Find the price the individual must have agreed to pay for the car.

Question

An individual agrees to pay $6,000 per year for three years to payoff a car loan. His payments are always
made at the end of each year. If the interest rate in year 1 is 3%, in year 2 is 3.5% and in year 3 is 4%,
and if compounding is done twice a year, how much did the car originally cost? (That is, work out the
price he must have agreed to pay for the car.)

Answer 1

So here we are being asked what is the PV or present value of the three payments of $6000, since that is the effective price of the car.

 

PV formula:

 

PV = FV / (1 + r/k)^n                      r = interest rate, k = number of compounds per year, n = years.

 

Let's start with the present value of the first payment of $6000, it is due in 1 year, and interest rates are 3%

 

So PV1 = 6000/(1 + 0.03/2)^1 = 5,823.97

 

But, that's just for the first payment. What about the next two. This is where it gets real. 

 

For the PV of the second payment we need to average the interest rates of year 1 and year 2.

 

So PV2 = 6000/(1+0.0325/2)^2 = 5,625.34

 

Same idea for PV3, but now we average interest rates of year 1, year 2 and year 3.

 

So PV3 = 6000(1 + 0.035/2)^3 = 5,406.86

 

 

Notice that the PV1 is greater than PV3, this is because the third payment takes place further into the future, so it has more time to compound and accrue interest.

 

 

We add up all the present values and boom we got the PV or effective price of the car:

 

PV1 + PV2 + PV3 = $16,856.17

 

Now we see that the value of paying $6000 each year for 3 years, with semiannual compounding and interest rates 3%, 3.5%, and 4% has the same value as paying $16,856.17 today.