# Math Help - Present Discounted Value of Payment

1. ## Present Discounted Value of Payment

Hi!

I have a question about the present discounted value of some amount of money, and how that info is used to make decisions.
Consider getting $1000 for 3 years when the annual interest rate is 10%. You get$1000 today, a year from now, and two years from now. The present discounted value of that is $1000 + 909.09 + 826.45. The total is clearly less than getting$3000 today. Why would someone value it at less than $3000 based on the interest rate today? I get that if you put 909.09 in the bank and let it sit for a year you would have$1000. If you put 826.45 in the bank today and let it sit for two years you would end up with $1000. Why would you value$3000 at less than $3000? Why are you valuing money as if you had it and would put it into a bank account to see how much interest you would make on it? I have the same question about business profits. Let's say that you get$1 million per year from your business (ignore costs, risk, inflation etc.). The value of those gains can be discounted when you consider an interest rate of 10%. Using the formula for perpetuities you get PV = 10,000,000. So from now until forever you value the profits at 10 million bucks today. This makes ZERO sense to me: 1 million dollars forever equals 10 million? What's the most confusing is that people use this to make decisions.

What person in their right mind says "oh, I'm going to get $1 million forever from a business, and the interest rate is 10%, so I'll get 10 million". To me, if you get 1 million every year, you get more than$10 million because you put that money in the bank and get interest on it. So as soon as you get that first million you put it in the bank and get the interest on it. Could someone please explain it to me intuitively? I get that someone might not value money tomorrow as much as money today because of uncertainty and impatience that goes along with waiting. But this interest rate stuff is so confusing! Why do you use the interest rate to see how much you value something in the present?

2. ## Re: Present Discounted Value of Payment

sorry, I don't know why the text is kind of weird ..

3. ## Re: Present Discounted Value of Payment

Originally Posted by mathbeginner97
Hi!

I have a question about the present discounted value of some amount of money, and how that info is used to make decisions.
Consider getting $1000 for 3 years when the annual interest rate is 10%. You get$1000 today, a year from now, and two years from now. The present discounted value of that is $1000 + 909.09 + 826.45. The total is clearly less than getting$3000 today. Why would someone value it at less than $3000 based on the interest rate today? I get that if you put 909.09 in the bank and let it sit for a year you would have$1000. If you put 826.45 in the bank today and let it sit for two years you would end up with $1000. Why would you value$3000 at less than $3000? Why are you valuing money as if you had it and would put it into a bank account to see how much interest you would make on it? I have the same question about business profits. Let's say that you get$1 million per year from your business (ignore costs, risk, inflation etc.). The value of those gains can be discounted when you consider an interest rate of 10%. Using the formula for perpetuities you get PV = 10,000,000. So from now until forever you value the profits at 10 million bucks today. This makes ZERO sense to me: 1 million dollars forever equals 10 million? What's the most confusing is that people use this to make decisions.

What person in their right mind says "oh, I'm going to get $1 million forever from a business, and the interest rate is 10%, so I'll get 10 million". To me, if you get 1 million every year, you get more than$10 million because you put that money in the bank and get interest on it. So as soon as you get that first million you put it in the bank and get the interest on it. Could someone please explain it to me intuitively? I get that someone might not value money tomorrow as much as money today because of uncertainty and impatience that goes along with waiting. But this interest rate stuff is so confusing! Why do you use the interest rate to see how much you value something in the present?
The second question is much easier to explain than the first. You have a business that generates 1 million a year in perpetuity without any risk etc. If you can buy a guaranteed perpetuity at 10%, how much would you have to invest to get exactly the same annual income as you get from the business. 10 million, right? So things that generate the same annual income with the same degree of risk have equal market value. Let's see what would happen if people could buy a risk-free perpetuity generating 1 million a year forever for 9 million and you wanted to sell for 10 million a business that will similarly have profits of 1 million a year. No one would buy at your asking price because they could get the same income for a smaller expenditure. With me so far? Similarly, if the market price of such a perpetuity was 11 million, people would be willing to buy the business for 11 million.

Now you do NOT need to utilize interest rates for valuing receipts or disbursements in the present. (Actually if you use the formula, (1 + r)^0 = 1 so you CAN use the formula; it's just a waste of time to do so.) Let's take an example. Suppose I ask you if you have change for a fifty. You do, and we swap my fifty for your two twenties and a ten. Interest rates have nothing to do with it. Now suppose we change the example. I ask if you have change for a fifty. You do. I say "OK, great. Give me your two twenties and a ten, and I'll give you a fifty ten years from today." Most people will not do such a transaction: they do not value money paid today as equal to money paid sometime in the future even if the future payment is absolutely guaranteed. Does this make sense to you? People simply do not value future money as highly as they do money right now. A bird in the hand is worth two in the bush.

OK. Now we get to the hard question. Why is the (risk-free) market interest rate relevant? Person A might be willing to give up two twenties and a ten right now for a guarantee of a hundred ten years from now, and Person B might be willing to give up two twenties and a ten right now only for a guarantee of two hundred dollar bills ten years from now. The "subjective time preference" of A and B differ. Nothing new here: some people like chocolate ice cream and some like vanilla. The market rate of interest is like any other price: it balances out the buyers and sellers of future payments of money. The current market rate is used in present value and future value formulas because it represents what it is practical to achieve in the current state of the market. You go to the grocery store to buy a rib roast. When you see the price, you say that is too much and you don't buy. That is your prerogative, but then you do not get the rib roast. If you want the rib roast, you have to pay the going market price. It is the same with financial markets. You can participate in them at the current structure of market rates, but you will do so only if your personal subjective time preference is consistent with those market rates. Does this help?

4. ## Re: Present Discounted Value of Payment

Thanks so much!

It makes intuitive sense!