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 \(B^{\prime}(t)=a B-m\) for \(t \geq 0\), with \(B(0)=B_{0}\). The constant a reflects the annual interest rate, m is the annual rate of withdrawal, and \(B_{0}\), is the initial balance in the account.

a. Solve the initial value problem with a = 0.05, m =$1000/yr, and \(B_{0}=\$ 15,000\). Does the balance in the account increase or decrease?

b. If a =0.05 and \(B_{0}=\$ 15,000\), what is the annual withdrawal rate m that ensures a constant balance in the account? What is the constant balance?

Problem 59E

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)= aB − m for t ≥ 0, with B(0)= B0. The constant a reflects the annual interest rate, m is the annual rate of withdrawal, and B0, is the initial balance in the account.

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

b. If a =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?

Solution:

Step 1

Solving for initial value problem,

Considering ,

Separating variables,

Integrating both sides,

Combining the constants,

Using initial condition ,

So, the final solution becomes,

Putting in the values a = 0.05, m =$1000/yr, and B0=$15,000, we get

The balance increases.