Study details and reference information

Finance

In this situation, the cost of the car is $25,000 and I have $10,000, meaning I am $15,000 short. The interest rate is 3%. The $5,000 that I have in cash will in three years be worth:

This will not be enough to purchase the car. However, collecting $5,000 three years instead of $5,000 today is absurd. The value of $5,000 in three years is going to be:

Thus, I would be giving away the difference either between this amount and $5,000 today, or the difference between $5,000 and $5,463.64 in the future.

As noted the value of $5,000 today, given three years at 3% interest, is $5,463.64. Thus, I would take $5,000 today and invest it, over taking $5,250 in three years' time.

Given that $5,000 invested today at 3% will be worth $5,463.64 in three years' time -- assuming annual compounding -- I would take $5,500 in three years' time over $5,000 today. The only reason I wouldn't is if I thought the interest rate was going to go up in that time. However, given a stable 3% rate, I would take $5,500 in three years' time.

1c. The time value of money reflects the fact that there is inflation and interest. The inflation rate and the interest rates are not the same, but both contribute to the change in the valuation of money over time. For example, over the next three years there will be inflation, and this inflation will reduce the buying power of that money. We saw in this scenario that the car did not increase in price. If the rate of inflation was at 1%, for example, the car's price would have changed to $25,757. Thus, the person seeking to save $25,000 would still not have enough to buy the car.

The same holds true for interest. Because money can be invested, the time value of money in hand is the interest foregone, should you keep the money in cash. For example, if this $5,000 is kept in cash, then in three years it will still be worth $5,000. If it is invested at 3% compounding annually, then it will be worth $5,463.64. The time value of money is therefore $463.64 in income that would be lost by not investing that money.

For somebody seeking to minimize their losses, they must invest money so that the interest on that money covers the rate of inflation. This is why, even if the rate of inflation is low, the money needs to be invested rather than staying in cash. If there is any inflation at all, the cash will gradually lose value over time. This loss is what is referred to as the time value of money.

Overall, what I would do is to take the 5% and invest the money. There are other ways that the time value of money can play out in this scenario as well. For example, the interest paid when car companies offer financing is also an example of the time value of money at work. However, sometimes dealers will offer financing that is below the fair value interest rate. They do this in order to entice buyers -- the interest rate on the financing comes from the automaker's finance arm and serves as a loss leader. In a case where the financing rate on a car is lower than what the market is offering, this should be considered. For example, buying a car with a 0.9% or 1.9% rate is clearly better than taking that money to the bank for three years at 3%, not the least because I would receive the car today. The cost of that money is lower than the time value of that money. So this is a good deal. It might be worth exploring this with the dealer. If there are no such deals, then I would simply take the $5,000 in cash, invest it at the 3% (assuming I am being offered to take less up front) and then in three years I will have enough money to purchase the car. I would, of course, have to cross my fingers than the price of the car does not increase with inflation.

2. The effective annual rate is the rate of interest when compounding occurs more than once per year. The annualized percentage rate is the rate expressed as a single percentage number that represents the actual yearly cost of funds over the term of the loan (Investopedia, 2012). This rate does not take compounding into account.

A good way of understanding the difference between the two is to use a credit card as an example. If the monthly rate on a credit card is 2%, then the EAR and the APR are going to be different. The APR is going to be 12 * 2 = 24%.

However, that assumes that each month the balance starts at zero. If that does not happen, then the balance will compound. To figure out that annual rate, the EAR would be used. In that case, it would be (1.02) ^ 12 = 1.268242 = 26.2842%.

As this example shows, the effect of compounding makes a big difference on the final rate that the customer actually pays. Thus, 2% per month might be advertised as an APR of 24%, but if the interest is applied to a balance that is never paid off, then the EAR is actually going to be 26.2842%.