What is the amount of each annuity payment if a 5 year ordinary annuity has a future value of 1000 with an interest rate of 8%?

1. The future value of the $100 to be received in a year is $100 × (1.01)2 = $102.01
2. The future value of the $100 to be received in 2 years is $100 × 1.01 = $101.00.
3. The future value of the $100 to be received in three years is $100.
4. The future value of the annuity is therefore equal to:

   $102.01 + $101.00 + $100.00 = $303.01

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   • What is the present value of a $1000 ordinary annuity that earns 8% annually for an infinite number of periods?
   • What is the present value of a 5 year ordinary annuity with annual payments of 200?
   • What is the formula to find the amount of ordinary annuity?
   • How do you calculate the future value of an annuity?

   This is equivalent to:

   $100(1.01)2 + $100(1.01)1 + $100(1.01)0
   = $100[(1.01)2 + (1.01)1 + (1.01)0]
   = $100∑t = 02(1.01)t

   The general formula can be written as:

   FVA = PMT∑t = 0nm - 1(1 + APRm)t

   A more useful formula is:

   FVA = PMT (1 + APRm)mn-1(APRm)

1. Example: What is the future value of a 15 year, 9% ordinary annuity of $75/year?

FVA = $75 × (1.09)1 × 15- 1(0.09) = $2,202 .07

Financial Calculator:

N = 15 I/Y = 9% PMT = -75

Hit FV key, or Compute Key then FV key

1. Annuity Due

FVADue = PMT [(1 + APRm )mn-1(APRm)](1 + APRm)

Example: What is the future value of a 15 year, 9%, annuity due of $75/year?

FVADue = $75 [(1.09)15- 1(0.09)] (1.09) = $2,400.25

Why is the FVDue larger than the future value of an ordinary annuity?

Financial Calculator - may have an annuity due button.

2. FV of an ordinary annuity when compounding occurs more than once a year.

Example: Suppose you plan to purchase an automobile upon graduating from EWU. You plan to graduate in twelve months, and would like to have a down payment saved up by then. If you save $40 every month in an account that pays 7.5 percent compounded monthly, how big will your down payment be?

FVA = $40 (1 + 0.07512) 12 × 1- 1(0.07512) = $496.85

Financial Calculator: I/Y = 0.625; PMT = 40; n = 12; Hit FV, or CPT and FV

1. Definition: The lump sum payment required today that would be equivalent to the annuity payments spread over the annuity period.

Formula: Take the FVA formula:

FVA = PMT (1 + APRm)mn-1( APRm)

The present value of an annuity can be written as: FVA/(1 + APR/m)nm

Therefore:

PVA = FVA(1 + APR m)mn= [PMT (1 + APRm)mn -1(APRm)](1 + APRm)mn= PMT[ (1 + APRm)mn-1][1(1 + APRm)mn ]APRmPVA =  PMT[1 - 1(1 + APRm)mn ]APRm                - or -PVA = PMT[1(APR m) - 1APRm(1 + APRm)mn]

2. Example: What is the present value of a 17 year annuity of $6200/year if the interest rate is 14% compounded annually?

PVA = $6,200 × [1 - 1(1.14)17] 0.14 = $39,511.73

Financial Calculator: N = 17; PMT = 6200; I/Y = 14; Hit PV, CPT then PV

Example: You have just won the lottery and you can choose between receiving $200,000 a year for four years with payments beginning in one year, or you can receive $200,000 now and receive payments of $75,000 per year at the end of each year for the next ten years. If the appropriate discount rate is 13 percent, which should you choose?

PVA = $200,000 × [1 - 1(1.13)4]0 .13 = $594,894.27PV = $200,000 + $75,000 × [1 - 1(1.13)10 ]0.13 = $606,968.26

3. When compounding occurs more than once a year:

Example: John pays a $137 car payment each month. He will have the loan paid off in 4 years if he continues to make his monthly payments. How much would John need to pay off his car loan today? Assume he has just made a payment and that the interest rate he is charged on this loan is 8.3 percent compounded monthly?
PVA = $137 × [1 - 1(1 + 0.08312)48 ]0.08312 = $5,579.54

Financial Calculator

Note: PVIFAr,n - always smaller than number of years the annuity runs. FVIFAr,n - larger than number of years assuming K > 0.
4. Present Value of an Annuity Due

PVADue = PMT[1(APRm) - 1APRm( 1 + APRm)mn](1 + APRm)

Example: What is the present value of an 17 year annuity due which pays $6,200 per year if the interest rate is 14 percent?

PVADue = $6,200[1(0.14) - 10.14(1.14) 17]×1.14=$45,043.37

1. Definition: An annuity that goes on indefinitely
2. PV of Perpetuity

PVP = PMT/(r/m)

This is just a special case of the present value of an annuity formula where n = ∞:

PVP = PVA = PMT [1(APRm)−1(APRm)(1+APRm)m∞ ]                        = PMT[1(APRm)−1( APRm)(1+APRm)∞]                        = PMT [1(APRm)−1(APRm)∞]                       = PMT [1(APRm)−1∞]                       = PMT[1 (APRm)−0]              PVP = PMT(APRm)
3. Example: What is the present value of a perpetuity which pays $7,000 per year if the discount rate is 3 percent?

PV = $7,000[1 - 1(1.03)1,000,000 0.03] = $233,333.33

PV = 7000/0.03 = $233,333.33

4. Example:

   One specific example of a perpetual Dutch annuity of the seventeenth century may be cited. In 1624 one Elsken Jorisdochter (Elsie, the daughter of George) invested 1200 florins in a bond issued for repairs to a dike. She received a bond of the Lekdyk Bovendams Company (chartered 1323), which was a semipublic enterprise with taxing power. The company and this bond survived at least to 1957. This perpetual bond originally paid 6¼% interest per annum, about the same rate then paid by the provinces. It promised no repayment of principal. At some time in the eighteenth century the then owner agreed to a reduction of interest to 2½%. In 1957 this bond was still paying 2½% per annum. The bond must be presented at Utrecht for interest payments at least once every five years, and payments are recorded on the back. The bond states that it is "free of all taxes, impositions or charges whichsoever, however called or disguised, with no single exception." In 1938 this bond was presented to the New York Stock Exchange, which collected interest as it became payable.

   A History of Interest Rates pp. 126-127

5. What is the future value of a perpetuity?

   FVP = FVA = PMT[(1+APRm)m∞−1(APR m)]                       = PMT[(1+APRm)∞ −1(APRm)]                       = PMT[∞ -1 (APRm)]                       = PMT[∞ (APR m)]               FVP = PMT × ∞

1. PV = Σ(PV of individual cash flow components)
2. FVn = Σ(FV of individual cash flow components)
3. Example:

   Let r = 4% compounded annually

   Year	PMT	PVIFr,n	PV
   1	$60	0.9615	$57.6900
   2	$30	0.9246	$27.7380
   3	$45	0.8890	$40.0050
   4	$50	0.8548	$42.7400
   		ΣPV	$168.1730

 

A financial calculator can be used to solve this problem.

4. Example: Future Value of a Series of Uneven Cash Flows

   Consider the following cash flow series:

   Quarter	Cash Flow
   1	$2,000
   2	$2,000
   3	$3,000
   4	$4,000
   5	$7,000
   6	$7,000

   What is the future value of these cash flows if the interest rate is 2 percent per quarter?

   FVEnd of Quarter 6 = $2,000 × (1.02)5 + $2,000 × (1.02)4 + $3,000 × (1.02)3 + $4,000 × (1.02)2 +
   $7,000 × (1.02) + $7,000 = $25,858.24993 or $25,858.25 rounded to the nearest penny

   VIDEO: Using the TI-84 and NPV to solve for the Future Value of Uneven Cash Flows

   VIDEO: Using the TI-BAII Plus and NPV to solve for the Future Value of Uneven Cash Flows

1. A loan that is paid off in equal periodic installments over time
2. Payment Determination

PVA = PMT[1(APRm)−1(APRm)(1+APRm) mn]PMT = PVA[1(APRm)−1(APRm )(1+APRm)mn]PMT = Principal[1(APR m)−1(APRm)(1+APRm)mn]

Each Payment part interest, part principal

Example: You borrow $12,000 to purchase a boat. The interest rate is 12½ percent compounded monthly, and the term of the loan is 3 years.

What are the monthly payments?

PMT = $12,000[1 - 1(1 + 0 .12512)360.12512] = $401.44
3. Amortization Schedule:

1. Breaking down the parts of a payment without a financial calculator or a spreadsheet:
1. Suppose we wanted to find the interest paid, principal paid, and ending balance for the payment from the example above.

2. The first thing we need to know is the beginning balance at the end of month (beginning of month ). To do this, find the present value of the payments from month to month ( payments) at the end of month :

   Loan Balance at the End of Month   = $ × [1()−1() (1+)]=$MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeitaiaab+ gacaqGHbGaaeOBaiaabccacaqGcbGaaeyyaiaabYgacaqGHbGaaeOB aiaabogacaqGLbGaaeiiaiaabggacaqG0bGaaeiiaiaabshacaqGOb GaaeyzaiaabccacaqGfbGaaeOBaiaabsgacaqGGaGaae4BaiaabAga caqGGaGaaeywaiaabwgacaqGHbGaaeOCaiaabccacaqG0aGaaeiiai aab2dacaqGGaGaaeijaiaabsdacaqGWaGaaeymaiaab6cacaqG0aGa aeinaiaabodacaqG1aGaaeiiaiabgEna0kaabccadaWadaqaamaala aabaGaaGymaaqaamaabmaabaWaaSaaaeaacaaIWaGaaiOlaiaaigda caaIYaGaaGynaaqaaiaaigdacaaIYaaaaaGaayjkaiaawMcaaaaacq GHsisldaWcaaqaaiaaigdaaeaadaqadaqaamaalaaabaGaaGimaiaa c6cacaaIXaGaaGOmaiaaiwdaaeaacaaIXaGaaGOmaaaaaiaawIcaca GLPaaadaqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaaicdacaGGUaGa aGymaiaaikdacaaI1aaabaGaaGymaiaaikdaaaaacaGLOaGaayzkaa WaaWbaaSqabeaacaaIZaGaaGOmaaaaaaaakiaawUfacaGLDbaacqGH 9aqpcaGGKaGaaGymaiaaicdacaGGSaGaaGioaiaaiEdacaaI2aGaai OlaiaaiIdacaaIZaaaaa@8184@

3. Now use the beginning balance at month to calculate the interest owed with the payment:

   $ × / = $

4. The principal paid is $ - $ = $
5. The ending balance is $ - = $

Let m = 10 billion; r = 1

(1 + 1/10,000,000,000)10,000,000,000 = 2.71828182832

1. Formula:

FVn = PVer(n) ; PV = FVne-r(n)

where:

r = stated annual interest rate

n = number of years

e = 2.7183

2. Example: What is the future value of $700 compounded continuously at 8% for 7 years?

FVn = 700e7(.08) = 1225.47	Compounded daily: FV = 700 (1 + .08/360)(360)7 = 1225.39
	Compounded monthly FV = 700 (1 + .08/12)12(7) = 1223.20

What is the present value of a $1000 ordinary annuity that earns 8% annually for an infinite number of periods?

What is the present value of a 5 year ordinary annuity with annual payments of 200?

What is the formula to find the amount of ordinary annuity?

How do you calculate the future value of an annuity?