Posted tagged ‘math’

The Easiest Hardest Question Ever

January 7, 2009

I was reading Felix’s post, recently (I know, I’m behind on everything… I’m posting on that soon too), where he cites a Tanta piece on negative amortizing loans. And it prompted me to have a very specific memory.

Here’s my email to Felix…


Subject: Here’s something funny-scary …

… that your post on option ARMS got me thinking about. No one, and I literally mean NO ONE, who works in securitized products knows very basic things about the loans, as you touched on. But the people securitizing the loans and selling the bonds don’t know very basic things that fall under the “you should be shot for not knowing something this basic” category… Here’s what I asked a whole bunch of these master’s of the universe and none of them knew the answer, they all guessed.

“When I, as an individual who has a mortgage on my home, have a fixed rate amortizing loan on a thirty-year amortization schedule, and I send in a curtailment (excess principal payment that doesn’t pay off the loan but reduces the principal balance faster than scheduled) what happens to all the subsequent payments?”

Here’s why this is tricky…

1. If you curtail the loan then your interest payment reduces. However, this means your payments are no longer “level” … They change from month to month. This is because amortization schedules set based on simple interest computation (rate*loan balance) but the principal is set to keep the payments level. When you curtail the loan, you destroy this balance.

2. If your interest payment reduces once, but the overall payment doesn’t change, then you have a loan that starts to amortize much faster than before. Why? Well, the bank can’t charge you interest for that month on a principal balance that is lower, right? So if your payment is “x” and you paid off 5% of your loan, because the interest portion of your payment is 5% lower, if the payment hasn’t changed that money that would have gone to interest on the paid off amount goes to principal repayment. This compounds the same problem for next month’s payment.

3. No one was aware of loans being recast. It doesn’t seem to be the case that loans are recast once someone sends in more than their payment, and it also doesn’t seem to be the case that loans are recast on any sort of schedule (annually, for example). Not a single person thought this happened.

Most common answer was “principal balance goes down” …. And once the details were asked? One usually got a hand wave and an answer of “Curtailments are so rare, this is unimportant.” Even the people modeling the actual cashflows didn’t understand what happens to loans when curtailments come in. They would model it as a partial prepayment of the pool, but not alter anything else (after all, curtailments are rare! why bother modeling them correctly?). Ha!



I did call some mortgage companies and it seems they do “turbo” the loan, essentially, by keeping payments level and applying more to principal … But this, obviously, makes it les than a 30-year loan. However, some mortgage companies will allow you to recast the mortgage totally for payments that are large enough.


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.