In my post on present value, I promised to explain how to turn a series of payments into a present value. This is the promised follow-up post.

# Tag Archives: discounting

# Pay Today or Pay More Tomorrow

I am 27 years old. I recently bought a life insurance policy with a face value of $100,000. This policy will last my whole life - in other words, no matter when I die, the payout happens. It cost me roughly $10,000 in today's money. If this is surprising to you, or you think the insurance company got a bad deal, then read this.

Everyone makes choices about whether they'd rather have something now, or something else later. Almost no one understands the economic concepts that describes these tradeoffs. they're called "present value" and "discount rate."

I will start by describing some simple examples that use these concepts, without using the jargon. Then I will explain what these all have in common. I'm not going to explain how to use these in real-life situations, but if you're interested, please let me know in the comments and I'll write a follow-up post.

**Return on Investment**

I'll start with a simplified example, with made-up numbers. Abby has a bank account with a bunch of money in it earning 2% guaranteed interest per year. She also owns a bond that would pay out $1,000 if she cashes it out now, or $1,030 if she cashes it out in a year. Should she cash it out now, or a year later?

Let's say that in any case she wouldn't use the money until a year from now. Then if she cashes out the bond now, she can immediately deposit the money, and in a year, she'll have $1,020. But that's less than the $1,030 she'd get if she held onto the bond for a year.

On the other hand, suppose she wants to use the money right now. Then if she cashes out the bond now, she has an immediate $1,000 to spend. On the other hand, let's say she holds onto the bond, and withdraws $1,000 from her bank account. Then in a year, she has $1,020 less in her account than she would have, but an extra $1,030 from the bond, putting her $10 ahead of the first strategy. So in this case too she should hold onto the bond for another year.

It should be easy to see that if the bond only returned $1,010 in a year, Abby comes out ahead by cashing out now, again regardless of whether she wants to use the money now or later. Because the bond gives her a lower return on investment (1%) than her savings account does (2%).

Then suppose the bond pays out $1,030 in a year, but her bank account offers 4% interest this year. Then Abby also comes out ahead by cashing out now, because the bond's return (3%) is less than the interest she gets on her bank account.

**Cost of Funds**

Brian doesn't have any savings - he a student. But he has a good credit rating and is able to borrow at 5% interest per year, and is allowed to pay off his loans at any time.

He is deciding whether to rent a textbook for $100, or buy it for $150 and sell it back used to his school's bookstore in a year for $55.

If Brian rents his textbook, then after a year, he will owe $105, including interest, and have no textbook. On the other hand, if he buys his textbook, then after a year, he will owe $157.50. He can then sell his textbook back to the bookstore for $55, use that to pay down his debt, and owe only $102.50. So buying the textbook is a better deal.

Suppose instead Brian can only borrow at 10% interest. Then if Brian rents his textbook, after a year, he will owe $110. On the other hand, if he buys his textbook, then after a year, he will owe $165-$55=$110. So he should be indifferent between the two alternatives.

If Brian has to pay 15% interest, then if he rents his textbook, after a year he owes $115, but if he buys, then after a year he owes $125, so he comes out ahead by renting.

On the other hand, suppose at the 5% rate of interest, Brian can only collect $50 for his textbook after a year. Then instead of owing $102.50 at the end of a year, he'd owe $107.50, more than the $105 he'd owe if he rented, so in that case renting again becomes more advantageous.

__Present Value__

In each of the above examples, a future amount of money was related to a present amount of money, by either how much money you'd have if you used the current money in the best way available (either investing or paying off debt), or how much money you would have to have now, to produce the future money. The first is called the "future value" of money, and the second is called the "present value" of money.

When Abby is choosing between $1,000 now and $1,030 in a year, the "future value" of $1,000 is how much money she'd have at the end of a year if she put the money in her bank account yielding 2%. To get this, you multiply by (100%+2%=1.00+0.02=1.02): $1,000 * 1.02 = $1,020. This is less than the one-year future value of $1,030 in a year, which is of course $1,030.

The "present value" of the year-later $1,030 is the amount Abby would need today to produce that amount in a year. To calculate the value a year in the past, you simply do the opposite of what you did when calculating the value a year in the future: you simply divide by (100%+2%=102%=1.02), to get $1,030/1.02=$1009.80, more than the present value of $1,000 today (which is of course $1,000).

Another way to show this is algebraically:

PV*1.02=FV

PV=FV/1.02

Now let's look at the first example involving Brian. Brian is comparing making a single payment today, with making a payment today plus receiving a payment in a year.

Since Brian has to pay 5% interest on money he borrows, the future value of the textbook rental expense is how much Brian will owe in a year if he borrows the money, or $100*1.05=$105. The future value of the purchase price of the textbook is $150*1.05=$157.50, and the future value of the $55 Brian will receive for his textbook in a year is just $55. So the net future value of Brian's textbook expenses if he buys is $157.50-$55.00=$102.50, less than the $105 future value of the rental fee.

The present value of the renting option, $100 today, is of course $100. The present value of the textbook's price today is also the same as the price, $150. The present value of getting $55 in a year is the amount of debt he'd have to pay off now, to owe $55 less in a year: $55/1.05=$52.38. So the present value of the cost of buying and selling back later is $150-$52.38=$97.62, less than the $100 textbook rental fee. So the buying option costs less, in present value terms, as well.

The key here is that by converting each value, whether positive or negative, into the equivalent value for a single time period - whether the present or the future - we end up with numbers that can be directly added and subtracted to find out which amount is higher on net.

Discount Rate

You may have noticed that in Abby's case we were using the rate at which she could expect return on her savings to equate future and present amounts, but in Brian's case we looked at the interest rate he'd have to pay to borrow money. These might seem like quite different things, but in finance, there's little difference between spending saved money and borrowing money; in both cases money in the future is worth more than money in the present, and we assume a fixed conversion factor. Instead of calling it a cost of borrowing sometimes and an expected return on investment at other times, economics abstracts this into the more general term "discount rate", which is basically the extra share you can demand if you get your money in a year instead of today, or the share of your money you should expect to give up if you get your money today instead of a year from now.

This is related to the economic concept of "opportunity cost," which I will cover in a future post.

I will also cover how to deal with a series of future payments in a future post - and in the process show you that if you believe in discount rates, the future isn't as big a deal as it seems.

Which means, of course, that this is the first post in a series.