To solve this, we need to calculate the present value (PV) of each payment using the formula for the present value of a single payment under compound interest:

$PV = \frac{FV}{(1 + i)^n}$

Where:

• $FV$ = Future value (payment amount)
• $i$ = Effective interest rate per period
• $n$ = Number of periods until the payment is made

Given Data:

• Payments: $1000$ in 1 year, $2000$ in 2 years, $3000$ in 3 years.
• Annual nominal interest rate = $7.5\% = 0.075$, compounded semi-annually.
• Effective semi-annual interest rate: $i = \frac{7.5\%}{2} = 3.75\% = 0.0375$.

Now, calculate the number of compounding periods:

• For $1$ year: $n = 2$ periods (since compounding is semi-annual).
• For $2$ years: $n = 4$.
• For $3$ years: $n = 6$.

Step-by-step Calculation:

1. For the $1000 payment in 1 year: $PV_1 = \frac{1000}{(1 + 0.0375)^2}$

2. For the $2000 payment in 2 years: $PV_2 = \frac{2000}{(1 + 0.0375)^4}$

3. For the $3000 payment in 3 years: $PV_3 = \frac{3000}{(1 + 0.0375)^6}$

Now, let's calculate these values. Given values future_values = [1000, 2000, 3000] # Future values (FV) interest_rate = 0.0375 # Effective semi-annual interest rate periods = [2, 4, 6] # Number of periods (n)

Calculate present value for each payment

present_values = [fv / (1 + interest_rate)**n for fv, n in zip(future_values, periods)]

Sum of present values

total_present_value = sum(present_values) present_values, total_present_value### Results:

1. Present value of the $1000 payment in 1 year: $929.02
2. Present value of the $2000 payment in 2 years: $1726.15
3. Present value of the $3000 payment in 3 years: $2405.43

The combined present value of the loan is: $\text{Total Present Value} = \mathbf{5060.59}$

Would you like a detailed explanation of any step?

Related Questions:

1. What is the significance of compounding frequency in present value calculations?
2. How does the interest rate affect the present value of future payments?
3. What would the combined present value be if interest were compounded quarterly instead of semi-annually?
4. How does increasing the repayment period (e.g., to 5 years) impact the present value?
5. Can this method be applied to continuous compounding? How?

Tip:

Always adjust the compounding periods and rates consistently with the problem's frequency to avoid errors in present value calculations.