Annuities : Future Value & Present Value Of An Ordinary Annuities

Annuities :Future value & Present Valueof an ordinary Annuitieshttp://science.utm.my/norhaiza/Department of Mathematical SciencesFaculty of ScienceSSCM4863Room: C10 336/C22 441Tel: 34321/34274/019-7747457

Annuities Definition Future value of an ordinary annuity Present value of an ordinary annuity Annuities due Perpetuities Deferred annuities Summary of annuities

Annuities An annuity is a sequence of periodic paymentso Often equal in amounto Made at equal intervals of time Example:o Monthly rent paymentso Annual premiums for a life insurance policyo Monthly housing loan repaymentso Regular deposits in a savings accountPayment period Time between successive paymentsTerm of an annuity Time from the beginning of the first payment period to the endof the last payment period

Types of AnnuitiesAnnuity Certain Has a specific statednumber of Payments Term of annuity is fixed Dates of the 1st and last paymentsare fixedEg. Housing loan repaymentOrdinary Annuity Payments made at the end of eachpayment periodEg. Loan repaymentContingent Annuity Has no fixed number of payments Term of annuity depends on someuncertain eventEg. Life insurance payments (stops with thedeath of the insured); Bond interest payment.Annuity Due Payments made at the beginning ofeach payment periodEg. Insurance premium

Annuities Definition Future value of an ordinary annuity Present value of an ordinary annuity Annuities due Perpetuities Deferred annuities Summary of annuities

Future Value of an ordinary annuityDefinition Amount due at the end of the termExample of ordinary annuity on a time diagram:Time0RRRR123n-1n Interest periodequals paymentperiod as unit ofmeasure R as the regularpayment at eachperiod1 periodtermFocal date

Future Value of an ordinary annuityWe can calculate the value of the annuity at the focal dateby repeated application of the compound interest formula

Example 1Find the future value of an ordinary annuity consisting of 4 annual paymentsof RM250 each at 3% paTime0RM250RM250RM250RM2501234Value of 2 ndpayment RM250 (2)at 4 250 1 0.03 / 𝑹𝑴𝟐𝟔𝟓. 𝟑𝟐3 rdValue of 1 stpayment RM250 (1)at 4 250 1 0.03 4 𝑹𝑴𝟐𝟕𝟑. 𝟏𝟖FV?Value ofpayment RM250 (3)at 4 250 1 0.03 . 𝑹𝑴𝟐𝟓𝟕. 𝟓𝟎Value of 4 th(final)payment RM200 (4)at 4 𝑹𝑴𝟐𝟓𝟎Thus, the future value of the ordinary annuity (based on end of term as focal date) RM273.18 RM265.32 RM257.50 RM250 RM1045.91

Future Value of an ordinary annuityAn alternative way to calculate the value of an ordinary annuity isusing the sum of geometric progression.Consider an ordinary annuity of n payments of RM1 each as shownbelowTime0RM1RM1RM1RM1RM1RM1123n-2n-1nFocal dateHere, to calculate the FV of this annuity, we need to accumulate eachpayment of RM1 to the end of the term of the annuity (ie. Focal date)and add them together (similar to Example 1)

Future Value of an ordinary annuity (cont’d)Time0RM1RM1RM1RM1RM1RM1123n-2n-1nFocal dateThe sum from the accumulation for each payment of RM1 to the focal date can beexpressed as𝐹𝑉 1 𝑖(C-.) 1 𝑖(C-/) 1 𝑖(C-4) 1 𝑖4 1 𝑖/ 1 𝑖. 1where the 1st payment at the end of the year earns interest for (n-1) years; the 2nd paymentfor (n-2) years etc. Reordering the order, we can express the equation above as:𝐹𝑉 1 1 𝑖. 1 𝑖/ 1 𝑖4 1 𝑖(C-4) 1 𝑖(C-/) 1 𝑖(C-.)

Future Value of an ordinary annuity (cont’d)𝐹𝑉 1 1 𝑖. 1 𝑖/ 1 𝑖4 1 𝑖(C-4) 1 𝑖(C-/) 1 𝑖(C-.)Geometric progression with n terms similar to the following expression.𝑆𝑛 𝑎 𝑎𝑟. 𝑎𝑟/ 𝑎𝑟4 𝑎𝑟(C-4) 𝑎𝑟(C-/) 𝑎𝑟(C-.)Hence, we have𝑎 𝑟𝑛 1𝐹𝑉 (𝑟 1)1 (1 𝑖)𝑛 1 ((1 𝑖) 1)1 𝑖 𝑛 1 𝑖Similar terms used:𝐹𝑉 𝑛 1 𝑖𝑖1𝑠𝑖𝑛1 𝑖 𝑛 1 𝑖𝐹𝑉𝐼𝐹𝐴𝑖 , 𝑛 𝑛1 𝑖 1𝑖𝐹𝑉𝐼𝐹𝐴 𝑖 , 𝑛 ie. Future value interest factor

Future Value of an ordinary annuity (cont’d)FV of an ordinary annuity of n payments of RM1 each,𝐹𝑉 𝑛 1 𝑖1𝑖è FV of an ordinary annuity of n payments of RM R each,𝐹𝑉 1 𝑖 𝑛 1𝑅𝑠 𝑅𝑖𝑖𝑛Eq.10Revisit Example 1Find the future value of an ordinary annuity consisting of 4 annual payments ofRM250 each at 3% pa𝐹𝑉 1 𝑖 𝑛 1𝑅𝑠 𝑅𝑖4250 𝑠 0.03 250 1 0.03 140.03𝑖𝑛

Example 2Find the future value at the end of 15 years of an annuity of RM100 payableat the end of each quarter if, 𝑗R 12%R 100; m 4;è i jm/m 0.12/4 0.03t 15èn mt 601 𝑖 𝑛 1𝐹𝑉 𝑅 𝑠 𝑅𝑖1 0.03 60 10.03100𝑠 60 100 0.03 𝑅𝑀16 305.34𝑖𝑛

Example 3A worker is saving RM1000 each year and depositing it into a bank. Howmuch money will she have at the end of 40 years for her retirement if theinterest rate is 9% pa?RM1KTime10RM1K2RM1K3RM1KRM1KRM1K383940R 1000; n 40; i 0.091 𝑖 𝑛 1𝐹𝑉 𝑅 𝑠 𝑅𝑖1 0.09 40 10.09𝑠1000 1000400.09 𝑅𝑀337 882.45𝑖𝑛The effect of compound interest earned over a long period is clearly evident.

Exercise1. A couple deposits RM500 every 3 months into a saving account which paysinterest at 6% convertible quarterly (i.e 1.5% per quarter). How much money be intheir account on 1 October 1999 immediately after their deposit, if the first depositwas made on 1 Jan 1992?(𝑅𝑀20 344.14)2. A frugal employee invests RM300 from his tax return each 31 Aug. After 10 suchpayment, he increases his deposits to RM400 p.a.Assuming that he has been earning 8% p.a. effective, what accumulation will therebe after 15 payments?(RM8732.29. Hint:all FV at focal date)

Annuities Definition Future value of an ordinary annuity Present value of an ordinary annuity Annuities due Perpetuities Deferred annuities Summary of annuities

Present Value of an ordinary annuityDefinition Amount due at the beginning of the term (i.e one period before the first payment)Example of ordinary annuity on a time diagram:Time01 periodPresent valueRRRR123n-1 R as the regularpayment at eachperiodn

Present Value of an ordinary annuityPV vs. FVTime0RRRR123n-1RRRR123n-1n1 periodPRESENT VALUETime0n1 periodFUTURE VALUENOTE: PV and FV are both values from the same set of payments but occur whencalculated at different valuation dates.

Present Value of an ordinary annuityThus, the relationship between PV and FV:PV FV x (1 i)-nè1 𝑖 𝑛 1 𝑅1 𝑖𝑖 -CWe define𝑎𝑖𝑛.- .dee1 1 𝑖 𝑖𝐹𝑉 𝑛Other notation:Present Value Interest Factor for an Annuity 𝑛𝑃𝑉𝐼𝐹𝐴𝑖 , 𝑛 𝑎Thus, the PV of the annuity:1 1 𝑖𝑃𝑉 𝑅𝑎 𝑖 𝑅𝑖𝑛1 𝑖 𝑛 1𝑅𝑠 𝑅𝑖𝑖𝑛 n𝑃𝑉 𝑅 𝑠 𝑖 1 𝑖𝑛 𝑅Recall 𝑛Eq.11𝑖𝑛1 1 𝑖 𝑖 𝑛

Example 4How much money is needed now to provide RM500 at the end of the year(first payment 1 year from now) for 15 years if the money earns interest at12% p.a. effective?R 500; n 15; i 0.121 1 𝑖𝑃𝑉 𝑅𝑎 𝑖 𝑅𝑖𝑛 𝑛1 1 0.12 500 𝑎 5000.12 𝑅𝑀3405.430.1215 15Note: the face value of 15 payments of RM500 each is RM7500. But only RM3405.43 isrequired NOW to provide these payments.The difference is due to the interest earned during the term

Example 5A student borrowed some money to purchase a car was to repay the loan withmonthly installments of RM150 for 3 years.Calculate the value of these repayments at the beginning of the loan if the interestrate was (a) 9% convertible monthly (b) 12% convertible monthly(a)R 150; t 3;m 12è n 36;j12 0.09; è i 0.0751 1 𝑖𝑃𝑉 𝑅𝑎 𝑖 𝑅𝑖𝑛 150 𝑎0.07536 𝑅𝑀4717.02(b) 𝑅𝑀4516.12 𝑛1 1 0.075 5000.075 36

Example 6En. Jo signed a contract that calls for a deposit of RM1500 and for the payment ofRM2000 a year for 10 years. Money is worth 10% p.a. effective.(a) What is the cash value of the contract?(b) If En Jo missed the first 2 payments, what must he pay at the time the 3rd paymentis due to bring himself up to date?(c) If En Jo missed the first 2 payments, what must he pay at the time the 3rd paymentis due to discharge his debt completely?

Example 6En. Jo signed a contract that calls for a deposit of RM1500 and for the payment ofRM2000 a year for 10 years. Money is worth 12% p.a. effective.(a) What is the cash value of the contract?(b) If En Jo missed the first 2 payments, what must he pay at the time the 3rd paymentis due to bring himself up to date?(c) If En Jo missed the first 2 payments, what must he pay at the time the 3rd paymentis due to discharge his debt completely?RM1.5K0RM2K1RM2K2RM2KRM2KRM2K3410?R 2000; t 10 yearsè n 10; i 0.121 1 𝑖𝑃𝑉 𝑅𝑎 𝑖 𝑅𝑖𝑛 2000𝑎0.1210 𝑛 1500 𝑅𝑀12 800.45 𝑅𝑀1500.- .df./2000f./-.f 1500

Example 6 (cont’d)En. Jo signed a contract that calls for a deposit of RM1500 and for the payment ofRM2000 a year for 10 years. Money is worth 12% p.a. effective.(a) What is the cash value of the contract?(b) If En Jo missed the first 2 payments, what must he pay at the time the 3rd paymentis due to bring himself up to date?(c) If En Jo missed the first 2 payments, what must he pay at the time the 3rd paymentis due to discharge his debt completely? (#exercise)RM1.5K0RM01RM0RM?23RM2K4This means En Jo has to pay the accumulated value of the 3 payments at the time of the 3rd paymentFuture Value of RM2000 annuity at time 3R 2000; t 10 yearsè n 3; i 0.12𝐹𝑉 𝑅𝑠 𝑅𝑖𝑛𝑛 1 𝑖1𝑖1 0.12 3 10.12𝑠 2000 10 2000 𝑅𝑀6748.800.12RM2K10

Example 7A company is considering the possibility of acquiring new computer equipment for RM600 000cash. The scrap value is estimated to be RM50 000 at the end of the 6-year life of the equipment.The company could lease the equipment for RM150 000 per year, payable at the end of eachyear. If the company can earn 16% p.a. on its capital, advise the company whether to buy or toleaseNet PV PV of Cash inflows – PV of cash outflowIf Lease:If Buy:0Debt1RM6000026RM50000PV of RM50K at time 0 𝑅𝑀50000(1 i) n 𝑅𝑀50000(1 0.16)-6 RM20522.11èNet PV PV of RM50000 at time 0 – PV of cash outflow RM20522.11 – RM60000 -RM579 477.89Should the company buy or lease? WHY?0RM150KRM150K121 1 𝑖𝑃𝑉 𝑅𝑎 𝑖 𝑅𝑖𝑛 150000𝑎 0.166RM150K6 𝑛1 1 0.16 1500000.16 𝑅𝑀552 710.39è Net PV PV of cash inflow – PV of cash outflow 0 – RM552 710.39 -RM552 710.39 6

Exercise1. En. Jo signed a contract that calls for a deposit of RM1500 and for the payment ofRM2000 a year for 10 years. Money is worth 12% p.a. effective.If En Jo missed the first 2 payments, what must he pay at the time the 3rd payment isdue to discharge his debt completely?RM15 876.312. An annuity pays RM500 p.a. for 5 years and then RM300 p.a. for 4 years. Calculatethe value of this annuity one year before the first payment using an annual interestrate of 11%.RM2400.303. A woman has an insurance policy whose value at age 65 will provide payments ofRM1500 a year for 15 years, first payment at age 66. If the insurance company pays9% pa on its funds, what is the policy’s value at age 65?RM12 091.03

Future value of an ordinary annuity Present value of an ordinary annuity Annuities due Perpetuities Deferred annuities Summary of annuities. . rate was (a) 9% convertible monthly (b) 12% convertible monthly R 150; t 3;m 12è n 36; j 12 0.09; è i 0.075 500 1 1 0.075 36 36 0.075 150 H 0.075 014717.02 .