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Computation of the Normal Yield Curve.

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Chapter I, Paragraph1:
Computation of the Normal Yield Curve.

"In this environment, long-term interest rates have trended lower in recent months even as the Federal Reserve has raised the level of the target federal funds rate by 150 basis points.

This development contrasts with most experience, which suggests that, other things being equal, increasing short-term interest rates are normally accompaniedby a rise in longer-term yields.

The simple mathematics of the yield curve governs the relationship between short- and long-term interest rates.Ten-year yields, for example, can be thought of as an average of ten consecutive one-year forward rates.
A rise in the first-year forward rate, which correlates closely with the federal funds rate, would increase the yield on ten-year U.S. Treasury notes even if the more-distant forward rates remain unchanged.
Historically, though, even these distant forward rates have tended to rise in association with monetary policy tightening.

In the current episode, however, the more-distant forward rates declined at the same time that short-term rates were rising. Indeed, the tenth-year tranche, which yielded 6-1/2 percent last June,is now at about 5-1/4 percent. During the same period, comparable real forward rates derived from quotes on Treasury inflation-indexed debt fell significantly as well,suggesting that only a portion of the decline in nominal forward rates in distant tranches is attributable to a drop in long-term inflation expectations"

Chairman Alan Greenspan
Federal Reserve Board's Semi-Annual Monetary Policy Report to the Congress.
Before the Committee on Financial Services, U.S. House of Representatives.
July 20, 2005



I am going to show that the normal yield curve is a function of the short term rates and of the volatility of interest rates. The obvious consequence is that you can't determine short-term rates (monetary policy) without taking in account long-term rates.

We are going to use a new model of long-term interest rate, which is rarely used: long-term rates as a geometric average of options on short-term rates. In fact

Interest rates as Options:

Fischer Sheffey Black first introduced this notion. It was his last work and was published after his death.

Interest Rate as Options, Fischer Sheffey Black, Journal of Finance, Vol. 50 No. 5, December 1995.

I propose a different use of the option point of view.

Forward Rates as Options:

R [t+ T] = Call [R[t], Vol R (t, t+T),0]

Where R [t] is the forward interest rate (annualized or continuouly compounded) for the period starting at t

Vol R (t, t + T) is the implied volatility of the interest for the period [t, t+T].

0 is the strike of the call option: obviously interest rates can't be negative.

The difference between R [t + T] and R[t] is hence the time value of an option.

The slope of the difference between two forward rates is hence the time value of an option.

The long-term rate being the composition of forward rates, the slope of the yield curve is hence a positive function of the time value of an option.

Locally if the option is undervalued the Market prefers the underlying asset rather than the option.

It prefers shorter-term maturity rather than longer term maturity.

This preference is local.

We derive from that our definition of normal yield curve steep yield curve and inverted yield curve.

These notions are global we will take them as approximate extensions of the local notions we have explored. We will hence be making an approximation: "I prefer to be roughly right than exactly wrong."

Normal Yield Curve:

A yield curve is normal if the options are fairly valued. There is Market indifference between the asset and the option. There is Market indifference between assets of different maturities.

Because time value of an option grows with its maturity, the normal yield curve is upward sloping. Because the time value of a call decrease as the value of the asset moves away from the strike price, the curve is convex.

The higher the implied volatility of interest rates the steeper the normal yield curve.

Inverted Yield Curve:

A yield curve is inverted if the option is undervalued. Long-term rates are too low compared to short-term rates. Because of the inverse relationship between present value and interest rate, there is Market preference for the short-term asset over asset of longer maturities. It is what Keynes termed liquidity preference.

Our definition says that a yield curve may be upward sloping and inverted. A flat yield curve is necessarily inverted because the time value of an option is always superior to zero (unless of course volatility is zero!!!).

Steep Yield Curve:

A yield curve is steep if the option is overvalued. Long-term rates are too high compared with short-term rates. There is Market preference for asset of longer maturity over asset of shorter maturities.


Because we saw that the yield curve can stay inverted for quite an extended period of time the arbitrage of the yield curve is not straight forward (punt intended.) However it is a risk free arbitrage which can be done by buying the spread long-term yield / short-term yield and shorting a complicated option on interest rates. I will let the Quants of the various hedge funds do that for us. It is their job to maximize the present value of future cash flaws.

All of This Stays True Until the Poor Becomes Richer Relatively to the Rich.

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