# Chapter 18: Interest Rate Models

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How would one be expected to approach the following question:

5. Suppose that the yield curve for zero-coupon bonds of a certain type is always flat. Show by means of an example that this can create arbitrage opportunities. Describe in full how the arbitrage would occur, and how this would tend to alter the yield curve.

Is it appropriate to draw a flat yield curve and use a mix of long term bonds and short term bonds to show that if the yield is the same for all the terms, then long term bonds would be cheaper than short term bonds thus resulting in arbitrage opportunities as people would sell short term bonds and buy the long term bonds. This would then result in the demand for long term bonds increasing hence the price of long term bonds would also increase. The yield curve would then be downward sloping because long term bonds would then have lower yields than the short term bonds.

As a follow up question, what would lead to a hump in our yield curve?

+1 vote
answered May 31 by (740 points)
edited May 31

(As background, recap the following:

If $$P_T=e^{-rT}$$ is the price of a zero coupon bond paying 1 at time $$T$$, the price change when the rate goes form $$r$$ to $$r+dr$$ is, by Taylor, approximately $$P_T(-T \cdot dr +1/2 T^2 dr^2)$$. For a collection of bonds (which can be broken down into a collection of zero coupon bonds) with value B, this translates to $$B(-D \cdot dr +1/2 C^2 dr^2)$$, where $$D$$ (for Duration) is some weighted average of term and and $$C$$ (for Convexity) is a weighted average of term squared.

Looking at this last equation, we see that if two bond portfolios have equal Duration, but their Convexities are different, the one with the higher Convexity will give a higher return, for a small parallel change in yields.

End of background)

For the question, the arbitrage results from the yield curve moving from one flat level to another flat level. Two portfolios with the same duration but different convexities will deliver different returns - the portfolio with the higher convexity will give a higher return. This result would tend to make longer-dated bonds more popular than medium-dated bonds, pushing up long-dated bond prices, pushing down long-dated yields. An opposite effect results in the medium-term space, causing a hump shaped yield curve.

For the example, construct one portfolio consisting of a medium term bond and the other portfolio consisting of two other bonds, one with a low term and one with a high term. Pick them such that their durations are equal.

(It would not be correct to suggest that lower prices of longer-dated bonds mean they are cheap.)

commented Jun 1 by (760 points)

Would holding an immunised portfolio with short positions in long-term bonds and long positions in shorter term bonds with positive convexity not work if there is an inversion of the yield curve? I understand it would work with a parallel rates shift but would something like an increase in the repo rate, pushing short term rates up from this point negatively affect this portfolio?

commented Jun 4 by (740 points)
edited Jun 4

The original question set-up just wanted to show that yield curve being flat and going up and down cannot describe an arbitrage-free interest rate space. A possible question then is what does an arbitrage-free interest rate space look like. (The one-factor models give some examples). I think the question you are addressing is, what kind of portfolio is "immunised" against changes in steepness and the absolute level. (?) One can try r=a+bT and see what the portfolio value does when a and b change (express change in terms of duration and convexity), i.e. derivatives with respect to a and b. Try it and let us know what you find.

commented Jun 4 by (760 points)

If $$P(t,t+\tau)=\exp(-(a+b\tau))$$ then $$\frac{\partial P(t,t+\tau)}{\partial a} \frac{1}{P(t,t+\tau)}=-a$$ and $$\frac{\partial P(t,t+\tau)}{\partial b}\frac{1}{P(t,t+\tau)}=-b\tau$$ while the duration is $$b$$ and convexity is $$b^2$$. So my thinking is that a portfolio such that $$\sum x_i \tau_i=0$$ would be protected against changes in the slope of the yield curve while if the duration of any portfolio regardless of the yield curve is 0 then it will be protected against vertical shifts in the yield curve. Not sure if my thinking makes sense though