Question

A good sell for $600 Cash and $400 per quarter thereafter for 6 payment. If the rate is 6%, what will be the equivalent cash price

Answer 1

[AI-generated answer — Bot GPT-4 Turbo]

To calculate the equivalent cash price, you need to determine the present value of the annuity (the $400 payments received quarterly over 6 periods) at the given interest rate, and then sum that value with the initial $600 cash payment.

In this scenario, the relevant formula to calculate the present value of an annuity is:

PV = P * [(1 - (1 + r)^-n) / r]

Here:
PV = Present Value of the annuity
P = Periodic payment amount
r = Periodic interest rate
n = Number of periods

First, allow me to convert the annual rate of 6% to a quarterly rate because the payments are quarterly.

Annual interest rate = 6% or 0.06 (as a decimal)
Quarterly interest rate = 0.06 / 4 = 0.015 (since there are 4 quarters in a year)

The number of quarterly payments (n) is 6.

Now plug these values into the formula:

P = $400
r = 0.015
n = 6

PV = $400 * [(1 - (1 + 0.015)^-6) / 0.015]

First, let's calculate the part within the brackets:

(1 + 0.015)^-6 ≈ (1.015)^-6 ≈ 1 / (1.015)^6

Now calculate (1.015)^6:

(1.015)^6 ≈ 1.093443

Now take the inverse to find the present value factor:

1 / 1.093443 ≈ 0.914844

The part in the brackets then becomes:

(1 - 0.914844) = 0.085156

Now plug this into the formula:

PV = $400 * (0.085156 / 0.015)
PV ≈ $400 * 5.6764
PV ≈ $2270.56

The present value of the annuity is approximately $2270.56.

Now, add the initial $600 cash payment to find the equivalent cash price:

Equivalent Cash Price = $600 + PV of annuity
Equivalent Cash Price = $600 + $2270.56
Equivalent Cash Price ≈ $2870.56

Therefore, the equivalent cash price, given a 6% annual interest rate with 6 quarterly payments of $400 after an initial $600 payment, is approximately $2870.56.