure 13.2 - Present value ofan annuity of 1 at 8% 3.50 3.00 2.58 2.50 1.78 2.00 1.50 .93 1.00 0.50 0.0012End of period13-193
Calculating Present Value of an Ordinary Annuityby Table LookupStep 1. Calculate the number of periods and rate perperiodStep 2. Find the percent page in the Math Handbook. Lookup the number of periods in the first column. At theintersection of the correct period row and the present valueof annuity column, you will find the needed table factor.Step 3. Multiply the withdrawal for a period by the tablefactor. This gives the present value of an ordinary annuity13-20
Table 13.2 - Present Value of an Annuity of 1Present value of an annuity of 1 9.71229.10798.55958.06077.6061
PeriodCompoundInterest(future value)PresentValueAmount ofAnnuity(future value)PresentValueof 1.62890.613912.57797.72170.07955%13-22Present Value of an Annuity at 5% and 10 periods
Present Value of an AnnuityDuncan Harris wants to receive a 5,000 annuity in 5 years.Interest on the annuity is 8%semiannually. Duncan will makewithdrawals every six months.How much must Duncan investtoday to receive a stream ofpayments for 5 years.Interest withdrawal- Interest withdrawal Interest withdrawal N 5 x 2 10R 8%/2 4%8.1109 x 5,000 40,554.5013-23Interest withdrawal Interest withdrawal End of Year 5 Manual Calculation 7.72(5,000.00)4,807.72192.315,000.03(5,000.00)0.03
Sinking Funds (Find Periodic Payments)BondsBondsSinking Fund Future x Sinking FundPaymentValueTable Factor13-24
Table 13.3 - Sinking Fund Table Based on 1Sinking fund table based on 1 0.05380.04990.04630.04300.03680.0315
PeriodCompoundInterest(future value)PresentValueAmount ofAnnuity(future value)PresentValueof 2.15890.463214.48666.71010.06908%13-26Sinking Fund at 8% and 10 periods
Sinking FundTo retire a bond issue, RandolphCompany needs 150,000 in 10 years.The interest rate is 8% compoundedannually. What payment must RandolphCo. make at the end of each year tomeet its obligation?N 10 x 1 10R 8%/1 8%Check 10,350 x 14.4866149,936.30*N 10, R 8%Use SinkingFund Column0.0690 x 150,000 10,35013-27* Off due to rounding
Calculating Future Value of an Ordinary Annuity by Hand (deposits at ends of years) Find the value of an investment after 5 years for a 2,000 ordinary annuity at 9% Manual Calculation End of Yr 1 180.00 int, yr 2 End of Yr 2 376.20 int, yr 3 End of Yr 3 590.06 int, yr 4 End of Yr 4 823.16 int, yr 5 End of Yr 5
