Dan has borrowed $800 today and will repay it in two equal payments: the first in 2 months and the second in 10 months. We need to calculate the size of the equal payments, given an interest rate of 5% per annum, and using today as the focal date.

Key Information:

• Loan Amount $P = 800$ (present value).
• Interest rate $i = 5\%$ per annum, or 0.05 annually. Monthly interest $i_m = \frac{0.05}{12} = 0.004167$.
• First payment is in 2 months, second payment is in 10 months.
• We are assuming the two payments are equal.

Step-by-step approach:

1. Interest adjustment for each payment: Since the present value $P = 800$, we will discount each payment to the focal date (today), using the monthly interest rate.

2. Discount factors:

   • For the first payment due in 2 months, the discount factor is $(1 + i_m)^2$.
   • For the second payment due in 10 months, the discount factor is $(1 + i_m)^{10}$.

3. Equation setup: The present value $P = 800$ must equal the sum of the discounted payments: $P = \frac{X}{(1 + i_m)^2} + \frac{X}{(1 + i_m)^{10}}$ where $X$ is the amount of each payment. Solve for $X$.

Calculation:

Let’s first compute the monthly interest rate and the discount factors:

• Monthly interest $i_m = 0.004167$.
• Discount factor for 2 months: $(1 + 0.004167)^2 = 1.00835$.
• Discount factor for 10 months: $(1 + 0.004167)^{10} = 1.04272$.

Now the equation becomes: $800 = \frac{X}{1.00835} + \frac{X}{1.04272}$

Simplify and solve for $X$: $800 = X \times \left( \frac{1}{1.00835} + \frac{1}{1.04272} \right)$ $800 = X \times (0.99172 + 0.95897)$ $800 = X \times 1.95069$ $X = \frac{800}{1.95069} \approx 410.10$

Final Answer:

The size of the equal payments is approximately $410.10.

Would you like more details or have any questions?

Here are 5 relative questions to further explore the topic:

1. What would happen to the payment size if the interest rate were 6% instead of 5%?
2. How would the payments change if the loan was repaid in 3 and 12 months instead of 2 and 10 months?
3. What if the payments were not equal? How would you approach the solution then?
4. How can you apply the time value of money concept to other financial problems?
5. What are some alternative methods for loan repayment?

Tip: When dealing with loan problems, always ensure you understand the timeline of payments and the method for discounting future payments to the present time.