Pure Expectation Theory

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Background:

The method of using bond yield curve to understand future interest rates is more meaningful as it allows us to focus primarily on what are the expectations of future interest rates when risk premium is ignored.

Explanation:

Evidence does suggest that there is a positive risk premium (MRP) but this assumption can be ignored if it is Term Structure-Pure Expectations Theoryassumed that the market is dominated by traders who trade securities of multiple maturities each day, without being concerned by maturity risk.  Thus a trader/investor would be willing to buy a 20-year maturity as much as they would be willing to hold a 3-month maturity if they both yielded the same profit.  Strict proponents of the yield curve thus suggest that the shape of the yield curve is determined solely by the market expectations of the future interest rate.  This approach has been called “Pure Expectation Theory.”  Thus this theory contends that bond traders establish bond prices and interest rates strictly on the basis of expectations of future interest rates and that they don’t factor in maturity risk premium (MRP) as they don’t view long term bonds being riskier than short term bonds.

Illustratively this means:

Option 1: Buy a 2-year security (bond) and hold it for the stipulated time frame

Option 2: Buy a 1-year security (bond) and then by the end of the year buy another 1-year bond

1-year security yield = 6%

1 year security yield 1 year from now = ?

Option1: Buy a 2-year security.  2-year security yield = 6.5%

Option 2: Buy a 1-year security @ 6% and then at the end of year 1 reinvest the proceeds in another 1-year security

As per Option 1:

Funds at the end of the 2nd year = (1.065) ^2 = $1.134225

As per Option 2:

Reinvestment rate (X) at the end of 2nd year       = $(1.06) *(1+X) = $1.134225

X = 7.0 %

Thus if expectation theory is correct then the 2nd year return 1 year from now would be 7.0% and points to an upward sloping yield curve.  In the absence of MRP, an upward sloping yield curve implies that the market has an upward sloping term structure.

Now what would happen if the term structure was downward sloping.  Thus investors expect the treasury rate to be 7.0% 1 years from now but the 2-year bond yields reduce to 6.25%.  This scenario offers arbitrage potential as:

Traders can buy a 2-year bond and then 6.25%

Invest in a series of 1 year securities

Rate of return of investment = (1.06*1.070) ^0.5-1 = 6.5%

This would enable the investor to earn a return of 6.5% when the cost of borrowing is 6.25%.  They would do this by selling of 1-year bond, which would increase supply of these bonds and reduce their interest rate and by demanding more of 2 year bonds thus increasing their demand and interest rates.  In time the arbitrage potential would diminish and the net effect would be that the market would re-equilibrate.

But mostly reality there is always a positive MRP.  For example, the MRP for a 2-year bond is 0.2%.  This premium means that the expected annual rate of return on a 2-year bond would be 6.3%:

Expected Rate of Return on a 2-year bond =       Rate of Return on a 2-year bond – MRP

6.5% – 0.2% = 6.3%

This scenario would reduce the 1 year forward rate as follows:

(1 + Y1) (1+Y2) = (I+Y12) ^2

(1+0.06) *(1+X) = (1+0.063) ^2

1.06X + 1.06 = 1.129969

X = 6.6%

Thus in the case of an MRP the 1 year forward rate reduces from 7% to 6.6%.

As per scenario 1 & 2 the yield on a 1 year is 6% and rises by 0.5% to give a yield in year 2 of 6.5%.  Of this increase in yield MRP constitutes 0.2% and the remaining 0.3% is attributable to in the expected 1-year rate next year.

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