Risk Latte - Interest Rates and Newton's Law of Cooling

Interest Rates and Newton's Law of Cooling

Team latte
Mar 11, 2006

Quite a few years ago one of our colleagues had attended a workshop on Interest rate modeling in the City of London . (He was a math grad student and had absolutely no interest in the banking world but somehow had accidentally landed up there, basically to meet up with one of his high school mates who was a banker in the City and was also attending the seminar.) The main issue of the seminar was as to whether interest rates in the economy were mean reverting.

The questions were flying all around: What is the half life of the short rate? What is the mean reversion speed of the short rate? How long will it take the current short rate to achieve the long term target? What volatility will cause the rate to achieve the long term rate? So on an so forth.

To him it appeared that all these guys were asking a very simple question: How long will it take a short rate to achieve a fraction of the value of the steady state value of the short rate?

The presenter, a quant from one of the leading financial institutions in the City, asked a question (something like this): say the short rate is 3% today and the long term value of the short rate is 12%. In six months time we observe that the short rate had inched up to 4%. How much longer will it take for the short rate to reach, say, 8% . Everyone in the room almost unanimously blurted out that this problem is incomplete or unsolvable because we don't know the volatility of the short rate and the mean reversion speed, etcetera, etcetera. Our man – the odd man out amongst the bankers and the quants – was fortunately carrying his HP scientific calculator and in under three minutes he stood up and said it would take the short rate another 3.44 years to reach 8%.

There was a stunned silence in the room. All eyes conveyed the same message: how the hell did you calculate this number? This is an incomplete problem, no volatility number, no mean reversion speed…what model did you use? When the speaker asked him how did he get the answer, he replied: simple, by Newton 's Law of Cooling. Here is how he formulated the problem:

If is the short rate and is the long term value of the short rate then the rate at which the short rate will move is given by:

where, is a constant and the solution of the above differential equation, in the limiting case, becomes the steady state equation for the short rate. Now, mind you, to all of you who are schooled in the theory and models of interest rates (the Ornstein-Uhlenbeck type of models such as Vasicek, CIR, etc) this equation will at best seem incomplete and to the fixed income evangelists even ridiculous. It is a deterministic equation with the stochastic term missing and therefore and completely inaccurate. Interest rates are a diffusion process, or so goes the theory, and the above equation is not a diffusion equation.

But for a moment assume that interest rates are totally deterministic and that the movement of short rate is completely defined by a steady state equation governed by other fundamentals parameters of the macro-economy, such as GDP, current account deficit, etc. Then the above equation is correct.

In fact, our man had no idea about these macro-economic variables, let alone think about a macro-economic state equation. He actually, thought about Newton’s Law of Cooling and substituted the short rate in that framework. Newton’s law of cooling describes the cooling of the temperature of a body. If a body with a certain temperature, say, is placed in an environment of temperature, say, then Newton’s law of cooling states that if such a body will cool according to the equation:

Our man just took a short cut and simply assumed that interest rates could be put in a "temperature equalization" model, a very simple steady state model. In fact, if we add a stochastic term, a Wiener process, to the above equation it will become a Vasicek model following Ornstein-Uhlenbeck process.

Therefore, if we want to make our lives very simple and assume that the interest rates are completely deterministic and given by equation (1) then integrating equation (1) above will give us the steady state equation for the short rate.

Using the above equation (2) we can answer the speaker's question easily. All that needs to be determined is the constant , which itself will be a deterministic value. We leave this simple piece of arithmetic to the reader.

Of course, the above model and the answer to the above question may or may not be right; and many quants will point out that this is a cheap trick to solve a problem, that otherwise cannot be solved within the established and conventional framework. And therefore totally incorrect! That may be the case, but the simple assumption helps us solve the problem, howsoever, inadequately.

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