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

Liquid premium is the extra return that a bond has to provide to the bond holder for holding a longer duration bond.

In the present case,

Interest rate for two year bond would be equal to (r1+r2)/2 based on re-investing.

For the two year bond, yield given is y2.

Therefore, for being indifferent between two options, the liquidity premium should be equal to

y₂ – (r₁+r₂)/2.

Same way, liquidity premium for three year bond over two year followed by one year bond would be

y3 – (2*y2+r3)/3.

Similarly, for n year bond, liquidity premium over n-1 year followed by one year bond would be

y_n – ((n-1)*y_n-1+r_N)/N.