Expectations Theory, Q2. Set out the formula for the five-year bond rate in term

Question

Expectations Theory,

Q2. Set out the formula for the five-year bond rate in terms of the one-year ra e today and one-year rates in the future using the Expectations Theory (ET a. b. Briefly explain whether the Expectations Theory (ET) can capture both ( the tendency of short and long-term rates to move together and (ii) the tender cy of the yield curve to flatten just prior to recessions? 1131 Assume now that risk averse investors require a risk premium over the re urn given by the pure Expectations Theory to hold long-term assets. Today's one year bond rate is 0.5%, the expected one-year rate in year 2 is 1%, in yea 3 it c. 2 is 1.5%, in year 4 t is 2% and in year 5 it is 2.5%. If today's 5-year bond rate is 2.5%, estimate the risk premium that investors require to hold a 5-year bond

Explanation / Answer

a)

Expectation theory assumes that bond traders establish bond prices and interest rates strictly on the basis of expectations about future interest rates, and they are indifferent to maturity because they do not view long-term bonds as being riskier than short-term bonds.

So essentially holding a bond for two years or holding it for one year and the reinvesting the amount at the end of year one will yield the same amount.

A five year bond rates can be thus made using this theory. The formula for the same will be as follows-

Here Zt represents the spot rate which we need to calculte which will be year 5 in our case.

Z1 is the present one year rate.

f are the forward rates which will prvail at the end of each year.

So if one knows the on year rate in the present and the future one year rates using this formula the five year rate can be easily calculated.

b)

According to the expectations theory the long and short term rates move together because the long term rate is just an average of the short term rates.

So, for example, if people expect that short-term interest rates will be 10% on average over the next two years, then the interest rate on 2-year bonds will be 10% too.

If the investors are anticipating a recession the short term rates will decline as there would be credit weekness and less demand. Now according to the expectation theory the lomg term rate is just the average of short term rates. so as the short term rates go sown the long erm rate will also be coming down leading to a flatening of the yield curve as the long term rates wil fall more than the short term rates.

c)

The investors requirfe a risk premium to hold a lomger maturity bond as there can be several risks that can arise during that period. This is called maturity risk premium. It is simplly he exra risk an investor would require to hold a lomger term instrument. In the above mentioned scenario the present one year rate is 0.05% and the 5 year rate is 2.5%. So essentially the investo gets an extra 2% to invest his money for 5 years. This is the risk premium which is not considered under the expectgation theory which states that investors are simplly indifferent to the maturity period and concerned only about the returns.

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