lecture 5_yield curve arbitrage

8/18/2019 Lecture 5_Yield Curve Arbitrage

http://slidepdf.com/reader/full/lecture-5yield-curve-arbitrage 1/6

Lecture 05: Yield Curve Arbitrage

This lecture concerns one of the most important topics in Finance, the yieldcurve. We said the yield was an attempt to look at an investment, and withoutpaying any attention to the market or anything outside the investment, to try toassess how attractive the investment was. This applyies that to a bond, since ithas cash ‡ows. You could also apply it to a hedge fund that is taking in moneyand paying out money, and the formula we came up with said:

C (0) + C (1)1 + y

+ C (2)(1 + y)2 + ::: +

C (T )(1 + y)T = 0 (1)

Some of the cash ‡ows (CFs) above might be negative and some of them mightbe positive. The formula solves for y, such that discounting all these CFs atrate y gives you 0. We de…ned this y as the yield of the investment.

We also 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, thatdoesn’t take into account that in some years, it’s got a lot more money. Soif those were the years that lost money, and the years when it hardly had anymoney were the years it made money, just taking the average, the multiplicativeaverage, the geometric average of all those yearly rates of returns, would give amisleading …gure.

However, the yield also may give a somewhat misleading …gure. Let’s givean example. Suppose that the cash ‡ows happen to be 1, -4, and 3. Now what’sthe yield to maturity? Well, there are two of them. You could have y = 0 ,because:

14

(1 + 0) 1 + 3

(1 + 0) 2 = 0 (2)

That equals 0, so the yield to maturity of 0 percent, makes this have presentvalue 0. But also I could try Y = 200% , and then I’d have:

14

(1 + 2) 1 + 3

(1 + 2) 2 = 0 (3)

which also equals 0. So is the yield to maturity, the internal rate of return 0percent or 200 percent? It’s ambiguous. So the yield to maturity can’t be theright way of doing things.

To go back to the hedge fund example, suppose that there was some period,at which point everyone had taken all their money out, so the hedge fund wasn’tactually doing anything for a bunch of years, maybe for a long time, and thenit started up and took money in and paid money. Well, because the gap in

time was very long with nothing happening, if you take a positive y, the stu¤ that happens in the second incarnation of the fund is hardly going to be makingany di¤erence, because by that time, it will all be discounted a lot. The yieldwill depend too sensitively on stu¤ early rather than stu¤ late. Therefore, eventhough the term is somewhat inappropriate, the word lives on, and part of thecommon vocabular.

1



Now what would Irving Fisher say you should do, if you had to summarizehow good an investment was? He would say: "Just look at the present value of all these cash ‡ows." However, to do that, you’d have to know, the market rateof interest with which to compute the present value? So Fisher would say, "It’sridiculous to evaluate how good an investment opportunity is just by looking atthe cash ‡ows. You’re throwing away too much information."

You know what the market is doing, you know what the interest rates are.Use the market interest rates and …gure out what the present value of all thecash ‡ows is. Let’s use an example. Suppose the U.S. Treasury issues a 1-yearbond, a 2-year bond, a 3-year bond, a 4-year bond and a 5-year bond, all in thesame day. Also suppose that the coupon it sets is 1 dollar for the 1-year bond,2 dollars for the 2-year bond, 3 dollars for 3 three-year bond, 4 for the 4-yearbond, 5 for the 5-year bond. and the face value is always 100. Further let’s say,when they actually market these, and supply equals demand in equilibrium, theprices turn out to be (1) = 100.1, (2) = 100.2, (3) = 100.3, (4) = 100.4,and (5) = 100.5. So 100.5 is the price the market is paying for the 5 year bond.Finding the yields for each bond is then straightforward. For instance, for the4-year bond:

100:4 = 4

1 + y(4) +

4[1 + y(4)]2 +

4[1 + y(4)]3 +

104[1 + y(4)]4 (4)

and y(4) can be found using something like the solver in Excel. However,the information that you want to deal with is the price, and what did thecoupon actually pay. But now, what does Fisher say you should do? The mostimportant thing to do is …nd the zeros interest rates. If you modernize Fisher alittle bit, the most important thing to do is …nd the prices of the zeros:

(i) today’s money price for 1 dollar at time i.

Okay, now why do you want to …nd these things? Because once you knowthese things, you’d be able to value any investment. For instance, the price orfair value of an investment is a sum of the product of the cash ‡owsand the ’s.:

P = C (1) (1) + C (1) (1) + ::: + C (T ) (T ) (5)

Now why is this the right price? Because if you can go in the market andbuy 1 dollar at time 1 for (1), and 1 dollar at time 2 for (2) , etc., you can buyall the cash ‡ows from this investment project by spending P . So if somebody iso¤ering you the investment opportunity at a higher price, it would be crazy todo it. You could have bought those cash ‡ows yourself by paying P . If he o¤ers

it to you at a lower price than that, then de…nitely you should do it, becauseit’s a bargain, because if you had to buy it yourself, it would be more expensive.In fact, if he’s o¤ering it to you at a lower price, you can make an arbitrage

pro…t. How could one realize this arbitrage pro…t? You buy his project forthe lower price P and then sell these very promises, C(1), C(2), C(T) on themarket. So you sell it in fact for a higher price. You make the di¤erence, and

2



when it comes time to keep your promises, the project is giving you the cash tokeep your promises, so you lock in a pro…t for sure. So if you knew the 0 s ,you would know for sure how to value any project where you knew for sure thecash ‡ows.

So far we said literally that (1) is the price you would pay today to buy 1dollar tomorrow. Now how could you go about buying 1 dollar tomorrow, giventhat the only things you can trade on the market are these Treasury bonds forwhich you know the price and coupons? You can trade, buy or sell any of theseTreasury bonds. The arguments involve the ideas of Replication , Pricing , andArbitrage .

(1) = 1101

(1) = 1101

100:1 = 0 :991 (6)

(2) = 2

102

1

101(1) +

1

102(2) = 0 :963 (7)

(3) = 3

103(1)

3103

1102

(2) + 1103

(3) = 0 :917 (8)

(1) is 1/101 of the price of the 1 year bond: to buy one 1-year bond itcosts 100.1 so (1) will cost 1/101 of the price of the bond. To get (2) you buy1/102 (since the 2-year bond pays 102 dollars) of the price of the 2-year bondand sell 1/102 of the present value of the $2 coupon that you don’t need butreceive in year 1.

In words, what we’ve done is we’ve said, there are things you can actuallytrade on the market. Those are the Treasuries. Those are our benchmarksecurities. Now what we’re interested in is some other maybe …ctitious securitiesor new securities. The price of the zeros, those are the basic building blocks

that will help us evaluate the present value of any investment. So the reasonwhy we know these prices is because we can replicate them by trading only thebenchmarks, only the Treasuries.

To get the 1-year zero, we just buy the correct fraction of 1 year Treasuries.To get the 2-year zero, we have to buy the correct fraction of 2 year Treasuriesand sell the correct fraction of 1 year Treasuries. So we’ve replicated the 2-yearzero by a portfolio consisting of being long the 2 year Treasury and short the1-year Treasury.

To get the 3-year zero coupon, we have to buy the 3-year Treasury, sell the2-year Treasury and do something complicated with the 1-year Treasury. Andthen we’ll just add up the cost of that portfolio that replicates this. The 0 s arethe prices of zero coupon bonds of various maturities, and those aren’t reallytraded directly in the market. What’s traded directly in the market, wherepieces of paper change hands, are the Treasury bonds. But everybody, everyday is calculating these zero coupon prices, because that’s what they need to doto evaluate every single project that they might conceivably do that day, anddecide whether it’s a good project or a bad project. Is it worth the price or notworth the price?

3



However, instead of the more complicated approach above there is a very fast algorithm that you can do almost instantly, and that’s why it’s such a triviality

to calculate these numbers ever day. So it’s called the prin iple of du lity .For instance, suppose someone is willing to pay me 93 cents to recieve $1

three years from now (i.e., pay 93 cets for a 3 year zero).Well, I’d say, "That’s wonderful." I’ll sell them this promise in year 3, of 1

dollar for 93 cents. Then with that 93 cents, I’ll only use 91.7 of those cents and I’ll go out and buy the 3-year Treasury. I’ll sell some of the 2 year Treasury and I’ll sell a little bit more of the 1 year Treasury. And that portfolio which I’ve done by doing that will pay me exactly 1 dollar in year 3, enabling me to keep my promise to him, but it will only have cost me 91.7 cents.

So I’ll have made a 1.3-cent pro…t for sure, with no chance–it’s a pure arbi-trage. I made a pro…t of 1.3 cents with no chance of losing any money, because I’ve done all the transactions today, and the government’s going to keep its promises. I don’t have to worry about the government giving me the money, and so I’ll be able to turn the money over to that guy in year 3. Meanwhile, he’s given me his 93 cents.

If you want to do an arbitrage and make your pro…t, you have to …gure out what the replicating portfolio is, and the replicating portfolio also tells you the price. But it takes a long time to …gure out what all these arbitrage-replicating portfolios are.

The principle of duality is required to replicate and …gure out what the (1) , (2), (3), (4), and (5) are. I can …nd those numbers now just by clicking a

button in Excel, trivially, without bothering to …nd the replicating portfolios. Then if some bad trader comes to me and o¤ers me 93 cents for the 3 year zero coupon, then I’ll …gure out the replicating portfolio and take advantage of that o¤er to make a pure pro…t for sure. The replicating formulas are below:

100:1 = 101 (1) (9)100:2 = 2 (1) + 102 (2) (10)100:3 = 3 (1) + 3 (2) + 103 (3) (11)100:4 = 4 (1) + 4 (2) + 4 (3) + 104 (4) (12)100:5 = 5 (1) + 5 (2) + 5 (3) + 5 (4) + 105 (5) (13)

The logic behind this approach is the following: we don’t know what (1)through (5) are, but if you did know them, you’d be able to price the verybonds that the market is trading. So you would know that 100.1 had to equal101 times (1) . And you’d know that 100.2, the 2 year zero Treasury bond,

whose price is 100.2, would have to be 2 times

(1) plus 102 times

(2):

Because (1) is the price you pay today for 1 dollar 1 year from now, 101dollars, 1 year from now, costs 101 (1) . If you knew (1) and (2) , you could…gure out the price of the 2 year Treasury bond, because 2 dollars at time 1cost 2 (1) and 102 dollars at time 2 cost 102 (2) . And then the 3 year is 100.3= 3 (1) + 3 (2) +103 (3) , etc.

4



So you don’t know the 0 s , but you do know the bond prices, because themarket tells you, and you know the payo¤s of all the bonds, because that’s just written on them, literally, so you can just read what the payo¤s are. Youknow the government’s going to keep its promise. So rather than doing thecomplicated computations, trying to …gure out the 0 s , you assume you hadthe 0 s . And then if you had the 0 s , they would tell you what the prices of everything were.

However, given that we have …ve equations and …ve unknowns we can useExcel to …gure out this system of linearly independent equations.

0.1 Forward Interest Rates

The forward interest rates 1 + i f t is the number of dollars at t +1 in exchange for

1 dollar at t . This is like the interest rate that you might pay at time t . You giveup a dollar at time t, how much do you get at time t + 1 ? This is the interestrate from t to t +1 we are agreeing upon today. When time t arrives, somebodyis going to hand over one dollar, and when times t + 1 arrives, somebody elseis going to give back a certain number of dollars. This is called the period tinterest rate forward, because we’re locking it in today for a forward period of time, but it’s really just the normal time t interest rate for one year:

1 + if t =

t

t +1(14)

The forward rate is a trade-o¤ between dollars at times t and t + 1 . Pleasenote that the forward rates will always be a positive number even if the yieldcurve is downward sloping (i.e., nominal interest rates cannot be negative).Suppose that the yield curve is uppward sloping. The Excel spreadsheet detailsall the calculations required to …nd the forward rates. One can notice thatforward rates increase at faster rate than the yield rates and than the zero rates(remeber that y(t) = tq 1

1+ t1 or t = 1

(1+ y ( t )) t ) . In other words, the 4-yearyield for example is in a sense averaging the payo¤s of the …rst 4 years, whilethe 5-year yield is averaging it over 5 years. So if the 5-year yield has gone up,it means that the forward rate in year 4 has gone up a lot to bring the long runaverage up.

0.2 Interest Rates Under Certainty

If the world were one of total certainty, so everybody trading today had aperfect forecast of what was going to happen in the future, the forward rates inthe market today would have to be exactly equal to the forward interest rate.

To say it backwards, if you assume everybody knows for sure what is going tohappen in the future, then the forward rates would be exactly equal to whateveryone is expecting to happen in the future. To say it slightly di¤erently,if you happen to be the one ignoramus in the world who didn’t know whatwas going to happen in the future, but you knew that everybody else who wastrading in the market did know what was going to happen in the future, and

5



you saw a forward rate of 5 percent, then you could deduce, even though youwere an ignoramus, that actually 2 years from now, the interest rate was goingto be 5 percent.

What are the reasons that the yield curve is upward sloping? On one hand,Irving Fisher has already told us that when the market gets more productive,then one is more optimistic about what is going to happen later and the realinterest rate goes up. And if in‡ation is constant, and the real interest rate goesup, the nominal interest rate has to go up.

The other possibility is that the real interest rate stays the same, but thereis in‡ation in the future. The real interest rate plus the in‡ation is the nominalinterest rate. That is another explanation for why people might expect thenominal interest rate to go up.

6

lecture 5_yield curve arbitrage

Documents