## Assume all Bonds are Spheres: Part I

*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.

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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 yieldFormula 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 – 1Algebra… 😦

1+ (r/12)) = (1+5%)^(2/12)Our answer!

r/12 = (1.05)^(2/12) – 1

r = 12 * ((1.05)^(2/12) – 1)

r = 9.79781526%

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.

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**Tags:** approximations, arbitrage, assumptions, bonds, conventions, daycount, imprecision, interest rates, loans, math, precision, profit, quantitative, spheres, yield

February 16, 2008 at 8:18 pm

I’m a student of the financial markets at university who found your blog through Felix Salmon’s. Just wanted to comment on how insightful and educational I’ve found the blog to be, keep up the good work.

February 17, 2008 at 6:40 pm

Thanks so much! Being new to this I love hearing any and all feedback. Please keep it coming. Also, let me know what topics you would like to hear about.