## Time Value of Money

In this post, I will help your understand the time value of money using a simple real world example.

Problem:

You have decided to buy a car, the price of the car is \$18,000. The car dealer presents you with two choices:

• (A) Purchase the car for cash and receive \$2000 instant cash rebate – your out of pocket expense is \$16,000 today.
• (B) Purchase the car for \$18,000 with zero percent interest 36-month loan with monthly payments.

Market interest rate is 4%.  Which option above is cheaper? How much do you save?

For the impatient reader, the correct answer is Option A. You save \$935.38 by going with Option A.

Explanation:

The most common mistake people make is comparing \$16,000 to \$18,000.  If you choose Option A, you are paying out \$16,000 now. If you choose Option B, you are paying monthly installments of \$500 for 36-months totaling \$18,0000.

In finance, the key thing to understand is you need to compare cost always at the same point in time.

In order to answer this question you need to understand the time value of money. This is where Present Value (PV) and Future Value (FV) come in.

## Present Value

If you have \$100 now, then it’s present value is \$100.

## Future Value

If the market interest rate is 5%, the future value of \$100:

• After 1 year  :
• FV = PV (Principal) + PV * r (Interest)
• FV = PV + PV * r = PV * (1 + r)
• FV = \$100.00 * (1 + 0.05) = \$105.00
• After 2 years:
• FV = PV * (1+r) * (1+r)  = \$100.00 * (1 + 0.05) * (1 + 0.05) = \$110.25
• After 3 years:
• FV = PV * (1+r) * (1+r) * (1+ r)  = \$100.00 * (1 + 0.05) * (1 + 0.05) * (1 + 0.05) = \$115.76

## Time Value of Money

Every financial problem is best understood with a timeline. Below is the timeline going from PV to FV:

If you know the FV and the interest rate, then it is very easy to calculate PV backwards. Above is the timeline going from FV to PV.  Now that you understand how to calculate FV from PV and vice versa, let us look the problem.

For Option 1, PV = \$16,000.  This is very straight forward. You need to pay \$16,000 today.

In order to compare Option 1 and Option 2, we need to calculate the PV for Option 2. In finance, “the key thing to understand is you need to compare cost always at the same point in time.”

For Option 2, you will be paying \$500 on a monthly basis for 36 months to pay the full amount of \$18,000.

• Market Interest Rate (r) = 4% – this is APR
• Monthly Market Interest Rate (r) = 0.04/12 = 0.0033
• Payment Term = 36 months (period)

Given this information, we can calculate the PV of each future payment manually as below:

• PV of first month’s payment = FV / (1 + r) = 500 / (1 + 0.0033) = 498.36
• PV of second month’s payment = FV / (1 + r) * (1 + r) = 500 / ((1 + 0.0033) * (1 + 0.0033)) = 496.72
• ….
• Finally add each of these PV’s

I showed the breakdown above for ease of understanding.  Fortunately, Excel makes it very easy to calculate PV and FV. When you type “=PV” in an Excel cell, you get something like what is shown in screenshot below:

• rate = 0.04/12
• nper = 36 (36 payments)
• pmt = \$500 (monthly payment)

If you type “=PV(0.04/12, 36, -500)” into an Excel cell, you will get \$16,935.38.

Finally,

• Option A: In today’s dollars (PV), you will pay \$16,000.00.
• Option B: In today’s dollars (PV), you will pay \$16,935.38.

Hence it is cheaper to go with Option 1. You end up saving \$935.38 in today’s dollars.

The beauty of finance if you are calculating PV from FV, you can verify your work by calculating FV back from the PV you arrived at.  This way you audit your own work.  If you understand the this basic principle, you will be able apply what you just learned and apply it to your daily life.

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