ECON 251: Financial Theory
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ECON 251 - Lecture 10 - Dynamic Present Value
Chapter 1. Dynamic Present Values [00:00:00]
Professor John Geanakoplos: Time to start. So this class and the next class and a half are going to be about Fisher’s theory of present value and the interest rate, and then we’re going to move to uncertainty.
So up until now what we’ve done is we found out first, if you know the whole economic system, how to solve for equilibrium. To figure out from the primitives of people’s tastes, their impatience, the technology, the economy, how to figure out the real rate of interest if provided there is no uncertainty in the world and people can forecast what is going to happen later.
We’ve found that once you’ve done that, the price of every asset, if people are rational and looking forward to the future, the price of every asset is going to be the present value of the future payments of the asset. So if you think of the payments as real payments, which is what Fisher always recommended, you discount by the real interest rate. If you think of them as cash payments then you discount by the nominal interest rate. So every asset corresponds to its present value of its dividends either discounted by the real rate or the nominal rate.
Now, this thinking is surprisingly powerful and leads you to unexpected conclusions. So the next two classes is about that. Mostly I’m going to talk about Social Security, but I’m going to begin today by finishing off a subject we didn’t quite get through last time.
So you see, if you realize that the price of every asset is just the present value of its dividends, and you also suppose you know what the dividends are going to be and what the future interest rates are going to be, then it follows that you know what the price of the asset is going to be, not just today, but next year, and the year after and the year after that. So your theory of asset pricing today, which was based on the assumption that you can forecast the future, necessarily implies the theory of asset pricing in the future. So you can tell something about how asset prices are going to change, and of course you can also test the theory because the theory implicitly is forecasting something about the future, and therefore you can test the theory.
So let’s just take a couple examples that are in the notes. So we said at the top, if you can read it, I hope it’s not too small–I want to move a little quickly so I could be writing this on the board, but if you can see it and you look at the top line it says that the present value of the assets is just the discount of future dividends. So maybe I’ll start one line up, actually, in these notes. So if you look in the middle here the present value today, suppose you have an asset, maybe I’m going to write on the board after all.
Present value of an asset, suppose you know the dividends are going to be paying C (1), C (2), and maybe C (T) at the end. Those are the dividends that are going to be paid, and you can ask what the present value of the asset is today, PV0 today, 0.
Well, Fisher would say if you think of these as nominal cash flows he would say what you need to do is you need to find out the price of zeros. So you recall that Fisher said that you could let pi1 = the price today of 1 dollar at time 1, and the price today piT = the price today of 1 dollar time T. It’s the price today of 1 dollar in the future. These were the crucial prices that Fisher said that you should always look for.
He said that you can always find these prices, pi1 through piT by deriving them from the yield curve. So every morning everybody looks at the yield curve, all financial analysts, they look at the prices of bonds traded in the market. The simplest form is you get a yield curve in the newspapers. You deduce from the yield curve what these present values pi1, piT are, and that allows you to price any asset like this one simply by multiplying by pi1, pi2 and at the end by piT.
So that’s step 1. But then step 2, we said, was that once you’ve got these prices you can figure out the forwards. So the forwards are going to be 1 + it, so 1 + it that’s by definition the interest from t to t + 1 that would be agreed today. It’s not the interest rate that’s going to prevail from time t, it’s the interest rate that people today at time 0 would agree to pay between time t and time t + 1. So that, of course, has to just equal pit over pit+1, because everybody today recognizes that the cost of a dollar time t is pit, and the cost of a dollar at time t + 1 is pit+1. So the tradeoff between t and t + 1 that’s effectively what an interest rate does is it trades off money at one time for another time. It has to be this ratio. So today people would agree on 1 + itF, forward, as the forward rate.
Now, you add to that the assumption that everybody is certain about the future, assume perfect forecasting. So I don’t have to actually assume people are right in their forecasting. I have to assume that they are completely sure of their forecasting. So if they’re completely sure of their forecasting then everybody must foresee that they think they’re sure that the future interest rate that will prevail at time t is also equal to this number.
So we can then rewrite this price. Hello. We can rewrite this price as 1 +i0F, forward; instead of pi2, pi2 is just going to be 1 over (1 + i0F) times (1 + i1F), and time piT is just going to be 1 over (1 + i0F) times (1 + i1F) … times (1 + it+1F). Because using the fact that (1 + it) is pit divided by pit+1 I just multiply pi1 divided by pi2 times pi2 divided by pi0 etcetera. If I take pi1 divided by pi0 times etcetera I’m going to get exactly this formula.
So pi0 is always 1, by the way. So if I want to take pi1 it’s just going to be 1 over (1 + i0F). If I want to take pi2 I can always do this, pi2, and so this is going to equal pi2, so pi2 is going to be 1 over (1 + iF0) times 1 over (1 + iF1) etcetera. And pi3, if I want to do pi3, I just multiply by pi3 over pi2 and so that’s equal to 1 over (1 + iF0) times 1 over (1 + iF1) times 1 over (1 + iF2). So instead of multiplying by the pis I may as well take the product of the forwards. It’s exactly the same thing. So the present value has this simple formula, but this is a formula that holds at every time t if you’re totally confident of your predictions of the Cs and of the future interest rates.
This is where we ended last time, basically with this formula.
Chapter 2. Marking to Market [00:08:49]
The key that you had to use in the problem set was to realize if you’re totally confident now about the future you have to be totally confident about doing this calculation in the future. So PV1 is going to be basically the same calculation.
So what’s going to happen at time 1? After you’ve gone down 1 year you’ve finished with this interest rate and you’re looking at everything 1 year later. So I may as well write the PV1 here and that’s going to equal C (2)–let’s put it a little bit lower–PV1 is going to equal the same C (2) as before, but now a year later. So it’s just going to be 1 + iF1 + C (T), and I don’t need the 1 + iF0 anymore. (1 + iF1) times … (1 + iFT-1).
So it’s exactly the same formula 1 year later, and 1 year later you no longer have to worry about the interest rate. That time already passed. So you just drop this at every time and you write the same formula, and of course you drop C (1) because that’s all finished. So the price at time 1 is just what’s left except you’ve chopped off the 1 + iF0 everywhere. So to write that the same way, so PV0 (I can write it the same way) =, PV0 = what?
It’s going to equal the interest payment you get, C (1), not the interest, the dividend you get, the cash flow you get plus PV1 divided by 1 + iF0. So this is a very important formula. So why is that?
Well, it’s a proof by formula, but another way of saying it is that PV0, when you get the asset at time 0 it gives you a cash flow at time 1, and then of course looked at from the point of view of time 1 you’ve got all the future cash flows, but that’s just what PV1 is. So basically the bond is going to give you a cash flow at time 1, and then the right to cash flows in the future, but this right to cash flows in the future at time 1 is worth PV1.
So it’s like you’ve got all this money at time 1 and so to get the value at time 0 you have to discount it back again. So this is a very famous and important formula which I want to pause a second and think about. So there’s a controversy today called marking to market. Does anybody know what that controversy is about? What does marking to market mean? Yep?
Student: I believe that it means that corporations and their accounting rules are currently required to value their assets at what they would fetch in the market if they were to sell them.
Professor John Geanakoplos: Right, but you know that they’ve been relieved of this responsibility recently.
Student: No, I didn’t know that.
Professor John Geanakoplos: Well, so they’ve been relieved of this responsibility lately. Now, why have they been relieved of it? Because the thought was that the market was so panicked in the crisis that the price that they could fetch by selling their things really had nothing to do with the value. So we’re going to come back to whether that was a good idea that Congress passed under tremendous pressure.
What is the point of marking to market? So let’s suppose that you really could anticipate the future and everybody was right about it. The market price then would really be PV0 today, and it really would be PV1 tomorrow, because if the interest rate’s do actually turn out to be what everybody expects, and everybody’s completely confident of their expectation, then as we saw the prices would have to be this today and this tomorrow, otherwise there’d be an arbitrage. You could make money for sure, or think you could make money for sure.
So if you’re marking to market what are you going to mark as the profits? Are you going to say the profit at time 1 is–so what is profit? If I had some room I’d put this somewhere. How would you define profit? I’ll get rid of this. I think we’ve got this straight. So how would you define profit at time 1? What would you write? So let’s say that this isn’t a bond. It’s an investment. You paid a bunch of money at time 0, maybe PV0 and in the future your investment is going to pay off some money.
Maybe it was a project you invested in. Maybe you bought a bunch of stuff and you’re selling it in the future and getting this money. Maybe you bought a bond which is paying these coupons. Whatever it is these are the cash flows of the project, and now of course your investors are very curious. In year 1, they want to know what profit have you made, how are you doing? So what’s a number that you’d think of giving? What’s the first number you’d think of telling them? Yep?
Student: Maybe the first cash flow payout.
Professor John Geanakoplos: So you might think of saying, C (1). That’s the cash you’ve gotten. So all these people could see you’ve gotten C (1). That’s your revenue, and I’m assuming this cash flow is net of expenses. So you could say here’s your revenue you got at time 1, C (1). Is that your profit?
If it were, people would then look and they’d say, “Well, we got profit of C (1) and we put an investment in of PV0 so is that the rate of return? Is that my rate of return?” Yeah?
Student: Can we do PV1 - PV0 and add it here?
Professor John Geanakoplos: So he’s exactly right. I’m going to repeat what he said, but I’m going to take longer to say it. This idea of writing C (1) over PV0 is actually what many people try to say is their profit, but it’s a bad number for profit.
Now, if in period 1 you stopped marking to market, and you had no idea, you didn’t have to report this number PV1, I mean, you didn’t know what it was or you didn’t have to report it, what else could you report but C (1) over PV0 as the profit? So marking to market, you see, is intimately connected to how you report profit. So if you don’t mark to market you don’t know, you don’t have to declare what the value of the assets are that are left. All you would do when you describe your profits is you’d say C (1) and people would, of course, in their heads, in fact they wouldn’t do it in their heads, you’d tell them, “Our rate of return was C (1) over PV0,” but why might that be a very misleading number?
Well, his point is it could well be that C (1) is a very high number. You’ve got a lot of cash flow this period, but he said it may turn out that PV1 is a terrible number. You may have gotten a lot of cash flow, but there’s nothing left going forward. So he doesn’t think that’s a very good measure of profit. So he’s suggesting why not report profit as C (1) + PV1 - PV0.
Now, suppose you did that? So he’s saying, look, you’ve got a certain amount of cash but the assets that you had also changed value and you ought to include that in your profit. So what if you did that? What would your rate of return be? And suppose PV0 and PV1 were calculated as this? And so you divided this by PV0 what would that equal? What number would I get if I did that? I can’t hear. What? Somebody speak up.
Professor John Geanakoplos: Yes. This would just be iF0, and you get the rate of return that you’re expecting to get. Now, how do I know that’s iF0? Because I multiplied by PV0 to this side, then I moved that PV0 to the other side. That gives me 1 + i0F PV0, then I divide by that 1 + iF0 and put it down here, and that’s this formula. So that is what you would get, iF0.
And you see that’s what you should expect to get because you haven’t done anything great. You’ve got a cash flow. You’ve got these cash flows at the beginning. You paid a fair price for them and next period everybody forecast that the values were going to go to PV1, and so it can’t be that you’re, you know–this was an investment you made. You put your money in. You bought this bond. You could sell the bond now at time 1 after having obtained the cash flows. It better be that your rate of return is the market interest rate of return otherwise there’d be some arbitrage, and in fact it is.
So if you can compute all the present values properly, the only fair thing to do–this is not a great investment. Even if C (1) is a huge amount of money it’s only huge because PV1 is a low amount of money. Because it has to be that the profit every period, the rate of profit, is exactly equal to the interest rate, because everything is always priced at its present value. So it’s a simple concept which somehow takes a long time to grasp.
It just seems to people that–here’s the money that’s come in. That’s the cash you can count in your hands. You’ve got it in your hands. That’s the number you should talk about as profit, but of course of the money coming into your hands–there’s this other hidden thing. The stuff that you own has suddenly fallen in value compared to what it was before. You’re really not putting the firm in any better position than it was before. You just earned your normal rate of return.
So you can see why, to give a fair description of how people are doing, the law was written, evolved to the point where people were forced not to just say C (1), they were forced to declare–mark to market, and add PV1 - PV0 to their profit to get a real rate of return, to get a more revealing rate of return.
And so what happened in the crisis is this PV1 became such a horribly low number that if you plug that into this formula you get PV1 is a really low number compared to PV0. You get a giant negative here. It would look like the rate of return was terrible. Everyone would panic and think the firms had fallen apart.
So especially the banks were the ones who didn’t want to do this, and so Congress didn’t want the public to be panicked so it simply said, “Okay, forget about it. We’re going to not hold you to writing PV1 anymore because we can’t count anybody figuring out what it is, and so you don’t have to tell us that. You just have to tell us this,” which happens to be a good number for the banks, but that doesn’t mean the banks are actually doing well because the assets they hold have been collapsing in value. Yeah?
Student: Does that <> say that assuming there’s no arbitrage opportunity if a firm pays out a high dividend its stock should go down?
Professor John Geanakoplos: Yes, right. That’s exactly what it says and that’s what does happen. So you always talk about the price, ex-dividend and stuff like that, precisely that. So if a firm pays a bigger than usual dividend, if it suddenly decides it’s going to pay itself a gigantic dividend then the price of the firm is going to go down, exactly, because there’s less value in the firm. You’ve paid it out here instead of keeping it into the firm. So this is a very simple idea, but it’s very easy to get confused about.
Are there any questions about it? So let’s do an example of it. The most basic example of it is the premium bond. So we talked about this before, the premium bond. So let’s say you have PV0 = 5 divided by 1.10 + 5 divided by 1.10 [correction: squared] + (maybe this was exactly the problem set, I can’t remember) 105 divided by 1.10 [correction: to the T].
So in other words the interest rate–I did that wrong–squared and T. So the forwards, in other words, 1 + iFt (small t) = 1 + .10 for all t, so all the forwards are 10 percent. The bond is paying–I can’t remember which one I did. Let’s do 20 here. I’m going to make it a premium bond, so it’s paying 20. So what’s going to be the price of the bond?
Well, in the first period it looks like you get this incredible profit. The interest rate is 10 percent. You’ve paid some present value and you’ve made 20. You’ve made a number much bigger–so the price of this thing, by the way what was the home work problem? What were the numbers? What was the interest rate? Did somebody do the homework? I did assign a homework on a premium bond, right? The interest rate was 5 percent and the coupon was 10, was that what it was? What? What was it? What was the homework problem?
Student: 8 and 6.
Professor John Geanakoplos: 8 and 6. So this was 8 percent. Might as well do this one, 6 here, yeah, 6 percent, and this one was 8. So what was the price PV0? What was PV0? You’ve done this homework problem, right? Does anyone remember what number they got? Ben, do you remember the number? No. I’m counting on my trusty class here to provide me all the numbers. It was what?
Professor John Geanakoplos: Was that what it was, 108.4? That’s all? This was the price just 108.4? So 108.4, right. That’s a high price. So it’s way above par. This is a bond that’s worth much more than 100. Why? Because the interest rate is 6 percent, but the coupon is 8 so this is called a premium bond. So if you look at the first year the rate of return on the first year is 8 over 108.4.
Now, is that more than 6 percent? It is way more than 6 percent, 8 over 108.4. It is way more than 6 percent because 7 percent of this is going to be less than 8. So it’s more than 7 percent, so this is greater than 7 percent, and so it’s certainly bigger than 6 percent.
So it sure looks like at the end of the first year like this bond was an ace bond. That’s the kind of calls you used to get. You still sometimes do get them if you’re a wealthy person and you actually answer calls like this, which you don’t of course, but you can get some cold call from a sales person saying, “We’ve got a fund and look how well it’s doing. On this much investment last year we got payments of 8 dollars. That’s more than 7 percent, and look, everybody can see the interest rate is only 6 percent and we did better than 7 percent. We’re doing great. Why don’t you invest in our fund?”
But actually, what the salesman hasn’t told you is that the value of his assets has gone down. So the present value PV1must be less than PV0, because we know from this formula, the formula right up here, that 6 percent is going to be the cash flow over PV0 which is more than 7 percent plus this difference, and this difference, therefore, is going to have to be negative. So what he didn’t tell you was that the fund lost value even though the first payment was better than 6 percent, the market rate of interest. So that’s the first example.
Now, let’s do another example. Suppose you have a–let’s see if I can write on this board. There’s a famous trade called the carry trade. Now, suppose the forwards–suppose 1 + iF0 is 2 percent, sort of like now, the 1 year yield, and 1 + iFt is 5 percent, for t greater than 1? So if you have a 1 year bond it’s going to pay–well, let’s say it’s 2 percent and this is equal to 1 +–can you see this or it’s disappearing. So let’s make the first two of them be 2 percent, and this is for t greater than or equal to 3. So the interest rate is 2 percent and then it’s going to jump to 5 percent.
All right, so if you have a 2-year bond, a 2-year Treasury, a 2-year bond might pay 2 and 102 and the present value of this equals 100, the present value of the 2-year bond. But now maybe you’ve got a longer bond that’s a 5-year bond which pays coupons 4, 4, 4, 104, so this is a 5-year bond, and maybe its price is close to 100. Actually, I haven’t worked out the number so you’re going to have to do that in the next problem set.
So is it possible, so I’ll ask it this way, is it possible for this coupon to be higher 4, 4, 4, 4, 4 and yet still have a present value of 100 than this one which is 2 and 102? Could you have a higher coupon and yet the same present value of 100 on a longer bond than on a shorter bond? How could this bond have a higher coupon and sell for the same price as this bond? Yep?
Student: You’re losing money on the assets you put on the principal value because you get it in a later period <>.
Professor John Geanakoplos: So the point is you’re discounting the first two payments by 2 percent a year, but you’re discounting these payments by 5 percent a year. So these things are going to be much worse than they look. And so even though the 4s are all better, this 100, you know, it’s 4, 4, 4, 4, 4 + 100, that 100, not to mention these three last 4s are getting discounted by a lot. So that 104 is going to be worth less than 100 back, you know, it’s going to be worth less.
So that’s why these things are going to go down a lot more than you think. They’re going to be worth less than 100. So even though these things are better than the 2 percent rate of return–this bond does great at the beginning, but does poorly thereafter, because in the beginning it’s paying 4 percent when the interest rate is 2 percent, but later it’s paying 4 when the interest rate is 5 percent, so clearly you could have some situation like that. So what’s the carry trade?
The carry trade is you buy the long bond and sell short the short bond, so the lower maturity–too many shorts–the lower maturity bond. So this is the, let’s say, 5 year and 2 year. Buy the 5-year bond and sell short the 2-year. The word, sell short, is different from the word, short bond. So you buy the 5-year bond and you sell the 2-year bond.
They both cost you 100, so what’s happening at the beginning? You’re making 4 dollars and you’re only having to pay 2 dollars. It looks like you’re making a profit for nothing, right? So that’s the carry trade. You buy long bonds with a high rate of interest. You sell short bonds with low coupons and it looks like you’re making a profit. But in fact, what’s really happening? So just repeat what you were saying before, but now in this context. What’s really happening?
Student: You’re paying off that difference, the loss of the value of your assets. You’re not getting back as much as you would if you had just bought the 2-year bond.
Professor John Geanakoplos: Right. So the long bond you got a cash flow of 4 the first year. You had to pay 2 because you sold the 2-year short, right? So you’re up 2 dollars, but the thing that you owe now at the end of the first year, since the interest rate is 2 percent, the next interest rate is 2 percent, the 2-year bond is still worth 100. So your negative position is still worth 100, but this positive position now, these 4, 4s and 4s it looks worse than it did before because you’ve moved up the yield curve.
Instead of having two 2 percent years you only got one 2 percent year before you shift into the 5 percents which is worse and worse. So what’s going to happen is you’re going to get a positive cash flow at the beginning, it looks like positive profits right at the beginning.
That C (1), the net C (1) looks really good, but right after that, the value of your assets is going to plummet compared to your liabilities, because now all of a sudden you’re discounting this at 5 percent and this thing is still, you know, it’s only got 1 year left where the interest rate is still 2 percent and the rest of this is discounted at 5 percent so it’s starting to go down in value. So if you didn’t have to mark to market what would you do?
If you didn’t have to declare to the world what your present value of your remaining assets are, because you could say, “Oh, it’s so hard to figure out. I don’t know what the stuff is worth, the present value of what’s going forward. I just know the cash that’s coming in.” This carry trade would look like a really good trade, wouldn’t it, and a lot of people would do it because then they would look–the public would think that they’re doing really well because they’re getting positive value, but in the future that positive value they’re getting is just disguising losses that are happening in their portfolios going forward. So any questions about that? Yeah?
Student: Can you explain where their losses <>?
Professor John Geanakoplos: I want you to explain it. So I didn’t calculate the numbers. It would have been better to do actual numbers. If we do 4, 4, 4 at 5 percent interest, or maybe I can do a–let’s just do a real number.
Well, let’s just do the one over here. So this one is–no, I can’t do it, because we’ve got 2 percent and then 4 percent. So we take a minute to do it on Excel and we’ve seen how great I am doing those on the fly. So the point is that you agree that this payment 4 is bigger than 2. So the first year if you buy this bond and you sell that bond you’re going to have a positive net 2 dollars.
However, it’s perfectly possible for this whole bond at the beginning to be worth 100 the same as this bond at the beginning. Now, how could that be? Well, the payments–4 is bigger than 2 so how could this whole thing have a higher present value than this thing? It has to be that starting at period 1 the present value of this 5 year bond (let’s call it this) in period 1 - the present value of the 2 year bond staring in period 1 has to be what?
If this present value of the 5 year bond, this is the 5 year bond at time 0, equals 100, and that’s exactly the same as this present value at time 0 of the 2 year bond, if that equals 100, so the present value from the beginning of this thing is the same as the present value of this thing from the beginning.
How could this thing have value 100? Well, it’s because it makes lots of high coupon payments, but the interest rate is going to jump up in the future so you’re discounting all these future cash flows at a big number. So all these things present-valued could well equal 100. So this could be 100 and this could be 100 even though this is paying off more at the beginning than this thing is.
But what does that formula that we just wrote down over there, what does this formula tell us? Where was this formula that I wrote down? This formula over here tells us if the present values of two instruments–I see I wrote it for 1 bond, but if you take 2 bonds with the same present value, one of which has a higher cash flow in the first period than the other, then that formula tells you the bond with the higher cash flow at the beginning has to have a bigger drop in the present value than the other bond, otherwise you can’t get them both equal to the same 100. So this bond and this bond have the same present value of 100. Because this payment is bigger than this payment, but the present values are the same, 100, it has to be that starting from this point on, this bond is worth less than that bond.
So what happens? You go long this bond, you short that bond, you say to the world, “Ah ha! I’ve made 2 dollars. I’m a genius,” and then you hide from them, you might hide from them if you didn’t have to tell them, they wouldn’t know that the present value of the bond you’re long going forward is actually lower by 2 dollars than the present value going forward–this is right after this period–the present value starting at this point of this bond going forward has actually dropped 2 dollars below the present value of this bond going forward, and it has to have dropped otherwise the present values wouldn’t be the same.
So to just say it much more simply, if you take 2 bonds with the same present value you could well have that situation where they have the same present value because one of the bonds pays a lot of stuff early and terrible stuff late. So in the beginning it looks like this bond is paying you more money than the other bond, but since they have the same present value it must mean that this bond is going to pay you more money at the tail than this bond is.
That’s why they had the same present value. So if one of them gets ahead at the beginning it has to be it’s going to fall behind the rest of the time. So this one got ahead at the beginning, it has to fall behind the rest of the time. That’s hard to see because it looks like this bond it’s paying a coupon that’s always bigger than that bond. So it’s easy to lose track of the fact that this bond, because its coupon is higher than this bond, how could it possibly ever fall below this bond?
Well, it falls below because it’s longer and the cash flows towards the end of it are being discounted by an interest rate–because everybody knows the interest rate’s going to go up.
And remember we saw last time the yield curve. The yield curve, remember today’s yield curve is practically 0 now because the government’s held it at 0 and it’s going to go way up to 4 percent or something in a couple years. So that’s the yield curve today. So everybody knows the interest rates are really low now and they’re going to get much higher later. So what it means is that every long bond that’s being issued now is going to be issued with a higher coupon than the short bonds, and if you go long the long bond and short the short bond you’re going to make money at the beginning but lose it back later.
But if you don’t have to report mark to market, you don’t have to say the value of what’s left over, the public’s just going to see that you’re making money at the beginning. It’s clear now? Any other questions about this? So a lot of this is going on right now.
Chapter 3. Mortgages and Backward Induction [00:39:53]
So let’s do one more application of this. Let’s explain how mortgages work, one more idea.
Since the present value is = to the cash flow + the present value at 1 times that, given this formula, that tells you one more thing, very important idea. How do you compute present value 0? Well, the way we’ve always computed present value at 0 is to do this calculation, this long calculation discounting all the future cash flows, but this formula tells you that there’s actually a more efficient way of calculating it by backward induction.
If I knew what the present value was at time 1 then I could get the present value at time 0 just by this formula. I’d add the present value at time 1 to the cash at time 1 and discount it. So in fact I don’t have to–now, I don’t know the present value at time 1, but if I go to the end, at the end of time, I know the present value of the bond is 0. There’s nothing coming up later.
At time T - 1 I know the present value’s very easy to calculate. It’s C (T) divided by 1 + iFT - 1. So I can go backwards by present value and compute. I can do backward induction. So that’s the word I want to describe which is going to play a very important role in the future of the class, backward induction.
It says if this formula is correct then a good way to do the computation is by working backwards from the end. Don’t just blindly take the present value. If you blindly take the present value all you’ve got is PV0. If I calculate by backward induction I start to the end and say, what would PVT-1 be? That’s a really simple thing.
Then I can very simply find out what PVT-2 is etcetera back to the beginning. I do basically the same calculation without having to take powers of–it’s a shorter calculation because here I’ve got exponents of interest rates multiplying each other. So it’s actually a shorter calculation, and on top of that I get much more information because now I’ve calculated out what the present value’s going to be at every time period. So a much more efficient way of calculating things is to do it by backward induction because it also tells me more. It tells me what the future path of the bond is going to be.
So let’s just see how that works on an example that I hope I worked out properly before. Of course, I did it last year, but let’s assume I did it right. So let’s take a mortgage, if you can see this. So this is something you have in your notes. So there’s a mortgage. Now, how does a–all right, well pay no attention to that. So let’s just say that–oh no, why did this happen? Let’s say that the payment is 8.
So I’ve made payments of 8 everywhere. I hope I haven’t screwed this up. So here’s the mortgage. It’s a 30-year mortgage. So these are the years 1, 2, 3, 4, 5, 6, 7, blah, blah, blah, blah, blah, blah up to year 30.
Now, suppose I happen to have a bond that pays a coupon of 8 dollars every year, and the interest rate, let’s say, is 7 percent. So it would be the interest rate on the mortgage, but let’s say the interest rate in the economy is 7 percent and the bond, however, is a premium bond. The bond is paying 8 percent every year.
Now, this is a mortgage so what does this mean? See the mortgage pays 8 every year until the very last year where it’s still paying 8. So it’s not paying 108, it’s just paying 8 every year. So it’s not a coupon bond. It’s a mortgage. A mortgage pays the same amount every year until the end.
So if this were a coupon bond the last payment would be 108 and of course the thing would be a tremendous–it would be a premium bond because the interest rate’s only 7 because it would pay a higher coupon of 8 and 108 at the end. But you see I’m not paying 108 at the end. I’m only paying 8 at the end.
So the first question is, what is the present value of this bond? So I could take 8 divided by 1.07 + 8 divided by 1.07 squared + (all the way) 8 over 1.07 to the thirtieth power and figure out that number. But a much better way of doing it is by backward induction. So what do I do?
I go to the end and I say this line is the remaining balance. Well, at year 30 there’d be no more payments of the bond so the present value of what’s left is obviously 0. Now, what is the present value of what’s left at time 29? Well, the only thing I’m going to get is I’m going to get this payment at time 8 right over there. Sorry. I’m going to get this 8 right there and so the present value of 8 at 7 percent interest is 7.47, I mean, 7 percent interest. So how did I figure that out? I just said, take that payment on the right and discount it by 7 percent, so it’s 7.47.
Now, what’s the present value here? Well, here you’re going to get two things. What are you going to get? You’re going to get a coupon payment. Just after the payment in year 28 what’s the present value of what’s left?
You’re going to get a payment in year 29 of 8 dollars. You’re also going to get one in year 30, but you don’t care about that. You just know that the value in year 29 is the coupon you get in year 29 plus the present value in year 29 of what’s left. The PV at time t, I can just put a t here, PV at time t is the coupon you get at time t + 1, + the present value at time t + 1 divided by the interest rate at time t.
So all I have to do is take 8 + 7.47 and discount that by 7 percent, and that gives me the present value as of time 28. So if I go back to time 26 I say, well, how do I figure out the present value at time 27? It’s time 27. I say, well, I’m going to get 8 at time 28, but the present value at that time, just after that payment, is 14.4. So I take 22.4 and discount it by 7 percent and I get about 21. So that’s how I can work backwards and figure out the present value today.
Now, in mortgages this number is very interesting. It’s called the remaining balance, which we’ll see in a second. But anyway, that number you can calculate it every t. It’s just the PVt worked backward from the end. So it’s a very simple calculation and it tells you the present value of the bond, of that 8 percent coupon, until year 30 and no principal at the end.
Well, a mortgage has to pay a coupon of 8 percent that gives 100 because if there were no uncertainty and there are no prepayments or anything, no uncertainty, the bank giving the coupon is not going to give you money unless its present value is equal to 100. So they’re going to ask you to pay a coupon.
So now you see what have I done over here? This is B 26 - B 30, so the original face it was supposed to be 100, but the present value of all those payments is only 99, so there’s a gap here of .72 and the square of the gap is that. The square of the gap is there, and we want to minimize that. So we’re going to find the payment every period, Solver. So I want to minimize B 32, minimize by choosing B 29 which looks good. That’s the first payment, and all the other payments are set to be equal to that, to minimize that difference. B 29 is the first payment, so why didn’t this work?
Student: You hardwired in C 29 and D 29 <>.
Professor John Geanakoplos: Is that what I did? So here’s 8 and then this over here. I hardwired that in, so I didn’t want to do that. So that’s got to be equal to what’s left, equal, left. So here I got–when I hardwire this then, okay, so that’s good. So let’s try the same thing now. Tools, glad I have you guys, Tools, so B 32 is what I’m minimizing. B 29 is the first payment and all the others have been set equal to it, and if I solve…
Student: C 29.
Professor John Geanakoplos: All right, let’s try. The first payment is year 1, right, so it should be–oh, C 29, thank you. Yeah, that’s why I’m confused, C 29, thanks. Tools, so let’s just look at this a second. So that’s what I’ve hardwired in. So Format, no Tools, Solver. Minimize B 32 subject to C 29 as you’ve told me 20 times, C 29 and now solve. So now we did it. So we got the right payment to make the balance exactly equal to 100.
So that’s how a mortgage works. You have to find the coupon payment such that if you take the present value of that same coupon payment forever it’s just going to be worth 100 which is that number. And how do you figure out the coupon payment? Well, you do it by backward inductions. Figure out the present value. Do it by backward induction just as we did.
Chapter 4. Remaining Balances and Amortization [00:50:42]
But we’ve gotten a lot of information. We’ve gotten this number at every period. So by doing it by backward induction instead of just doing the long exponential calculation, by doing it by backward induction we’ve produced the present value at every time in the future.
Now, that’s an incredibly important number in mortgages. It’s called the remaining balance. Why is that such an important number in mortgages? This will play a very key role in the rest of the course. What is the remaining balance and why is it so important? Does anyone know how a mortgage works? Yep?
Student: Is that the mark to market value or just whatever you’d be able to sell that mortgage for to someone?
Professor John Geanakoplos: Not what you’d be able to sell it for, but close. So when the bank gives you a mortgage it says–so how did mortgages work?
It used to be in the old days that the mortgages were coupon bonds. They’d pay 8, 8, 8, 8, 8, 108, and then what would happen is just before the 108 payment everyone would default. So in the Depression the people who defaulted were people who defaulted just before their big principal payment, so bankers got wise after the Depression. They said, “Well, that’s a terrible thing to do. We should make the payment be constant and that way there’s no reason for the guy to default right at the end and we’re not going to get stuck with 100 dollars that’s not paid.” So it’s constant.
But of course if it’s constant that means the present value of what’s left is going down all the time, so that’s why this number is going down all the time. If it’s 108 at the end, that’s why the present value would stay the same. So it’s going down all the time, so that’s why it’s called an amortizing mortgage. It’s because what’s left in the mortgage is getting smaller all the time. So a fixed rate mortgage pays the same coupon every year. The present value of what’s left, therefore, must be going down every year and that’s why it’s called amortizing.
And bankers wanted that to happen because that way their risk is going down every year. Every year the house is presumably still worth 120 or whatever it was at the beginning and the amount owed is getting lower and lower. So the bankers are feeling more and more secure every year because the house is backing a smaller and smaller loan, or to put it another way, if you’re uncertain about what the price of the house will be in the future, you want to make sure that what is owed is going down in the future. If the price gradually goes down of the house what’s owed is gradually going to go down.
But the main reason why the number is so important is you have to realize the purpose of a mortgage is you take out a loan using the house as collateral. If you don’t make your payment they can take your house. Well, what happens if you want to move?
If you want to move you’re going to sell the house. The house is no longer collateral, so if you want to move you have to undo the promise to the bank. So if in year 5 just after you’re making your payment of 8 dollars and 5 cents in year 5 you decide to move, you say to the bank, “I want to cancel the mortgage.” How much will they ask you to pay? Well, the remaining balance, 93.91.
So that’s why this remaining balance is such an important number. So it allows people to leave their house and pay off their mortgage by paying off the remaining balance. You wouldn’t want them to pay 100 if they left because they’ve made payments.
You notice that the payment here is 8 dollars. That’s bigger than 7 percent, right, because if it’s a coupon bond you’d pay 7 all the way to the end and pay 107. That would have a present value of 100. If you’re not paying the 100 at the end, but just making a level payment all the way through, the payment you have to make every year better be more than 7, so it’s 8. So this 8 - 7 is sort of what you’re paying down of your mortgage, right? The interest is only 7. You’ve paid 8. That’s why you owe less than 100. You’ve paid 1 dollar 6 extra.
That’s why you only owe 98.94. So every year you’re paying part of your principal down. That means it’s amortizing, and that means if you want to get out of the mortgage you can get out of it by paying less and less. It means the lender, the mortgage lender, is more and more protected by the house every year because what you owe is less and less. So that’s how a mortgage works. Any questions about that?
Chapter 5. Weaknesses in the U.S. Social Security System [00:54:52]
All right, so what’s the point? The point is that by simple present value thinking you can start to understand the main instruments in the economy, how mortgages work, why they’re called amortizing, why the amount you have to pay to get out of your mortgage goes down every year, and exactly how much it goes down every year etcetera.
So we can do a lot more examples like that, but I want to change, shift the discussion now to a much bigger subject, a subject of tremendous policy interest in the country, namely Social Security.
What should we do about Social Security? Now, it will turn out that you can analyze the situation the same way we’ve just analyzed these bonds. It’s very simple to figure out what the problem is and what went wrong, yet very few people understand it, including most of our politicians and, I’m sorry to say, a lot of our economists.
So I want to describe now in the next class and a half the Social Security problem and how to solve the problem, but also how to understand the problem. You can’t figure out the right solution until you’ve understood what the problem is. So were there any questions?
Before I start this, were there any questions? I should have paused. Are there any questions about the mortgage, or present value, or how present value changes through time? So those ideas, and marking to market, those are very important ideas I think once you think about them not so hard.
I’m now going to take exactly those ideas and apply it to Social Security where the public is totally baffled, but all you have to do is apply the same thinking. So Social Security is supposed to be in a terrible crisis. That’s what they always tell you. It was a big campaign issue in 2000, you probably were too young to remember that, but there were three debates between Gore and Bush in which Gore grimaced and everybody thought he wasn’t a good guy and so they voted for Bush. He was mostly grimacing about Social Security.
And then in 2000–well, I’ll get to the future. So anyway, in those debates three mistakes were made. So Bush argued that the returns on Social Security were disastrously low. He said the whole program is in a terrible crisis and we’ve got to privatize to save the system.
How did it get into the crisis? Well, it wasn’t clear exactly how it got into the crisis, but it seems like the baby boomers had something to do with it. They’re all getting old and they’re going to have to get these huge Social Security payments, and that’s why we’re in the crisis because the baby boomers are getting old. It’s all my fault, or my generation’s fault.
Then the third mistake was Gore said, “Well, it’s impossible to privatize.” Privatize means take the money that your parents are paying in Social Security and that you’ll start to pay; instead of putting it into the fund that’s being used somehow, you don’t probably know exactly how, instead of doing that, take that money and say it’s the taxpayer’s money. It’s your money. You can put it in the stock market if you want. So that’s what Bush wanted to do, privatize Social Security, say your tax contributions should go into a stock market with your name on them.
And Gore said, “Well, that’s impossible. If you privatize, what are the old people today going to do? Where are they going to get the payments? You can’t privatize Social Security and take today’s young tax contributions to Social Security and say, ‘You young guys, there’s your money. You can keep them in the stock market,’ and at the same time pay the old retirees, so Bush must not know what he’s talking about.” So that also was wrong.
So those three things, that Social Security is going to give terrible returns, it must be it’s wasting money, something’s horribly wrong with it and the only way to save it is to privatize it. That’s Bush’s main claim. Blame it all on the baby generation, that’s everybody’s claim, and Gore saying you can’t privatize without screwing today’s young [correction: today’s retirees]. All three of those things sound pretty convincing and yet they’re all three wrong.
So I want to explain the system to you and help you understand it, and then I have a policy recommendation you’ll get next class which most people don’t agree with, so you probably won’t either, but I’ll warn you when we get to a point that’s controversial. So everything I’m going to say in the first 90 minutes is going to be uncontroversial. Not everybody knows it, but I think it’s obviously just a matter of logic. And then my conclusion about what to do, you can criticize it. I think it’s a matter of logic too, but I admit most people don’t agree with it.
So now, in 2005 if nothing else Bush was tremendously consistent. So whatever he told you he was going to do no matter how wrong-handed it was he did it. So he said in the debates that he wanted to privatize Social Security and sure enough he kept his word.
He launched a huge program. That’s how he started right after the 2004 election. His first initiative, you might remember, was we’ve got to privatize Social Security. He went on a 60-day, 60-city tour to kick off his second term. So after the 2004 election privatizing Social Security was a huge issue.
In the 2008 election it was still a big issue. McCain said sort of what Bush said. “I want young workers to be able to if they choose, to take part of their own money which is their taxes, their money that’s getting taxed and getting put into Social Security, I want them to have their own account and put it into the stock market with their name on it.” And Obama said he’s totally against that. That was 2008.
Recently, of course, the public has made another mistake. So everybody is saying now, “Oh, the financial crisis. We better not talk about Social Security anymore.” Nobody’s talking about Social Security. They’re saying, “Well, could you have imagined what would have happened had we privatized as Bush wanted us to do and the stock market collapsed? Everybody would have lost all their money. What a disaster that would have been, and it would have ruined the old.”
This sounds pretty persuasive. So Krugman wrote a column saying, “A bullet dodged. What would have happened if Bush had succeeded? All the old people would be broke now.” And Robert Reich who was in the Clinton Administration sort of more or less said the same thing that it’d be a disaster.
Now, the Obama Administration, by the way, hasn’t stopped talking about Social Security. So their director, he’s the Director of the White House Office of the Budget, so this is an incredibly important position, Peter Orszag. He’s the son of–there’s a math professor here, Orszag, his father. So he’s a friend of mine, the son. He said once healthcare reform is in place the U.S. can then focus on other important things, especially Social Security. So Obama wants to do something about it, but he just doesn’t want to privatize. So it’s a big problem and everything everybody says about it seems plausible.
Now, just to continue along the plausibility of it, so Bush says what he really wants to do is–why private accounts are a better thing is that if you put the money in private accounts it can grow. You can get a greater rate of return than the current system which is terrible. If you were a young person by putting your Social Security tax money aside in a private account you’ll be able to get a better rate of return on your money than the government could get you on your money.
So why would you just want to give it to the government? It’s for your retirement and you’ll be able to pass that money along to your children and grandchildren if you want at the end. And best of all the money’s yours and the government can’t take it away. So that’s what Bush says and he’s said many times.
So there’s one sense in which he’s right. So let’s look at the returns people got on Social Security. Now, what is a return? We know what the rate of return is. How do they calculate it? You can go back to people born starting in 1878. Social Security, as you’ll see in a second, I’m going to give you the history, began 1939. So these people in 1878 they’re 60 when Social Security’s beginning, if my arithmetic is right. So they’re 60 at that point.
So you can look at all these people and you can say, for every generation in the past, you can say how much money did they pay when they were young? They paid taxes. So they got negative 12.4, that’s the tax rate. Negative 12.4, they did that a bunch of years when they were young and then they started getting payments when they were old, 24, 24, 20 something like that. Those are the payments when they’re old. So there are negative ones at the beginning and positive ones at the end.
You can calculate the internal rate of return, the yield, the thing that makes this present value 0. We know how to do that. We’ve done it. So people have done this. So there’s a guy named Limmer, Limmer or Leemer from the Social Security Administration who did these calculations, prompted a little bit by me.
So I should say that a lot of the reason I got started thinking about this is I got put on a Presidential Panel to study Social Security Reform in the Clinton Administration and every Democrat was matched with a Republican. So the two chairmen, one Democrat, one Republican, all the way along there was Democrat and Republican, and after Bush got elected all the Democrats got kicked off the committee. But anyway, so in any case here are the rates of return.
For people who are old when the program began, they got sensational rates of returns, 40 percent, 30 percent, incredible rates of return. As the generations get younger and younger the returns go lower and lower and they’re down now to 2 percent. These are a forecast of these rates of returns. And so let’s see, you’re 20, say, something on average. You were born in 1990, something like that or around there.
So here’s your rate of return. It’s down here, right? It’s 2 percent, and there it is blown up. It’s under 2 percent. So these are the people from 1924 to 2002, so you’re right at the end here. That’s your rate of return blown up.
You can only expect 1 and a half or 2 percent. So George Bush is right. The rate of return on Social Security is terrible. If you look at the taxes that you put in, and you look at the benefits that you can expect to get your generation is getting totally screwed.
So now if you look historically–this was done in 1994, so the number’s a little bit–it’s not quite as dramatic, but it hasn’t changed as much as you think. From money put in the stock market between ‘26 and 2004, that’s before the crash of ‘29, so I should have gotten the number after the recent crash. It doesn’t change that much. Before the crash of ‘29 you looked at keeping your money in the stock market, just leaving it there for all that time, after inflation you’d get 9.1 percent return.
On treasury bonds you get 2.7 percent return and yet on Social Security you’re going to get under 2 percent. So George Bush says, “Look, put it in the stock market. Get 9 percent.” We had a little disaster here, so maybe it’s 7 percent. “Put it in the stock market and get 7 percent. Why be satisfied with 2 percent? There’s something wrong. We’ve got to privatize, put an end to this.”
So there seems to be something to what he is saying. So now you look at what’s going on with Social Security. So what happens in Social Security? I’m going to explain the whole history and how it works here.
What happens is you pay taxes, 12.4 percent tax. You probably know that. Everybody’s paying taxes on the money they make and then there are benefits that are being paid. Now, the benefits are not–there’s a formula for benefits which doesn’t have anything to do with the amount of taxes being paid, so at the present time the taxes are bigger than the benefits. So where does the extra money go?
It goes into the Social Security Trust Fund. So here’s the Social Security Trust Fund which is now around 2 trillion, which is going to keep growing because contributions are going to be bigger than benefits until 2020 or so, and at that point contributions–that’s the baby boom generation, that’s me retiring, or a little later in my case, but anyway the baby boom generation–in fact since I’m a Yale professor it’s out here somewhere. So anyway, my generation’s going to start retiring here and then the benefits–so we’re going to be not working, not paying taxes but we’re collecting benefits and there are so many of us that the benefits are going to be less than the taxes and the trust fund’s going to go down, and down, and down.
And then the year 2030, at a little bit more than 2030, the trust fund’s going to go to 0, and then after that the taxes are still going to be smaller than the benefits. And so what’s going to happen? They’ll be no money to pay these people and there’s going to have to be a big drop in what people are getting. So it looks like the system is not only paying a horrible rate of return, it’s not even going to pay. It’s going to run out of money and go broke in 2040. So it seems like a total disaster.
So that’s the setting of the question. George Bush said it’s a disaster. It looks, at first glance, like a disaster. The newspapers tell you all the time it’s a disaster. How did it get to be so bad and what should we do about it?
All right, now it’s going to turn out that by using the concept of present value it can be very simple and straightforward to understand, and it’s going to be the exact opposite of what everybody seems to be saying. So I’m going to play you a clip next time of Roosevelt announcing the Social Security program and this lady behind him is Frances Perkins, the first woman to ever be in the Cabinet, and she played a tremendous role in shaping Social Security. So after Roosevelt started Social Security in 2004–I’m wondering whether I really have time to do this, so I’ll just say this.
So Roosevelt started the program in 1938-‘39. He started the program in ‘38-‘39. It was one of the cornerstones of the New Deal. It’s an incredibly famous program and it seems to be in incredible trouble. So we need to find out and get to the bottom of why that is and we’ll start doing that next class.
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