To solve this problem, we need to determine the equivalent value of the payments at a common focal date, which is 30 days from now. We will use the given interest rate of 9.25% per annum to discount or accumulate the payments to this focal date.

1. **Convert the annual interest rate to a daily interest rate:**
$$
\text{Daily interest rate} = \left(1 + \frac{0.0925}{365}\right) - 1 \approx \frac{0.0925}{365} = 0.00025342
$$

2. **Calculate the value of the $3,000 payment 40 days ago at the focal date (30 days from now):**
$$
\text{Number of days from 40 days ago to 30 days from now} = 40 + 30 = 70 \text{ days}
$$
$$
\text{Future value of } \$3,000 = 3000 \times (1 + 0.00025342)^{70}
$$
$$
\text{Future value of } \$3,000 \approx 3000 \times (1.0178) \approx 3053.40
$$

3. **Calculate the value of the $2,900 payment in 50 days at the focal date (30 days from now):**
$$
\text{Number of days from 50 days to 30 days from now} = 50 - 30 = 20 \text{ days}
$$
$$
\text{Present value of } \$2,900 = 2900 \times (1 + 0.00025342)^{-20}
$$
$$
\text{Present value of } \$2,900 \approx 2900 \times (0.9949) \approx 2888.21
$$

4. **Calculate the value of the $3,400 payment today at the focal date (30 days from now):**
$$
\text{Number of days from today to 30 days from now} = 30 \text{ days}
$$
$$
\text{Future value of } \$3,400 = 3400 \times (1 + 0.00025342)^{30}
$$
$$
\text{Future value of } \$3,400 \approx 3400 \times (1.0076) \approx 3425.84
$$

5. **Determine the second payment (P) in 30 days:**
$$
\text{Total value of original payments at focal date} = 3053.40 + 2888.21 = 5941.61
$$
$$
\text{Total value of new payments at focal date} = 3425.84 + P
$$
$$
3425.84 + P = 5941.61
$$
$$
P = 5941.61 - 3425.84 = 2515.77
$$

Therefore, the second payment must be $2,515.77.

Question

To solve this problem, we need to determine the equivalent value of the payments at a common focal date, which is 30 days from now. We will use the given interest rate of 9.25% per annum to discount or accumulate the payments to this focal date.

1. **Convert the annual interest rate to a daily interest rate:**
   $$
   \text{Daily interest rate} = \left(1 + \frac{0.0925}{365}\right) - 1 \approx \frac{0.0925}{365} = 0.00025342
   $$

2. **Calculate the value of the $3,000 payment 40 days ago at the focal date (30 days from now):**
   $$
   \text{Number of days from 40 days ago to 30 days from now} = 40 + 30 = 70 \text{ days}
   $$
   $$
   \text{Future value of } \$3,000 = 3000 \times (1 + 0.00025342)^{70}
   $$
   $$
   \text{Future value of } \$3,000 \approx 3000 \times (1.0178) \approx 3053.40
   $$

3. **Calculate the value of the $2,900 payment in 50 days at the focal date (30 days from now):**
   $$
   \text{Number of days from 50 days to 30 days from now} = 50 - 30 = 20 \text{ days}
   $$
   $$
   \text{Present value of } \$2,900 = 2900 \times (1 + 0.00025342)^{-20}
   $$
   $$
   \text{Present value of } \$2,900 \approx 2900 \times (0.9949) \approx 2888.21
   $$

4. **Calculate the value of the $3,400 payment today at the focal date (30 days from now):**
   $$
   \text{Number of days from today to 30 days from now} = 30 \text{ days}
   $$
   $$
   \text{Future value of } \$3,400 = 3400 \times (1 + 0.00025342)^{30}
   $$
   $$
   \text{Future value of } \$3,400 \approx 3400 \times (1.0076) \approx 3425.84
   $$

5. **Determine the second payment (P) in 30 days:**
   $$
   \text{Total value of original payments at focal date} = 3053.40 + 2888.21 = 5941.61
   $$
   $$
   \text{Total value of new payments at focal date} = 3425.84 + P
   $$
   $$
   3425.84 + P = 5941.61
   $$
   $$
   P = 5941.61 - 3425.84 = 2515.77
   $$

Therefore, the second payment must be $2,515.77.

Knowee AI · Accepted Answer

To solve this problem, we need to determine the equivalent value of the payments at a common focal date, which is 30 days from now. We will use the given interest rate of 9.25% per annum to discount or accumulate the payments to this focal date.

1. **Convert the annual interest rate to a daily interest rate:**
   $$
   \text{Daily interest rate} = \left(1 + \frac{0.0925}{365}\right) - 1 \approx \frac{0.0925}{365} = 0.00025342
   $$

2. **Calculate the value of the $3,000 payment 40 days ago at the focal date (30 days from now):**
   $$
   \text{Number of days from 40 days ago to 30 days from now} = 40 + 30 = 70 \text{ days}
   $$
   $$
   \text{Future value of } \$3,000 = 3000 \times (1 + 0.00025342)^{70}
   $$
   $$
   \text{Future value of } \$3,000 \approx 3000 \times (1.0178) \approx 3053.40
   $$

3. **Calculate the value of the $2,900 payment in 50 days at the focal date (30 days from now):**
   $$
   \text{Number of days from 50 days to 30 days from now} = 50 - 30 = 20 \text{ days}
   $$
   $$
   \text{Present value of } \$2,900 = 2900 \times (1 + 0.00025342)^{-20}
   $$
   $$
   \text{Present value of } \$2,900 \approx 2900 \times (0.9949) \approx 2888.21
   $$

4. **Calculate the value of the $3,400 payment today at the focal date (30 days from now):**
   $$
   \text{Number of days from today to 30 days from now} = 30 \text{ days}
   $$
   $$
   \text{Future value of } \$3,400 = 3400 \times (1 + 0.00025342)^{30}
   $$
   $$
   \text{Future value of } \$3,400 \approx 3400 \times (1.0076) \approx 3425.84
   $$

5. **Determine the second payment (P) in 30 days:**
   $$
   \text{Total value of original payments at focal date} = 3053.40 + 2888.21 = 5941.61
   $$
   $$
   \text{Total value of new payments at focal date} = 3425.84 + P
   $$
   $$
   3425.84 + P = 5941.61
   $$
   $$
   P = 5941.61 - 3425.84 = 2515.77
   $$

Therefore, the second payment must be $2,515.77.

Question

Solution

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