Chapter D1.4, Problem 42E

Endowment model An endowment is an investment account in which the balance ideally remains constant and withdrawals are made on the interest earned by the account. Such an account may be modeled by the initial value problem B′(t) = rB – m, for t ≥ 0, with B(0) = B₀. The constant r > 0 reflects the annual interest rate, m > 0 is the annual rate of withdrawal, B₀ is the initial balance in the account, and t is measured in years.

a.    Solve the initial value problem with r = 0.05, m = $1000/year, and B₀ = $15,000. Does the balance in the account increase or decrease?

b.    If r = 0.05 and B₀ = $50,000, what is the annual withdrawal rate m that ensures a constant balance in the account? What is the constant balance?