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ANNUITY - Quipper

ANNUITY - Quipper
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ANNUITYby: LLOYD PSYCHE T. BALTAZAR

LESSON OBJECTIVESAt the end of the lesson, you are expected to:a. distinguishes between simple and general annuities;b. finds the future value and present value of both simple annuities andgeneral annuities; andc. calculates the present value and period of deferral of a deferred annuity.

DEFINITION OF TERMSANNUITY- a series of equal payments made at equal intervals of timePAYMENT INTERVAL- the period of time between successivepayments (e.g. monthly, quarterly)FUTURE VALUE- sum of future values of all the payments tobe made during the entire term of theannuity.TERM OF ANNUITY- the length of time between the beginningof the first payment period and the end ofthe last payment period.PRESENT VALUE- sum of present values of all the paymentsto be made during the entire term of theannuity

DEFINITION OF TERMSORDINARY ANNUITY- a series of payments where each periodic paymentis made at the end of the payment interval.SIMPLE ANNUITYGENERAL ANNUITY- is an annuity whose interest conversionperiod is equal to the payment interval.- is an annuity whose interest conversionperiod is not equal to the payment interval.Example:Monthly payments, and the interestis compounded monthlyExample:Monthly payments, but the interestis compounded semi-annually

Formula for Simple Ordinary AnnuityFV R(1 𝑖)𝑛 1𝑖1 (1 𝑖) 𝑛𝑖PV Rwhere:FV future value of simple ordinary annuityPV present value of simple ordinary annuityR amount of periodic paymenti rate of interest per conversion periodπ‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ οΏ½οΏ½π‘Ÿπ‘ π‘–π‘œπ‘› π‘π‘’π‘Ÿπ‘–π‘œπ‘‘n number of payment (π‘π‘œπ‘›. π‘π‘’π‘Ÿπ‘–π‘œπ‘‘ π‘‘π‘’π‘Ÿπ‘š π‘œπ‘“ π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘¦)

ILLUSTRATIVE PROBLEMREGULAR/PERIODICPAYMENT (R)A person made a deposit of 2,000 at the end of each six monthsfor 2 years at 5% compounded semi-annually. How much is in hisaccount at the end of 2 years?RATE OF INTERESTCONVERSION PERIODTERM OF ANNUITYPAYMENT INTERVAL

Prob 1: A person made a deposit of 2,000 at the end of each sixmonths for 2 years at 5% compounded semi-annually. How much is inhis account at the end of 2 years?Given:To find i,𝑅 2,000π‘Ÿ 0.05term of annuity 2𝑖 π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ οΏ½οΏ½π‘Ÿπ‘ π‘–π‘œπ‘› π‘π‘’π‘Ÿπ‘–π‘œπ‘‘π‘– 0.052To find n,𝑛 π‘π‘œπ‘›. π‘π‘’π‘Ÿπ‘–π‘œπ‘‘ π‘‘π‘’π‘Ÿπ‘š π‘œπ‘“ π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘¦πΉπ‘‰ 𝑅(1 𝑖)𝑛 1𝑖𝐹𝑉 𝑅(1 𝑖)𝑛 1𝑖𝐹𝑉 2,000(1 0.025)4 10.025𝐹𝑉 2,000(1.025)4 10.025𝑖 0.025con. period 2payment int. 2Solution for FV:𝑛 2 2𝑛 4𝐹𝑉 8,305.03

Prob 2: A television (TV) set is for sale at 13,499 in cash or oninstallment terms, 2,500 each month for the next 6 months at 9%compounded monthly. If you were the buyer, what would you prefer,cash or installment?Given:To find i,𝑅 2,500π‘Ÿ 0.09term of annuity 61π‘œπ‘Ÿ122𝑖 π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ οΏ½οΏ½π‘Ÿπ‘ π‘–π‘œπ‘› π‘π‘’π‘Ÿπ‘–π‘œπ‘‘π‘– 0.0912𝑃𝑉 𝑅1 (1 𝑖) 𝑛𝑖𝑃𝑉 𝑅1 (1 𝑖) 𝑛𝑖𝑃𝑉 2,5001 (1 0.0075) 60.0075𝑃𝑉 2,5001 (1.0075) 60.0075𝑖 0.0075con. period 12payment int. 12Solution for PV:To find n,𝑛 π‘π‘œπ‘›. π‘π‘’π‘Ÿπ‘–π‘œπ‘‘ π‘‘π‘’π‘Ÿπ‘š π‘œπ‘“ π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘¦1𝑛 12 2𝑛 6𝑃𝑉 14,613.99Therefore, it is wiser to buy the televisionat P13,499 in cash

Prob 3: Mr. Tanjiro paid 200,000 as down payment for a car. Theremaining amount is to be settled by paying 16,200 at the end of eachmonth for 5 years. If interest is 10.5% compounded monthly, what is thecash price of his car? (Cash price down payment present value)Given:To find i,𝑅 16,200π‘Ÿ 0.105term of annuity 5𝑖 π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ οΏ½οΏ½π‘Ÿπ‘ π‘–π‘œπ‘› π‘π‘’π‘Ÿπ‘–π‘œπ‘‘π‘ƒπ‘‰ 𝑖 0.10512𝑃𝑉 𝑖 0.00875con. period 12payment int. 12𝑃𝑉 𝑅1 (1 𝑖) 𝑛𝑖Solution for PV:To find n,1 (1 𝑖) 𝑛𝑅𝑖1 (1 0.00875) 6016,2000.00875𝑃𝑉 16,2001 (1.00875) 600.00875𝑃𝑉 753,702.20𝑛 π‘π‘œπ‘›. π‘π‘’π‘Ÿπ‘–π‘œπ‘‘ π‘‘π‘’π‘Ÿπ‘š π‘œπ‘“ π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘¦π‘π‘Žπ‘ β„Ž π‘π‘Ÿπ‘–π‘π‘’ 200,000 753.702.20𝑛 12 5𝑛 60π‘π‘Žπ‘ β„Ž π‘π‘Ÿπ‘–π‘π‘’ 953,702.20

Prob 4: Suppose that you vow to save 500 a month for the next threeyears, with your first deposit one month from today. If your savings canearn 3% converted monthly, determine the total in your account 3 yearsfrom now.Given:To find i,𝑅 500π‘Ÿ 0.03term of annuity 3𝑖 π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ οΏ½οΏ½π‘Ÿπ‘ π‘–π‘œπ‘› π‘π‘’π‘Ÿπ‘–π‘œπ‘‘π‘– 0.0312𝐹𝑉 𝑅(1 𝑖)𝑛 1𝑖𝐹𝑉 𝑅(1 𝑖)𝑛 1𝑖𝐹𝑉 500(1 0.0025)36 10.0025𝐹𝑉 500(1.0025)36 10.0025𝑖 0.0025con. period 12payment int. 12Solution for FV:To find n,𝑛 π‘π‘œπ‘›. π‘π‘’π‘Ÿπ‘–π‘œπ‘‘ π‘‘π‘’π‘Ÿπ‘š π‘œπ‘“ π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘¦π‘› 12 3𝑛 36𝐹𝑉 18,810.28

Prob 5: A retired employee wished to get 15,000 every month for 10years from her savings deposit. If the money is worth 12% compoundedmonthly, how much should her money be in the account in order to getthe desired amount?Given:To find i,𝑅 15,000π‘Ÿ 0.12term of annuity 10𝑖 π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ οΏ½οΏ½π‘Ÿπ‘ π‘–π‘œπ‘› π‘π‘’π‘Ÿπ‘–π‘œπ‘‘π‘ƒπ‘‰ 𝑖 0.1212𝑃𝑉 𝑖 0.01con. period 12payment int. 12𝑃𝑉 𝑅1 (1 𝑖) 𝑛𝑖Solution for PV:To find n,𝑛 π‘π‘œπ‘›. π‘π‘’π‘Ÿπ‘–π‘œπ‘‘ π‘‘π‘’π‘Ÿπ‘š π‘œπ‘“ π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘¦π‘› 12 10𝑛 1201 (1 𝑖) 𝑛𝑅𝑖1 (1 0.01) 12015,0000.01𝑃𝑉 15,0001 (1.01) 1200.01𝑃𝑉 1,045,507.83

GENERAL ANNUITY

STEPS IN SOLVING GENERAL ANNUITY1. convert the regular/periodic payment in general annuity intoits equivalent regular/periodic payment in simple annuity withrespect to the interest period;2. then use the formula for simple annuity.

Formula for General AnnuityB R𝑖(1 𝑖)π‘˜ 1FV R(1 𝑖)𝑛 1𝑖PV R1 (1 𝑖) 𝑛𝑖where:B Periodic Payment (in simple interest)R Periodic payment in the given general annuity problemi rate of interest per conversion period π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ οΏ½οΏ½π‘Ÿπ‘ π‘–π‘œπ‘› π‘π‘’π‘Ÿπ‘–π‘œπ‘‘k interest conversion period in a year divided payment interval also in a yeark π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘π‘œπ‘›π‘£π‘’π‘Ÿπ‘ π‘–π‘œπ‘› οΏ½οΏ½ π‘–π‘›π‘‘π‘’π‘Ÿπ‘£π‘Žπ‘™

Prob 6: Nami deposits 1,000 at the end of each quarter in her savingsaccount earning interest rate of 3.6% compounded monthly. How muchwill she have in 5 years?Given:To find i,𝑅 1,000𝑖 π‘π‘œπ‘›π‘£π‘’π‘Ÿπ‘ π‘–π‘œπ‘› π‘π‘’π‘Ÿπ‘–π‘œπ‘‘π‘Ÿ 0.036𝑖 term of annuity 5𝑖 0.003con. period 12payment int. 4B R𝑖(1 𝑖)π‘˜ 1Solution for B:π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘0.03612To find k,π‘˜ π‘π‘œπ‘›π‘£π‘’π‘Ÿπ‘ π‘–π‘œπ‘› οΏ½οΏ½ π‘–π‘›π‘‘π‘’π‘Ÿπ‘£π‘Žπ‘™π‘˜ 124π‘˜ 3B R𝑖(1 𝑖)π‘˜ 1B (1,000)(0.003)(1 0.003)3 1Solution for FV:𝐡 𝑅 332.34To find n,𝑛 π‘π‘œπ‘›. π‘π‘’π‘Ÿπ‘–π‘œπ‘‘. π‘‘π‘’π‘Ÿπ‘š π‘œπ‘“ π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘¦π‘› 12 5𝑛 60𝐹𝑉 𝑅(1 𝑖)𝑛 1𝑖𝐹𝑉 332.34(1 0.003)60 10.003𝐹𝑉 332.34(1.003)60 10.003𝐹𝑉 21,812.01

Prob 7: Sanji Vinsmoke borrowed an amount of money from Luffy. Heagrees to pay the principal plus interest by paying 38,973.76 each yearfor 3 years. How much money did he borrow if interest is 8%compounded quarterly?Given:To find i,𝑅 38,973.76𝑖 π‘π‘œπ‘›π‘£π‘’π‘Ÿπ‘ π‘–π‘œπ‘› π‘π‘’π‘Ÿπ‘–π‘œπ‘‘π‘Ÿ 0.08𝑖 term of annuity 3𝑖 0.02con. period 4payment int. 1B R𝑖(1 𝑖)π‘˜ 1Solution for B:π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘0.084To find k,π‘˜ π‘π‘œπ‘›π‘£π‘’π‘Ÿπ‘ π‘–π‘œπ‘› οΏ½οΏ½ π‘–π‘›π‘‘π‘’π‘Ÿπ‘£π‘Žπ‘™π‘˜ 41π‘˜ 4B R𝑖(1 𝑖)π‘˜ 1B (38,973.76)(0.02)(1 0.02)4 1Solution for PV:𝐡 𝑅 9,455.96To find n,𝑛 π‘π‘œπ‘›. π‘π‘’π‘Ÿπ‘–π‘œπ‘‘. π‘‘π‘’π‘Ÿπ‘š π‘œπ‘“ π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘¦π‘› 4 3𝑛 12𝑃𝑉 𝑅1 (1 𝑖) 𝑛𝑖𝑃𝑉 9,455.961 (1 0.02) 120.02𝑃𝑉 9,455.961 (1.02) 120.02𝑃𝑉 100,000.00

Prob 8: Monkey D. Luffy started to deposit 300 monthly in a fund thatpays 6% compounded quarterly. How much will be in the fund after 15years?Given:To find i,𝑅 300𝑖 π‘π‘œπ‘›π‘£π‘’π‘Ÿπ‘ π‘–π‘œπ‘› π‘π‘’π‘Ÿπ‘–π‘œπ‘‘π‘Ÿ 0.06𝑖 term of annuity 15con. period 4payment int. 12B R𝑖(1 𝑖)π‘˜ 1Solution for B:π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘0.064𝑖 0.015To find k,π‘˜ π‘π‘œπ‘›π‘£π‘’π‘Ÿπ‘ π‘–π‘œπ‘› οΏ½οΏ½ π‘–π‘›π‘‘π‘’π‘Ÿπ‘£π‘Žπ‘™π‘˜ 4121π‘˜ 3B B Solution for FV:R𝑖(1 𝑖)π‘˜ 1(300)(0.015)1(1 0.015)3 1𝐡 𝑅 904.49To find n,𝑛 π‘π‘œπ‘›. π‘π‘’π‘Ÿπ‘–π‘œπ‘‘. π‘‘π‘’π‘Ÿπ‘š π‘œπ‘“ π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘¦π‘› 4 15𝑛 60𝐹𝑉 𝑅(1 𝑖)𝑛 1𝑖𝐹𝑉 904.49(1 0.015)60 10.015𝐹𝑉 904.49(1.015)60 10.015𝐹𝑉 87,025.19

DEFERRED ANNUITY

TERMINOLOGYDeferred Annuity- an annuity in which the first periodic payment is made after a certain interval of time,known as the deferral periodOrdinary Deferred Annuity- when deferral period ends one payment interval before the first periodic payment.The future value of a deferred annuity- is the accumulated value of the stream of payments at the end of the annuity period. Thisis the same procedure as future value of an ordinary annuity (both simple and generalannuity).The present value of a deferred annuity- is the discounted value of the stream of payments at the beginning of the deferral period.

Ordinary Deferred AnnuityPVdef Rwhere:1 (1 𝑖) 𝑛𝑖(1 𝑖)𝑑FORMULA(1 𝑖)𝑛 1FVdef R𝑖PVdef present value of deferred annuityFVdef future value of deferred annuityR amount of periodic paymenti rate of interest per conversion periodn number of paymentd number of deferred period

DETERMINE THEPERIOD OF DEFERRAL(Assume the annuities are Ordinary) Payments of 1,000 at the end of each year for ten years with the firstpayment made three years from now.period of deferral 2 Payments of 5,000 at the end of every 6 months for 15 years with thefirst payment made 5 years from nowperiod of deferral 9

DETERMINE THEPERIOD OF DEFERRAL(Assume the annuities are Ordinary) A second hand car sells for 120,000 down payment and 24 monthly paymentsof 7,000 each, the first payment being due at the end of the 6th month. Find thecash price if the interest rate is 8% compounded monthly.period of deferral 5 A quarterly payment of 8,500 at 6% compounded quarterly, the first payment isdue in 1 year and 6 months and the last payment is at the end of 5 years.period of deferral 5

Prob 9: A second hand car sells for 120,000 down payment and 24monthly payments of 7,000 each, the first payment being due at theend of the 6th month. Find the cash price if the interest rate is 8%compounded monthly. (Cash price down payment present value)Given:To find i,𝑅 7,000π‘Ÿ 0.08term of annuity 2con. period 12payment int. 12𝑑 5PVdef R1 (1 𝑖) 𝑛𝑖(1 𝑖)𝑑𝑖 π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ οΏ½οΏ½π‘Ÿπ‘ π‘–π‘œπ‘› π‘π‘’π‘Ÿπ‘–π‘œπ‘‘π‘– 0.0812Retain i, since the answeris repeatingSolution for PV:PVdef R1 (1 𝑖) 𝑛𝑖(1 𝑖)𝑑PVdef 7,0000.08 241 1 120.080.08 51 1212PVdef 149,716.28To find n,𝑛 π‘π‘œπ‘›. π‘π‘’π‘Ÿπ‘–π‘œπ‘‘ π‘‘π‘’π‘Ÿπ‘š π‘œπ‘“ π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘¦π‘π‘Žπ‘ β„Ž π‘π‘Ÿπ‘–π‘π‘’ 120,000 149,716.28𝑛 12 2𝑛 24π‘π‘Žπ‘ β„Ž π‘π‘Ÿπ‘–π‘π‘’ 269,716.28

Prob 10: If money is worth 9% compounded semi-annually, find thepresent value of 6 semi-annual payments of 10,000 each, the firstpayment is due in 4 years.Given:To find i,𝑅 10,000π‘Ÿ 0.09term of annuity 6con. period 2payment int. 2𝑑 7PVdef R1 (1 𝑖) 𝑛𝑖(1 𝑖)𝑑𝑖 π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ οΏ½οΏ½π‘Ÿπ‘ π‘–π‘œπ‘› π‘π‘’π‘Ÿπ‘–π‘œπ‘‘PVdef R𝑖 0.092PVdef 10,0001 1 0.045 120.045 1 0.045 7PVdef 10,0001 1.045 120.045 1.045 7𝑖 0.045To find n,𝑛 π‘π‘œπ‘›. π‘π‘’π‘Ÿπ‘–π‘œπ‘‘ π‘‘π‘’π‘Ÿπ‘š π‘œπ‘“ π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘¦1 (1 𝑖) 𝑛𝑖(1 𝑖)𝑑Solution for PV:𝑛 2 6𝑛 12PVdef 67,005.93

Prob 11: Find the present value of 24 annual payments of 20,000each, the first payment is due after 3 years and the interest rate is 9%compounded annually.Given:To find i,𝑅 20,000π‘Ÿ 0.09term of annuity 24con. period 1payment int. 1𝑑 2PVdef R1 (1 𝑖) 𝑛𝑖(1 𝑖)𝑑𝑖 π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ οΏ½οΏ½π‘Ÿπ‘ π‘–π‘œπ‘› π‘π‘’π‘Ÿπ‘–π‘œπ‘‘PVdef R𝑖 0.091PVdef 20,0001 1 0.09 240.09 1 0.09 2PVdef 20,0001 1.09 240.09 1.09 2𝑖 0.09To find n,𝑛 π‘π‘œπ‘›. π‘π‘’π‘Ÿπ‘–π‘œπ‘‘ π‘‘π‘’π‘Ÿπ‘š π‘œπ‘“ π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘¦1 (1 𝑖) 𝑛𝑖(1 𝑖)𝑑Solution for PV:𝑛 1 24𝑛 24PVdef 163,397.22

Ordinary Deferred Annuity - when deferral period ends one payment interval before the first periodic payment. The future value of a deferred annuity - is the accumulated value of the stream of payments at the end of the annuity period. This is the same procedure as future value of an ordinary annuity (both simple and general annuity).

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