Annuities and Income Streams MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics Summer 212
Objectives After completing this lesson we will be able to: determine the value of a given annuity, and calculate present and future values of income streams.
Annuities Definition An annuity is a sequence of equal payments made at regular time intervals over an extended period of time (for example, payroll savings plans, mortgage payments, individual retirement accounts). The amount (or future value) of an annuity is its total value (the sum of payments and earned interest). For an ordinary annuity the payments are made at the end of each time period (e.g., mortgages), while for an annuity due the payments are made at the beginning of the time period (e.g., rent).
Example At the beginning of each month you deposit $25 in a savings account earning 3.5% interest compounded continuously. What is the value of the annuity at the end of the sixth month?
Example At the beginning of each month you deposit $25 in a savings account earning 3.5% interest compounded continuously. What is the value of the annuity at the end of the sixth month? A = 25e.35(6) + 25e.35(5) + 25e.35(4) + 25e.35(3) + 25e (.35(2) + 25e.35(1) = 25 e.35(6) + e.35(5) + e.35(4) + e.35(3) + e.35(2) + e.35(1)) = 25(6.7944) = $1698.51
Riemann Sum Approach Suppose instead of a lump sum of $25 deposited at the beginning of each month, $25/3 $8.33 was deposited each day. What is the value of the annuity at the end of the sixth month?
Riemann Sum Approach Suppose instead of a lump sum of $25 deposited at the beginning of each month, $25/3 $8.33 was deposited each day. What is the value of the annuity at the end of the sixth month? A = 25 3 ( e.35(6) + e.35(5.96667) + + e.35( = 25 3 (21.412) = $1678.44
Riemann Sum Approach Suppose instead of a lump sum of $25 deposited at the beginning of each month, $25/3 $8.33 was deposited each day. What is the value of the annuity at the end of the sixth month? 6 A = 25 3 = 25 3 (21.412) = $1678.44 [ 25e.35(6 t) dt = 25.35 e.35(6 t) = $1669.13 ( e.35(6) + e.35(5.96667) + + e.35( ] 6
Future Value of an Annuity Definition The amount (or future value) of an annuity at the end of N time periods is approximated by the definite integral N Pe r(n t) dt = P r (ern 1), where P is the number of dollars invested each time period, and r is the interest rate (expressed as a decimal) per time period.
Income Stream Definition If R(t) represents a continuous income function in dollars per year (where t represents years), if T represents the term of the annuity in years, and if r represents the interest rate compounded continuously, then the amount of an annuity is A = T R(t) e r(t t) dt.
Example Suppose you deposit $(2, + 1t) per year in a savings account earning 4% interest, how much will be in the account after 15 years?
Example Suppose you deposit $(2, + 1t) per year in a savings account earning 4% interest, how much will be in the account after 15 years? A = = T 15 R(t) e r(t t) dt (2 + 1t) e.4(15 t) dt = (2 + 1t)( 25)e.4(15 t) 15 15 + [ = 875 ( 9115.9) + 25.4 e.4(15 t) = 365.94 ( 625 ( 113882)) = $54, 988.4 25e.4(15 t) dt ] 15
Present Value Suppose you will need amount A on a day t years in the future. If you can deposit an amount P today at a fixed annually interest rate r compounded n times per year, what should P be? The quantity P is known as the present value of A. ( P = A 1 + n) r nt. If the interest is compounded continuously, P = Ae rt.
Example If you estimate that a reliable used car can be purchased in 22 for $2, how much money should be put into savings today at 5% compounded continuously so that you can purchase the car?
Example If you estimate that a reliable used car can be purchased in 22 for $2, how much money should be put into savings today at 5% compounded continuously so that you can purchase the car? The present value of $2, in 212 is P = 2e.5(8) = $13, 46.4.
Present Value of an Income Stream Definition If R(t) represents a continuous income stream in dollars per year and the annual rate of interest is r (expressed as a decimal), then the present value of the income stream over T years is PV = T R(t)e rt dt.
Example Find the present value of an annuity which pays $5, per year for 2 years. Assume the rate of interest is 3% per year.
Example Find the present value of an annuity which pays $5, per year for 2 years. Assume the rate of interest is 3% per year. 2 5e.3t dt [ = 5 ] 2.3 e.3t = 914686 ( 1666666) = $751, 981
Example A company expects its income during the next 1 years to be R(t) = 25, t for t 1. Assuming the annual interest rate is 4%, what is the present value of the income?
Example A company expects its income during the next 1 years to be R(t) = 25, t for t 1. Assuming the annual interest rate is 4%, what is the present value of the income? PV = 1 25te.4t dt = ( 25)(25t)e.4t 1 = 41895 ( 625) 1 + (25)(25)e.4t dt [ 625.4 e.4t = 265 (14738 15625) = $961, 749 ] 1