Chapter 3Mathematics of FinanceMathematics of FinanceSection 3Future Value of an Annuity; Sinking Fundsy; gDefinition of Annuity  An annuity is any sequence of equal periodic payments. Anordinary annuityis one in which payments are made atAn ordinary annuityis one in which payments are made at the end of each time interval. If for example, $100 is deposited into an account every quarter (3 months) at an interest rate of 8% per year, the following sequence illustrates the growth of money in the account after one year: ()0.08100 100 1 100 1.02 (1.02) 100(1.02)(1.02)(1.02)4⎛⎞+++ +⎜⎟⎝⎠2234100 100(1.02) 100(1.02) 100(1.02)⎝⎠++ +3rdqtr 2ndquarter 1stquarterThis amount was just put in at the end of the 4th quarter, so it has earned no interest. General Formula forFuture Value of an Annuity1+i()n−1whereFV = future value (amount)PMT = periodic payment FV = PMT1+i()1i3i = rate per periodn = number of payments (periods)Note: Payments are made at the end of each period.ExampleSuppose a $1000 payment is made at the end of eachSuppose a $1000 payment is made at the end of each quarter and the money in the account is compounded quarterly at 6.5% interest for 15 years. How much is in the account after the 15 year period?4ExampleSuppose a $1000 payment is made at the end of eachSuppose a $1000 payment is made at the end of each quarter and the money in the account is compounded quarterly at 6.5% interest for 15 years. How much is in the account after the 15 year period?  Solution:(1 ) 1niFV PMTi⎛⎞+−=⎜⎟⎝⎠⎛⎞54(15)0.0651141000 100,336.680.0654FV⎛⎞⎛⎞+−⎜⎟⎜⎟⎝⎠⎜⎟==⎜⎟⎜⎟⎝⎠Amount of Interest Earned How much interest was earned over the 15 year period?How much interest was earned over the 15 year period? 6Amount of Interest EarnedSolution How much interest was earned over the 15 year period?How much interest was earned over the 15 year period?  Solution:Each periodic payment was $1000. Over 15 years, 15(4)=60 payments were made for a total of $60,000. Total amount in account after 15 years is $100,336.68. Therefore, amount of accrued interest is $100,336.68 -$60 000 = $40 336 687$60,000 = $40,336.68. Graphical Display 8Balance in the Account at the End of Each Period9Sinking FundDefinition:Any account that is established forDefinition:Any account that is established for accumulating funds to meet future obligations or debts is called a sinking fund.  The sinking fund payment is defined to be the amount that must be deposited into an account periodically to have a given future amount. 10Sinking Fund Payment FormulaTo derive the sinking fund payment formula we useTo derive the sinking fund payment formula, we use algebraic techniques to rewrite the formula for the future value of an annuity and solve for the variable PMT: (1 ) 1niFV PMTi⎛⎞+−=⎜⎟⎝⎠11(1 ) 1niFV PMTi⎝⎠⎛⎞=⎜⎟+−⎝⎠Sinking FundSample ProblemHow much must Harry save each month in order to buy a newHow much must Harry save each month in order to buy a new car for $12,000 in three years if the interest rate is 6% compounded monthly? 12Sinking FundSample Problem SolutionHow much must Harry save each month in order to buy a newHow much must Harry save each month in order to buy a new car for $12,000 in three years if the interest rate is 6% compounded monthly? Solution:(1 ) 1006niFV PMTi⎛⎞=⎜⎟+−⎝⎠⎛⎞13360.061212000 305.060.061112pmt⎛⎞⎜⎟⎜⎟==⎜⎟⎛⎞+−⎜⎟⎜⎟⎝⎠⎝⎠Approximating Interest RatesExampleMr Ray has deposited $150 per month into an ordinaryMr. Ray has deposited $150 per month into an ordinary annuity. After 14 years, the annuity is worth $85,000. What annual rate compounded monthly has this annuity earned during the 14 year period?14Approximating Interest RatesSolutionMr Ray has deposited $150 per month into an ordinaryMr. Ray has deposited $150 per month into an ordinary annuity. After 14 years, the annuity is worth $85,000. What annual rate compounded monthly has this annuity earned during the 14 year period?Solution: Use the FV formula: Here FV = $85,000, PMT = $150 and n, the number of payments is 14(12) = 168. Substitute these values into the formula. Solution is15Substitute these values into the formula. Solution is approximated graphically. Solution(continued) Graph each side of the (1)1niFV PMT⎛⎞+−⎜⎟plast equation separately on a graphing calculator and find the point of intersection. ()FV PMTi=⎜⎟⎝⎠14(12)168(1 ) 185,000 15085,000 (1 ) 1iii⎛⎞+−=⎜⎟⎝⎠⎛⎞+−=⎜⎟16150 i=⎜⎟⎝⎠168(1 ) 1 85,000566.67150xyx⎛⎞+−===⎜⎟⎝⎠Solution(continued)Graph of y = py566.67Graph ofy =168(1 ) 1xx+−17The monthly interest rate is about 0.01253 or 1.253%. The annual interest rate is about