Question

Suppose that in an economy, 1-year spot rate is currently at 6 percent. Due to economic slump, the 1-year spot rate one year from now will be 4 percent, the 1-year spot rate two years from now will be 3 percent; and the 1-year spot rate three years from now will be 2 percent. Under the unbiased expectations theory, what must today's four-year spot rate be? Suppose the four-year spot rate is actually 4.75 percent, how could you take advantage of this? Explain.

Answer 1

Assume $1 invested today at 5percent.

Value after one year =$1.06

If it is reinvested for another one year at spot rate4%

Value after TWO years=1.06*1.04=$1.1024

Againt it is reinvested at spot rate of 3% two years from now.

Future Value at end of THREE years=1.1024*1.03=$1.135472

At the end of year 3 spot rate=2%

If the amount is reinvested at spot rate for one year<

Future Value at end of 4 years=1.135472*1.02=$1.158181

Assume four year spot rate now is=R

If $1 is invested at rate =R for four years,

Future Value at end of Year4=(1+R)^4

Under the unbiased expectations theory:

Both Values should be same (Investing and reinvesting one year at a time or investing for four years at a time should yield the same Future Value)

(1+R)^4=1.158181

1+R=1.158181^(1/4)=1.037395

R=Four year spot rate now=0.037395=3.74%(Rounded to two decimals)

If the four year spot rate is actually higher at 4.75%;

One can take advantage by borrowing at one year spot rate and investing at 4 year rate.

For Example,

You borrow $10,000 at current spot rate and invest at 4.75%

After one year you borrow at 4% and Repay the earlier loan

After two years you borrow at 3% and repay old loan.

Finally you borrow at 2% and repay the old loan