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Financial Theory: Lecture 9

Professor John Geanakoplos: Okay, so I think I should start. Hello. All right, so last time we began, or maybe two times ago, we began a discussion of various vocabulary and facts you have to know about the markets if you want to think about finance. Today we're going to deal mostly with the most important one, the most basic one, the yield curve. And last time, we introduced this word, "yield."

Now, yield is an extremely common expression in finance, and it turns out not to be that well defined, often, or that useful. But the word is so important and has been used so often that it still hangs around, even when probably we should use different concepts.

So remember the yield was an attempt to look at an investment, and without paying any attention to the market or anything outside the investment, just looking at the investment itself, try to assess, give a number, quantifying how attractive the investment was. So we said you could apply that to a bond — it has cash flows. You could apply it to a hedge fund that's taking in money and paying out money, and the formula we came up with said that if the cash flows are given by C(1), C(2), the net cash flows, C(T) over the course of the period, and its price is some P(0), maybe it's a negative cash flow, so C(0). So some of these cash flows might be negative and some of them might be positive, then we should just look at the number Y, such that discounting all these things at rate Y gives you 0. The Y that did that was what we defined as the yield of the investment.

So we saw that that had some advantages. For example, in a hedge fund, if you just look at the rate of return it makes on its money every year, that doesn't take into account that in some years, it's got a lot more money. So if those were the years that lost money, and the years when it hardly had any money were the years it made money, just taking the average, the multiplicative average, the geometric average of all those yearly rates of returns, would give a misleading figure.

Well, the yield also gives a somewhat misleading figure, and I don't want to spend too much time on why it might be misleading, but I'll give you just an example. Suppose that the cash flows happen to be 1, -4, and 3. Now what's the yield to maturity? Well, there are two of them. You could have Y = 0, because 1 over (1 + 0), + - 4 over (1 + 0), + 3 over (1 + 0), is just 1 - 4 + 3. That equals 0, so the yield to maturity of 0 percent, the yield of 0 percent, makes this have present value 0. But also I could try Y = 200 percent, and then I'd have Y — I'd put a + 2 and a + 2 squared here, and I'd have + 1 - 4 thirds + 1 over 3 squared is 3 over 9, so it's + 1 third. So it would be 1 minus 4 thirds, plus 1 third, which also equals 0. So is the yield to maturity, the internal rate of return 0 percent or 200 percent? It's ambiguous. So yield to maturity can't be the right way of doing things.

To go back to the hedge fund example, you know, the hedge fund was taking in money, paying out money, taking in more money, paying out money, and we calculated the yield to maturity. Well suppose that there was some period, you know, here, at which point everyone had taken all their money out, so the hedge fund wasn't actually doing anything for a bunch of years, maybe for a long time, and then it started up and took money in and paid money out and stuff.

Professor John Geanakoplos: Okay, so I think I should start. Hello. All right, so last time we began, or maybe two times ago, we began a discussion of various vocabulary and facts you have to know about the markets if you want to think about finance. Today we're going to deal mostly with the most important one, the most basic one, the yield curve. And last time, we introduced this word, "yield."

Now, yield is an extremely common expression in finance, and it turns out not to be that well defined, often, or that useful. But the word is so important and has been used so often that it still hangs around, even when probably we should use different concepts.

So remember the yield was an attempt to look at an investment, and without paying any attention to the market or anything outside the investment, just looking at the investment itself, try to assess, give a number, quantifying how attractive the investment was. So we said you could apply that to a bond — it has cash flows. You could apply it to a hedge fund that's taking in money and paying out money, and the formula we came up with said that if the cash flows are given by C(1), C(2), the net cash flows, C(T) over the course of the period, and its price is some P(0), maybe it's a negative cash flow, so C(0). So some of these cash flows might be negative and some of them might be positive, then we should just look at the number Y, such that discounting all these things at rate Y gives you 0. The Y that did that was what we defined as the yield of the investment.

So we saw that that had some advantages. For example, in a hedge fund, if you just look at the rate of return it makes on its money every year, that doesn't take into account that in some years, it's got a lot more money. So if those were the years that lost money, and the years when it hardly had any money were the years it made money, just taking the average, the multiplicative average, the geometric average of all those yearly rates of returns, would give a misleading figure.

Well, the yield also gives a somewhat misleading figure, and I don't want to spend too much time on why it might be misleading, but I'll give you just an example. Suppose that the cash flows happen to be 1, -4, and 3. Now what's the yield to maturity? Well, there are two of them. You could have Y = 0, because 1 over (1 + 0), + - 4 over (1 + 0), + 3 over (1 + 0), is just 1 - 4 + 3. That equals 0, so the yield to maturity of 0 percent, the yield of 0 percent, makes this have present value 0. But also I could try Y = 200 percent, and then I'd have Y — I'd put a + 2 and a + 2 squared here, and I'd have + 1 - 4 thirds + 1 over 3 squared is 3 over 9, so it's + 1 third. So it would be 1 minus 4 thirds, plus 1 third, which also equals 0. So is the yield to maturity, the internal rate of return 0 percent or 200 percent? It's ambiguous. So yield to maturity can't be the right way of doing things.

To go back to the hedge fund example, you know, the hedge fund was taking in money, paying out money, taking in more money, paying out money, and we calculated the yield to maturity. Well suppose that there was some period, you know, here, at which point everyone had taken all their money out, so the hedge fund wasn't actually doing anything for a bunch of years, maybe for a long time, and then it started up and took money in and paid money out and stuff.

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