Posted tagged ‘arbitrage’

Assume all Bonds are Spheres: Part I

February 14, 2008

This segment is a regularly occurring feature. It gets its name from a joke that is commonly made about technical people. Usually a very simple problem is presented about horses (or sometimes cows) and a physicist/engineer/mathematician is asked to provide a solution. The solution is made complex because incorporating the nuances of the animals in question is extremely difficult, hence the highly technical person is required. The solution then starts with the technical person saying, “I made two assumptions. The first assumption is gravity. The second assumption I made was that all horses are spheres.” The crux of the joke, which is often found in the markets, is that approximating is both easier and “accurate enough.” Hence the naming of this series on mathematical financial tricks and other interesting tidbits.

Loans are simple enough. I tell you that I would like to originate a loan, which I will securitize, and your interest rate on a loan is going to be 10%. Easy! You just send in 10% of the loan amount every year. Well, not really:

  • If I followed convention, I quoted you an Act/360 rate (it accrues yearly based on the actual days, but assumes a 360 day year)–the interest you pay is really 10.14% (~ 365/360 * 10%). (Note that the lower the interest rate, the lower the impact of converting to Act/360.)
  • If this was a 10 year loan, those 14 basis points (bps, 1/100th of a percent) are worth about 1.11% of the total notional of the loan (don’t forget, that’s 14 bps extra one pays per year, for 10 years–take the present value of all those cashflows).

Now, let’s look at what happens when I securitize the loan.

  • I monetize the 14 bps because the bonds I sell accrue on a 30/360 basis (market convention), so those extra days of interest never get paid to bondholders–I keep it.
  • Another nuance: The bonds are priced to the market convention, which is assuming a semi-annual yield. Investors will demand, say, a 10% coupon on their bonds. The bonds, though, match the loan–they pay monthly. What is the 10% worth on a monthly basis? About 9.80% (see below). What do I earn on arbitraging the difference? Well, about 20 bps per year, which, present valued, is worth about 1.59% of the loan amount.

Let’s review: I quoted you a loan at 10%. The 10% tuned out to be more than 10%. I then sold the loan using a completely different set of assumptions and, doing nothing at all, managed to pocket 2.7% of the loan amount. On $10 million dollars that’s $270,000 I made just for arbitraging various conventions and the difference in how bond investors think and how lenders generally think. Would you have been better off to take the 10.05% loan from the insurance company, quoted on a 30/360 basis? Yep. But the rate was lower … and all bonds are spheres.


The voodoo behind the 9.80% monthly equivalent coupon is easy. The first insight is to recognize that a yield is essentially a discount rate. To find a monthly equivalent to a semi-annual yield one needs only to find the interest rate that, when compounded twelve times a year, equals the yield that assumes compounding twice a year (the semi-annual yield).

We start by saying that an annualized yield, assuming compounding x times a year is

     (1 + i/x)^(x) – 1 = annualized yield       Formula that codifies the above intuition.

For our example of a 10% semi-annual yield, we get the following:

ann. yield (12 pmts.) = ann. yield (2 pmts)
(1 + (r/12))^12 – 1 = (1 + (10%/2))^2 – 1 
1+ (r/12)) = (1+5%)^(2/12)                      
Algebra… 😦
r/12 = (1.05)^(2/12) – 1
r = 12 * ((1.05)^(2/12) – 1)
r =  9.79781526%   
                                   Our answer!

And that’s how I arrived at that solution. I’m sure I’m missing something minor that excel would do for me, but you get the idea.


Mindset Arbitrage

February 6, 2008

One interesting observation about the securitization process is that many of the securitized products are meant to aggregate a risk so it can be distributed to capital market participants (hedge funds, opportunity funds, insurance companies that buy securities, and money managers and banks).

For example, Commercial Mortgage Backed Securities are used to source risk from real estate investors that own and operate real estate assets. This is a cash flow based market where people that manage buildings have a well defined plan for extracting more dollars to the bottom line (raise rents, lower expenses, re-capitalize the entity that owns the building, etc.). When your day-to-day business is servicing tenants, collecting rents, staffing a building, and maintaining the property you aren’t use to thinking about duration or what rating the rating agencies have assigned your loan. Things like day count conventions and their effect on the yield vs. the coupon isn’t your expertise. These are all esoteric things that some traders and bankers spend years not understanding in any rigorous way.

Residential Mortgage Backed Securities source risk from homeowners and mortgage originators. Here people’s assets secure loans and allow risk to be quantified by analyzing statistical data which, in aggregate, should hold closely to some historical trends. One can view how high LTV loans made to borrowers with a FICO score between 600 and 620 have performed in a given state over the past five years and extrapolate the performance of other loans with the same characteristics (recent problems arose when people didn’t care about the loan characteristics, just that they could get bonds or what rating the bonds had).  I don’t think I have to explain that, given all the examples in the media trying to explain CDOs (but really securitized products in general), it should be Q.E.D. that people taking out mortgages didn’t fully understand how the risk was being laid off nor how it would affect them (even the banks got sloppy).

Even corporate CDOs took instruments from a market where securities are valued based on the fundamental credit quality and technical factors and sold instruments that were valued using correlation trading models.  Obviously this is a bit different since these are all sophisticated (theoretically) institutional accounts. However, no one was prepared for, nor did they understand fully, the implications of things like the Ford and G.M. downgrades to junk or how the price movement in CDO tranches was going to show a disconnect because of the types of accounts that held them. (What this means is that the lowest tranches, which take losses first, become the worst risk in the pool. Essentially, then, these “equity tranches” became proxies for Ford and G.M. Since these equity tranches were held by accounts that have to mark to market, they dropped in price to reflect liquidity concerns. The tranches above them were generally held by insurance companies, or non–mark to market accounts. Dealers, proprietary trading desks, and hedge funds had all put on trades that went long equity tranches and short the tranches above them assuming the relative spread would move predictably. Unfortunately, for them, there was no price movement in the tranches they were short, since those accounts didn’t have to mark to market, and thus didn’t need to sell to prevent further losses while the equity tranches dropped in value.)

Even starting a fund is an exercise in mindset arbitrage. Some manager looks to buy assets, like mezzanine debt, which generally trades on price or as a spread to a benchmark interest rate, by raising equity for a fund. All they are doing is taking the returns they know are out in the capital markets, adding leverage, and going to investors that are very focused on R.O.E. (return on equity). This is probably the worst example here, despite being quite valid, because the level of sophistication can be quite high. However, with lots of money chasing high returns, sometimes due diligence and other metrics of reliability and quality fall by the wayside.

It should become apparent, now why there is money to be made re-packaging risk–essentially you are outsourcing the placement and structuring of that risk so that the person who is willing to get paid the least for taking a particular risk gets it. This is why it is possible for one to sell securitized products at a net interest rate less than the underlying assets.