Question

13. (10pts) Suppose that an account is governed by a quadratic accumulation function. If $800 invested at time 0 grows to $805 at time 1 and $4000 invested at time 0 grows to $4080 at time 2. a. Find the accumulation function. b. Determine if the accumulation function is legitimate

Answer 1

The amount A you get from investing k dollars after time t is given by

A(t) = k a(t)

where a(t) is the accumulation function which is given in the question to be a quadratic equation.

Hence , let the accumulation function be a(t) = αt² + βt + γ

Note that in interest theory a(0) = 1. Thus γ = 1 (You get it when you put t=0 in the formula: a(0) = 0*α + 0*β + γ

If you invest k = $800 and accumulate $805 at t = 1 then we have the equation

805 = 800 a(4)

Similarly, if you invest k = $4000 and accumulate $4080 at t = 2 then we have the equation

4080 = 4000 a(2)

We have two equations:
4080 = 4000(4α + 2β + 1); ---eq (1) and

805 = 800(α + β + 1)----eq(2)

Solving for α and β, we get:
α = 30/8000 = 0.00375

β = 20/8000 = 0.0025

b. Putting these values in eq(1) and eq(2) we see that both the equations are satisfied, hence it is a legitimate accumulation function.