To solve this problem, we'll break it into two parts:

Part 1: Calculate the value of the investment at age 17.

Yumi's grandparents gave her $20,000, which was invested for 5 years at an interest rate of 2.5% compounded monthly.

The formula to calculate the future value of an investment compounded monthly is:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

Where:

• $A$ is the future value of the investment,
• $P$ is the principal amount (initial investment),
• $r$ is the annual interest rate (in decimal form),
• $n$ is the number of times interest is compounded per year,
• $t$ is the time in years.

Given:

• $P = 20,000$,
• $r = 2.5\% = 0.025$,
• $n = 12$ (monthly compounding),
• $t = 5$.

Substituting the values:

$A = 20000 \left(1 + \frac{0.025}{12}\right)^{12 \times 5}$

We'll calculate the future value $A$.

Part 2: Determine the equal annual withdrawals.

Once we have the future value at age 17, Yumi plans to withdraw the funds over 4 years with an interest rate of 3%, compounded annually. We will use the annuity formula to determine the equal annual installment amount.

The formula for the present value of an annuity is:

$PVA = \frac{C}{r} \left( 1 - \frac{1}{(1 + r)^t} \right)$

Where:

• $PVA$ is the present value of the annuity (the amount available at age 17),
• $C$ is the annual installment (what we're solving for),
• $r$ is the interest rate per period (3% = 0.03),
• $t$ is the number of periods (4 years).

Rearranging the formula to solve for $C$:

$C = PVA \times \frac{r}{1 - (1 + r)^{-t}}$

We'll calculate the value of each annual installment $C$.# Given values for Part 1 P = 20000 # Initial investment r = 0.025 # Annual interest rate for first 5 years (2.5%) n = 12 # Compounded monthly t = 5 # Duration of investment in years

Future value calculation (A)

A = P * (1 + r/n)**(n*t) AThe value of Yumi's investment at age 17 is approximately $22,660.02.

Part 2: Determine the Equal Annual Withdrawals

Now, we need to calculate the size of each annual installment over 4 years, with an interest rate of 3% compounded annually. We will use the annuity formula for this.

Given:

• Present value $PVA = 22,660.02$,
• Interest rate $r = 0.03$ (3% annually),
• Number of periods $t = 4$.

We'll now calculate the annual withdrawal amount $C$.The size of each annual installment Yumi can withdraw over the next 4 years is approximately $6,096.16.

Would you like further details or have any questions?

Here are 5 related questions:

1. How would the installment change if the interest rate was higher?
2. What happens if Yumi wanted to withdraw the money over 5 years instead of 4?
3. How does compounding frequency affect the growth of the investment?
4. What if Yumi made withdrawals semi-annually instead of annually?
5. How would inflation affect the real value of the installments?

Tip: When dealing with interest calculations, always be clear about the compounding frequency as it can significantly impact your final results.