Risk Latte - Yield on Long Term Bonds - A Mathematical Mystery!

Yield on Long Term Bonds - A Mathematical Mystery!

Rahul Bhattacharya
April 8, 2005

Recently there was a heated debate amongst some senior strategists in top banks in Hong Kong about whether the crude price hike would unleash a 70s style inflationary scenario in the world and whether the long bond yields would go through the roof. One of our analysts was present in that rather informal Ivy League meeting of the titans purely by accident (this analyst of ours understands nothing about financial economics, if there is anything to be understood at all, but knows a lot about food and wine and a little bit math).

Anyway, these guys were fixated with long bond yields and its impact of the economy (or in other words, their jobs). But something they said caught our man's attention. Someone in that august gathering mentioned that the yields on long term bonds could remain exceptionally high ˇV much higher than the short term rates ˇV for a long period of time. There was also the mention of a future long run average interest rate and the suggestion that the long bond rates could remain above the future average rates as well for a long time.

Perhaps, the above fact is obvious or trivial to a lot of economists and strategists working in the industry, but to us this was a bit baffling. It is an interesting point indeed and we were convinced that the answer lies not in any macro economic theory but rather in simple bond mathematics.

For the last ten days we have ransacked our online Stacks (research library) and in house library, gone on the internet to look for answers and discussed with some very illuminated minds in the banks here in Asia. But to no avail.

Then all of a sudden, by chance, one of us came across a brilliant paper by Neil D. Pearson where the research of Dybvig and Marshall (1996) was quoted. We then looked up the original research by these guys.

All economists and financial theorist assume perfect knowledge about models and the parameters that determine those models. But investors don't understand and know models, they behave according to their own set of rules, preferences and utility; traders buy and sell bonds (for that matter any asset) based on the market conditions and their preferences and utility. The math of any ˇ§modelˇ¨ is important but that math is in the head of a trader or an investor. And more importantly, the math is the ˇ§math of probabilityˇ¨ ˇV the probability of a certain price being achieved, or the probability of hitting a certain interest rate.

So as far as the long bond yields are concerned, let's look at some of the simple math behind the bonds. Uncertainty on models and parameters mean that interest rates follow probabilistic paths. The simplest of a probability path is a binomial tree model.

Assume a binomial model with probabilistic outcome (a binomial tree is where the price of an asset can go up or down) where the price of a long-term zero coupon bond can go up to a value x or go down to a value y where y > x (y is greater than x). Let the probability of the up move for the bond price be q and the probability of the down move be (1-q).

We are making an assumptions that market participants lack the perfect (or any) knowledge about the term structure model of interest rates and how the yields are distributed across maturities. This is a fair assumption.

Therefore, the price of a long term zero coupon bond (assume $1 notional) will be:

P(T) = qe-XT + (1-q)e-yT

Because y is greater than x as T (time to maturity of the zero coupon bond) becomes large, the first term in the above equation becomes dominant. In particular, if we take the limit of the ration of the two terms we get infinity as shown below.

Therefore, in the limiting case the yield on a long term zero coupon bond would be simply x, as shown below.

Now since y is always greater than x, the yield on the long bond x is less than the expected interest rate (average interest rate) as shown below.

The above demonstrates that the price of long term zero coupon bond is dominated by interest rate paths with the lowest average interest rate. Also, uncertainty about the model and its parameters cannot explain high yields on long term bonds.

In fact the above proves that uncertainty (or in math terms probability) about the model and its parameters necessarily implies that the yield on long bonds should be less than the expected (average) future interest rate.


Reference: Fixed-Income Subtleties and the Pricing of Long Bonds by Neil D. Pearson; Advanced Fixed Income Valuation Tools by Bruce Tuckman & Narasimhan Jegadeesh; Dybvig and Marshall (1996)

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