Question

The probability of state in boom economy is 20%. The rate of stock return is 20% and the bond rate of return is 10%.

The probability of state in normal good growth economy is 25%. The stock rate of return is 10% and the bond rate of return is 5%

The probability of state in normal growth economy is 15%. The stock rate of return is 3% and the bond rate of return is 5%

The probability of state in normal bad growth economy is 25%. %. The stock rate of return is 0 and the bond rate of return is 5%.

The probability of state in recession economy is the rest of probabilities. The stock rate of return is -15%(negative) and the bond rate of return is - 2%(negative).

Q1: what is the probability of state in recession economy? (percentage)

Q2: what is the expected rate of return on both assets.?

Q3: what is the variance of both assets?

Q4: what is the standard deviation of both asset?

Answer 1

(a) We are given the details regarding the state probabilities and the rate of return on stocks and bonds. Using the details given, we calculate the probability of the state in recession economy.

State of Economy	Probability of state	Stock Return (in %)	Bond Return (in %)
Boom Economy	0.20	20	10
Normal Good Growth	0.25	10	5
Normal Growth	0.15	3	5
Normal Bad Growth	0.25	0	5
Recession Economy	0.15	-15	-2

Since the total of probabilities equals one, we have the probability of state in recession economy as follows

Thus, we have the probability of state in recession economy as 0.15 or 15%

(b) The expected return on an asset is the sum of the products of the return in each state and the probability.

We know that E(X + Y) = E(X) + E(Y)

Thus, the expected return on both assets is

Therefore, the expected return on both the assets together is 9.65%.

(c) We calculate the Variance using the following Formula

We simplify the above formula as below.

where X is the return on the assets and so E(X) has been calculated above as the expected return on the individual assets. We calculate E(X²) as follows:

Similarly, we calculate for the Bond

Using these values, we calculate the variance of the assets individually.

(rounded to 2 decimal places)

We also find the covariance as below:

We calculate the first term on the right hand side as below

We know that Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y)

Thus, the variance is 118.01%% sq for Stock, 12.35%% for Bond and 202.33%% for both the Assets.

(d) The Standard Deviation is calculated as the square root of Variance.

(rounded to 2 decimal places)

Thus, the standard deviation is 10.86% for Stock, 3.51% for Bond and 14.22% for both the Assets.