Question

Suppose there is a mean-variance optimizer looking to invest for two periods in bonds. They
can invest in a two year bond with annual yield y2 or they can invest in the one year bond at r1 and then,
a year later, invest in another one year bond at currently uncertain rate r2. What must be the liquiditiy
premium in order for them to be indifferent between the two options?
Now suppose they are looking to invest for three years. What is the liquidity premium to make them
indifferent between the three year bond or getting a two year bond, then a one year bond after?
What if they want to invest for n years. Derive the liquidity premium to make them indifferent between the
n year bond and the n − 1 year bond followed by a one year bond.

Answer 1

Soln : Liquidity premium is a premium of risk involved of liquidation risk between 2 bonds. Like 2 year bond is illiquid in 1 year but 1 year bond can be liquidated after 1 year and again invest the same in 1 year bond.

Liquidity premium is a combination of expectation hypothesis theory, which says that yield on long maturity bond is ana average of yeild on short term bonds, and segmented market hypotheses, which says the exact opposite of expectation hypothesis.

So, as per liquidity premium theory, the yield of long term bond is combination of average yield of short term bonds and a premum called as liquiduty premium.

Here, yield of 2 year annual yield bond is given as y₂ and for 1 year bond as r1, a year later one year bond is r2

So, as per the liquidity premium theory we can say that, Let R be the liquidity premium

y2 = (r₁+r₂)/2 + R or R = y2 -(r₁+r₂)/2

Similarly if we it is required for 3 year bond and 2 year bond with invest in 1 year bond thereafter

Again using the same equation we can say that let y3 be the yield for 3 year bond

R = y3-(2*y2+r₃)/3

Similarly for n year bond and n-1 year bond followed by a one year bond

R_n = y_n -((n-1)*y_n-1 + r_n)/n