*Submitted by Omer Rosen of Legedermath*, *who will gladly answer any reader questions related to the matter presented in the comment section.*

**Legerdemath II: Anatomy of a Banking Trick**

In my previous article, “Legerdemath: Tricks of the Banking Trade,” I made brief mention of Treasury-rate locks:

Most brazenly, we taught clients phony math that involved settling Treasury-rate locks by referencing Treasury yields rather than prices.

A number of readers expressed a doubt that using a settlement method based on Treasury prices was appropriate. What follows is as good an explanation of Treasury-rate lock settlements as 2,000 words will allow. I have simplified some of the bond math and concepts and will end with an analogy that I hope will elucidate what the math did not. However, as this post hardly qualifies as an easy read, feel free to ask questions in the comments section. Confession: I fudged the word count a few sentences ago to increase the likelihood of you reading on.

Forget for a moment, everything you have heard or think you know about Treasury bonds. Taken in isolation, the purchase of a Treasury bond is nothing more than the purchase of a fixed set of future cash flows. If you find the term “cash flows” confusing, think instead of the following: buy a bond today, receive predetermined amounts of money on predetermined dates in the future.

In this column I will be referencing a 10-year Treasury bond paying a coupon of 5.00%, with a notional amount of $100. For convenience, I will christen this bond “Bondie.” Sans jargon, the fixed set of cash flows received when purchasing Bondie would be $2.50 every 6 months for 10 years and an additional $100 at the end of the 10th year.

There are two basic ways to describe the value of this fixed set of cash flows, either by price or by yield. Price answers a simple question: How much would it cost you to purchase this fixed set of cash flows? This price will change over time, in much the same way that the price of a stock changes over time. Yield expresses the return earned by purchasing these cash flows at a certain price.

If you had to pay $100 in order to receive the fixed set of cash flows I described above, then your yield would be 5.00%. If you had to pay more to purchase these same cash flows, say $105, then the return you would be earning (the yield) would be lower than 5.00% – it would be 4.3772%. Intuitively this should make sense – the more you have to pay for a given set of cash flows the lower your return will be. Or, more simply, when prices go up, yields come down. Conversely, if you had to pay only $95 for these same cash flows, the yield earned would be higher than 5.00% – it would be 5.6617%.

Algebraically speaking, price and yield are linked by an equation where all the other variables are known. Therefore, if you know the yield of a given bond you can calculate the price of that bond and vice versa. In plain terms, saying you are willing to pay $100 for Bondie is the same as saying you are willing to buy Bondie at a yield of 5.00% (i.e. at a price that will allow you to earn a return of 5.00%). It is similar to how one can describe the speed of a car either by the number of miles per hour it is traveling at or by the time it takes it to travel one mile – if you know one you can solve for the other, and if one goes up the other comes down.

To belabor the point, if a car is traveling around a 1-mile track at an average speed of 1 mph then it is easy to solve for the time needed to complete a single lap: 60 minutes. Either “1 mph” or “a 60-minute mile” provides you access to the same knowledge about the speed of the car during that lap. And, if the car’s speed were to increase, the time it would take to complete another lap would decrease (At 2 mph a mile would only take 30 minutes). The same inverse relationship holds true between prices and yields.

Now back to Treasury-rate locks. When a company puts on a Treasury-rate lock, it is doing nothing more than taking a short position in a Treasury bond. A short position is a bet that will pay off for the company if Treasury prices go down and go against them if prices go up. Why would they do this? That is a subject for another column and I ask that you accept as an article of faith that sometimes this bet, rather than being a gamble, reduces risk and uncertainty for a company.

The short position can be viewed as an agreement under which the client will sell the bank Treasury bonds at a certain price on a set date in the future. This price is determined based on current market conditions. For example, let us say, that based on what current market conditions dictate, the client agrees to sell Bondie to the bank at $95 one month hence. A month passes and Bondie is now trading at $100. The client will have to go into the market, buy Bondie at the current price of $100, and then sell it at a loss of $5 to the bank at the previously agreed upon price of $95. For expediency’s sake, the client just pays the bank the $5 it has lost and the bank takes care of all the buying and selling behind the scenes. The calculation of $5 in the above manner – subtraction – is an example of the price-settlement method of Treasury-rate locks.

However, when it comes to bonds, corporate clients do not think in terms of price; they think in terms of yield because yield is expressed in the language of interest rates, the same language companies are familiar with from business concepts such as rates of return and borrowing costs. In theory, this should add only a simple step to the settlement process. The company locks in a sale of Bondie at the same level as before, $95, but rather than quoting them that price the bank quotes them the corresponding yield of 5.6617%. We can refer to this yield as the locked-in yield.

A month passes and the Treasury rate lock is settled. Rather than telling the client that Bondie is now trading at $100, the bank tells them that the yield is now 5.00%, having fallen by 0.6617%. But 0.6617% is not a dollar value that can be paid out as a settlement. To calculate the settlement, both yields, 5.6617% and 5.00%, need to first be converted back to their respective corresponding prices, $95 and $100. Taking the difference between the two prices results in the same settlement value we calculated before: $5.

But the client is never shown how to settle based on prices. Instead they are introduced to a nonsensical and more complicated method called yield settlement. The sole purpose of this settlement method is to trick the client into allowing the bank extra profit.

Whereas price settlement asks the question, “By how much did Treasury prices change?” yield settlement asks, “By how much did Treasury yields change?” As mentioned in the previous paragraph, the yield decreased by 0.6617%. But how does one convert 0.6617% into a dollar value that can be paid out?

First, a unit conversion is necessary. For clarity and convenience, finance makes use of a unit called a basis point. Each basis point is equal to 0.01%. Using this new unit, the above decrease of 0.6617% can be expressed as 66.17 basis points. Of course, this solves nothing, only modifying our most recent question slightly: now we ask, how much is each of the 66.17 basis points worth in dollar terms?

At this point the client is introduced to a concept called DV01 (Dollar Value of One Basis Point). DV01 is defined as the change in price of a bond for a one basis-point change in yield. For example, if the yield on a bond changes from 5.00% to 5.01% or from 5.00% to 4.99%, by how much would the corresponding price of that bond change? This change in price is the DV01. If yields shifted by 66.17 basis points, DV01 will answer the question of how much each of these basis points is worth.

The starting point for this calculation is the yield at the time of settlement. In our example, the yield at the time of settlement is 5.00%. At this yield, the corresponding price of Bondie is $100. If the yield were to rise by one basis point to 5.01%, the corresponding price of the bond would fall to $99.922091, a decrease of 7.7909 cents. If instead the yield were to decrease by one basis point to 4.99%, the corresponding price would rise to $100.077983, an increase of 7.7983 cents. By convention, the average of these two changes in bond prices is taken to be the DV01. So, at a yield of 5.00%, the DV01 would be 7.7946 cents per one basis-point move ((7.7983 7.7909) ÷ 2). If the yield changes by one basis point, price is said to move by 7.7946 cents. Or, in more plain terms, each basis point has been assigned a value of 7.7946 cents.

The DV01 is then multiplied by the difference between the current yield and the locked-in yield. In our example the difference between 5.00% and 5.6617% is 66.17 basis points. From the previous paragraph we know that each of these 66.17 basis points is worth 7.7946 cents. Multiplying 66.17 by 7.7946 cents we arrive at a settlement value of $5.1577. This is the yield-settlement method of Treasury-rate locks.

Apart from being confusing, the yield-settlement method has resulted in a settlement value that is greater than the $5 calculated using the price-settlement methodology. For a good-sized rate lock, say $500 million dollars worth of 10-year Treasuries, the client would pay the bank an extra $788,500 (500 million x (5.1577 – 5.00) ÷ 100) when settling using the yield-based methodology. This “extra” is profit for the bank.

I ask that you stop reading here for a moment. I have stated from the beginning that yield settlement is incorrect. However, when reading the explanation of yield settlement, did you find yourself agreeing with the logic? At what point, if any, did you spot the flaw? And can you guess what happens if prices had gone the other way? If prices had gone down instead of up, say to $90, the bank would have owed the client money. However, yield settlement would have allowed the bank to earn a profit by paying the client less than it actually owed them. No matter what happens to prices, yield settlement allows the bank to earn extra profit.

Now picture yourself as a client receiving a tutorial on Treasury-rate locks. You are being instructed by a banker on a matter that seems procedural, in a manner that seems advisory and helpful, without any warning that something might be amiss. You are led through the yield-based settlement process and taught how the DV01 is calculated. If you have access to a Bloomberg terminal you are shown where the DV01 can be found on the relevant Treasury bond’s profile page. Perhaps presentation materials are sent over detailing the mechanics of rate locks and different possible outcomes depending on various possible market movements. And all this is part of a larger interaction, a relationship even, during which the banker is nothing but genuinely friendly and informative. Furthermore, there is a good chance that someone from a different part of the bank, someone who has advised you before, was the one that introduced the two of you in the first place. Would you question your banker?

Clients, among them some of the largest corporations in the world, never did. Confident in the tools provided them and blinded by specious logic, the client never even thinks to question the underlying methodology. And, especially since the client is never made aware of price settlement, the methodology does sound logical: Check to see by how many basis points Treasury yields moved. Calculate the dollar value of each basis point. Multiply the two and arrive at a settlement value.

However, this methodology is an approximation that always works out in the bank’s favor. Why? Because each of the 66.17 basis points has erroneously been assigned the same value of 7.7946 cents. The DV01 calculated at a certain yield is only valid for a one basis-point move away from that yield. Therefore, while the first basis-point shift away from 5.00% is indeed worth 7.7946 cents, successive ones are not.

Put another way, DV01 at 5.00% is different than DV01 at 5.01% is different than DV01 at 5.02% is different than DV01 at every other yield. And so the value of the basis-point change from 5.00% to 5.01% is different than the value of the basis-point change from 5.01% to 5.02% is different than the value of all successive basis-point changes. In fact, even the original DV01 is inaccurate because it was taken to be an average of two different movements. Multiplying the 66.17 basis-point change by a single DV01 ignores all this and assumes that the relationship between changes in yield and changes in price is constant – that each one basis-point move results in a fixed change in price no matter what the yield. Yield settlement takes the graphical representation of the relationship between prices and yields – a curve – and flattens it into a straight line.

Admittedly, all this can be a bit confusing. After all, if price and yield are both valid ways of expressing the value of a bond, shouldn’t you also be able to measure the change in value of a bond by looking at either the change in its price or the change in its yield? The math says no. Resorting to hyperbole, teaching the client yield-based settlement is akin to selling them on time travel.

Return for a moment to the example of a car driving along a 1-mile track (a conceptual, though not mathematical, equivalent to rate lock settlements). In this analogy, “mph” will play the role of “yield” and “travel time” will play the role of “price.” Assume the car is traveling at a speed of 1 mph. If the car speeds up to 2 mph, the time required to travel a mile decreases from 60 minutes to only 30 minutes – a 30-minute decrease in travel time. This 30-minute change plays the role of “DV01″.

Now assume that the car is traveling at a speed of 120 mph. If again the car’s speed increases by 1 mph, here to 121 miles per hour, does the time needed to travel a mile again decrease by 30 minutes? Since a mile only takes 30 seconds to complete at a speed of 120 miles per hour, short of a DeLorean and some lightning, reducing the completion time by 30 minutes would be impossible. The actual reduction in travel time – the “DV01″ – would be only a fraction of a second at this high speed. “DV01″ is not a constant in this analogy either.

To extend the analogy, calculating a rate lock settlement would be akin to calculating the difference in travel times for each of two laps. If lap 1 were completed at a speed of 120 mph and lap 2 at a speed of 1 mph, how would you calculate the difference in travel time between the first and the second lap? Would you take the difference between 120 mph and 1 mph and multiply that difference by the 30-minute “DV01″ calculated above? Doing so would imply an impossibly high difference between the two lap times: 3,570 minutes ((120 – 1) x 30). This calculation is the parallel of the yield-settlement method.

For makes and models without a flux capacitor, you would simply look at the difference between the times the car took to complete each lap. If a stopwatch is not handy, the following quick math provides the answer: a 120-mph lap takes 30 seconds to complete and a 1-mph lap takes 60 minutes to complete. The difference in travel time between the two laps is therefore 59.5 minutes. This calculation is the parallel of the price-settlement method. As you can see, the 3,570 minutes calculated using the other method is far off the mark.

In price/yield relationships the same problem exists – that problem being the realities of math. Yet banks I encountered almost always instructed clients to use the yield-based settlement method. And so a product that is meant to return the difference between two Treasury prices, a matter of elementary subtraction, is perverted for profit.

If yields change by very little, this profit does not amount to much. Fortunately, depending on one’s point of view, banks have other tricks for profiting from rate locks and do not rely solely on yield-based settlement. In fact, miseducating clients with yield-based settlement is almost an afterthought, just a bonus that pays off with large movements in yield. Because as yields move by more and more basis points two things happen: First, there are more basis points to infect with an erroneously constant DV01. Second, the constant DV01 becomes an even worse approximation for the proper DV01 of each basis point.

In behavior that might be considered yet more sinister, sometimes banks had to implicitly agree with one another to use yield settlement. This transpired if a client decided to divvy up a single rate-lock transaction, with each bank getting a piece of the deal and each bank knowing that settlement of the rate lock would have to be a coordinated affair.

All this mathiness is hidden in plain sight. Some examples of yield settlement can be found online. Or you can just ask a company that put on a rate lock to dig up some trade confirmations and see what settlement methodology was used. There are hundreds, if not thousands, such documents in corporate offices around the country, each one part of an unwarranted transfer of millions of dollars from clients to banks.