To find the equivalent cash price of the payment plan described, we need to calculate the present value of the cash flows.
- The initial cash payment is $600.
- The subsequent payments are $400 each, made quarterly for 6 quarters.
Step 1: Present Value of the Quarterly Payments
Given:
- Quarterly payment (PMT) = $400
- Number of payments (n) = 6
- Interest rate per quarter = 6% annually = \( \frac{6%}{4} = 1.5% = 0.015\)
The present value (PV) of an annuity can be calculated using the formula:
\[ PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]
Where:
- \( PMT \) is the payment per period
- \( r \) is the interest rate per period
- \( n \) is the total number of payments
Substituting the values:
\[ PV = 400 \times \left( \frac{1 - (1 + 0.015)^{-6}}{0.015} \right) \]
First, we calculate \((1 + r)^{-n}\):
\[ (1 + 0.015)^{-6} \approx 0.9138 \]
Now, substituting this back into the formula:
\[ PV = 400 \times \left( \frac{1 - 0.9138}{0.015} \right) \] \[ PV = 400 \times \left( \frac{0.0862}{0.015} \right) \] \[ PV = 400 \times 5.7470 \approx 2298.80 \]
Step 2: Total Present Value (Cash Price)
Now we need to add the initial cash payment to the present value of the subsequent payments:
\[ \text{Total Cash Price} = 600 + 2298.80 \approx 2898.80 \]
Conclusion:
The equivalent cash price is approximately $2898.80.