Question

An obligation of $10 000 is due one year from now with interest at 10% compounded semi-annually. The obligation is to be settled by a payment of $6000 in 6 months and a final payment in 15 months. What is the size of the second payment if interest is now 9% compounded monthly?

Answer 1

To find the size of the second payment, you can use the formula for compound interest. In this case, the size of the second payment is approximately $4,482.4.

Step-by-step explanation:

To find the size of the second payment, we can use the formula for compound interest:

PV = FV / (1 + r/n)^(nt)

Where:

• PV is the present value or initial amount
• FV is the future value or final amount
• r is the interest rate
• n is the number of times the interest is compounded per year
• t is the number of years

In this case, the present value (PV) is $4,000 ($10,000 - $6,000) because the first payment has already been made. The future value (FV) is $10,000, the interest rate (r) is 9% or 0.09, the number of times the interest is compounded per year (n) is 12 (monthly compounding), and the number of years (t) is 1.25 (15 months ÷ 12).

Plugging in these values, we can solve for the size of the second payment:

4000 = 10000 / (1 + 0.09/12)^(12 × 1.25)

4000 = 10000 / (1 + 0.0075)^(15)

4000 = 10000 / (1.0075)^(15)

4000 = 10000 / 1.1206

4000 × 1.1206 = 10000

4482.4 = 10000

Therefore, the size of the second payment is approximately $4,482.4.

Answer 2

The size of the second payment is $3784.2.

Step-by-step explanation:

To find the size of the second payment, we need to calculate the value of the obligation at the time of the final payment. The first payment is $6000 at 6 months. Using the compound interest formula, we can find the future value of this payment:

FV = P(1 + i/n)^(nt)

where FV is the future value, P is the principal amount, i is the interest rate, n is the number of times interest is compounded per year, and t is the time in years.

Plugging in the values, we have:

FV = $6000(1 + 0.10/2)^(2*0.5) = $6000(1.05)^1 = $6000(1.05) = $6300

The remaining obligation is $10,000 - $6300 = $3700. With a new interest rate of 9% compounded monthly, we can use the formula again to find the size of the second payment:

FV = $3700(1 + 0.09/12)^(12*0.25) = $3700(1.0075)^3 = $3700(1.0226) = $3784.