USING INTEGRATION TO CALCULATE FUTURE VALUE OF A CONTINUOUS INVESTMENT STREAM

BASIC FORMULA (Continuous compounding) Future value after t years of P = Pert Suppose total time of investment is T. If I(t) is deposited at time t, time that amount is in account = T-t Future value of I(t) = I(t)e(r(T-t))

Example T = 10 years I(t) = Amount deposited at time t = 3000+5t FV(t) = Future value of I(t) in 10-t years t I(t) FV(t) 1 3005 4712.78 2 3010 4490.39 3 3015 4278.49 4 3020 4076.57 5 3025 3884.18 6 3030 3700.85 7 3035 3526.17 8 3040 3359.72 9 3045 3201.12 10 3050 3050.00 Total = 38280.27

Chart of data Remind you of Riemann sum?

Graph of I(t)er(T-t) Graph of (3000+5t)e(0.05(10-t)) Total future value is the area under graph, given by integral from 0 to 10

Integral by integration by parts Also in separate file on update page

6.2 Numerical Integration Let Δx = (b-a)/n. Here a = 0, b = 10, n = 10, so Δx = 1. Let x1 = 0, x2 = 1,…, x10 = 9. Area of each rectangle is Δx times f(xi) The left end-point approximation gives: sum = Δx (f(x1)+f(x2)+…+f(x10)) = 1(f(0)+f(1)+…f(9)) = 40176.43

Right end-point sum As before Δx = (b-a)/n = (10-0)/10=1. Let x2 = 1, x3 = 2,…, x10 = 9, x11=10. Area of each rectangle is Δx times f(xi) The right end-point approximation gives: sum = Δx (f(x2)+f(x3)+…+f(x11)) = 1(f(1)+f(2)+…f(10)) = 38280.27

Average of right and left sums Right endpoint sum = 38280.27 Left end-point sum = 40176.43 Average = (38280.27 + 40176.43)/2 = 39228.35 Actual value = 39223.64

Trapezoidal Method Basically, use trapezoids instead of rectangles Sum = (Δx/2)(f(x1)+2f(x2)+…2f(xn-1)+f(xn)) Here the sum becomes (1/2)(f(x1)+2f(x2)+…2f(x9)+f(x11)) = 39228.35 Why is this same as the average of right and left sums?

Diagram for trapezoidal rule