1. USING INTEGRATION TO CALCULATE FUTURE VALUE OF A CONTINUOUS INVESTMENT STREAM

2. 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))

3. 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 2
3010 3
3015 4
3020 5
3025 6
3030 7
3035 8
3040 9
3045 10
3050
Total =

4. Chart of data Remind you of Riemann sum?

5. 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

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

7. 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)) =

8. 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)) =

9. Average of right and left sums
Right endpoint sum =
Left end-point sum =
Average = ( )/2 =
Actual value =

10. 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)) = Why is this same as the average of right and left sums?

11. Diagram for trapezoidal rule