DSE 2010

[Swati Kanodia]

Sir please explain in detail how to solve the following question:

Suppose a random variable X takes values -2,0,1 and 4 with probabilities 0.4, 0.1, 0.3, and 0.2 respectively.

a) The unique median of the distribution is 1
b) The unique median of the distribution is 0
c) The unique median of the distribution lies between 0 and 1
d) The distribution has multiple medians

[Amit Goyal]

Let X be the random variable.
We define median in the following way:
The real number m is called the median of random variable X if the following is true
Pr(X<=m)>=0.5
and
Pr(X>=m)>=0.5
Check that the above conditions are true for multiple ms. In particular it is true for both the numbers 0 and 1.

[Ashima Mathur]

Hi Sir,

Can you please explain how to solve the following question:

Suppose there is only one future period and (the presently unknown) state of the world in that period can be either s1 or s2. The future return on a share of a given company is 5 in state s1 and -1 in state s2. The future return on a government bond is independent of the state. Suppose a third asset is offered on the market whose return is 3 in state s1 and 0 in state s2. The current prices of the stock and the bond are 3 and 1 respectively. If the price of the new asset rules out the possibility of any arbitrage profit, what is the price of the new asset?
a. It depends on probabilities of the future states
b. Strictly between 2 and 3
c. Strictly between 1 and 2
d. 2

[Amit Goyal]

Return vector on a share of a given company is (5,-1)
Return vector on a bond is (1,1)
If you hold a portfolio of s shares and b bonds then your return from the portfolio is (5s+b,-s+b) where 5s+b is the return in s_1 and -s+b is the return in s_2. Solving for the portfolio of the above two assets which is equivalent to the third asset in terms of the returns it generates i.e. (3,0), we get b=s=0.5.
Since the prices of the stocks and bonds are 3 and 1 respectively, cost of this portfolio is 2. And by no arbitrage condition this must also be the price of the third asset.

[Ashima]

Thank you sir:)

[ravian]

by multiple medians, does it mean that 0 and 1 are the only possible medians..because the probability of x taking any value other than -2,0,1,4 is 0...

[Amit Goyal]

Well using the definition you can also say that any number between 0 and 1 is also the median (besides 0 and 1) and there is no other median.