Question

A certain portfolio’s value increases by 30% during a financial boom, increases by 5% during normal times and it decreases by 2% during a recession. Suppose each scenario is equally likely and you invested $200 in this portfolio.

1.Create a probability model for your net gain. (6 pts)

2.What is the expected net gain when you invest $200 in this portfolio? (3 pts)

3.What’s the standard deviation of your net gain when you invest $200 in this portfolio? (3 pts)

Answer 1

Solution :

Let us consider the given information;

A certain portfolio’s value increases by 30% during a financial boom, increases by 5% during normal times and it decreases by 2% during a recession.

Let we denote these three scenarios as follows; B = financial boom, N =normal times and R = recession.

Since each scenario is equally likely. We can write P(B) = P(N) = P(R) = 1/3.

Suppose we invested $200 in this portfolio.

(a). We first creat a probability model for net gain as follows;

Scenario	Probability (p)	Net gain(X)
Boom (B)	1/3	130%
Normal time (N)	1/3	105%
Recession (R)	1/3	98%

(b). The expected net gain is given by;

Scenario	Probability (p)	Net gain(X)	p*X
Boom (B)	1/3	130%	43.33%
Normal time (N)	1/3	105%	35%
Recession (R)	1/3	98%	32.67%
Total	1	---	111%

The expected value is given by;

So E(X) = 111%

And if we have invested $200 then the expected net gain is; 200*111% = $ 222.

(c). To get standard deviation of net gain;

Scenario	Probability (p)	Net gain(X)	p*(X-expected)^2
Boom (B)	1/3	130%	120.3333
Normal time (N)	1/3	105%	12
Recession (R)	1/3	98%	56.3333
Total	1	---	188.6667

The variance of net gain is given by;

And hence standard deviation of net gain = square root ( variance ) = 13.7356%

So if we have invested $ 200 to the portfolio then standard deviation of net gain is; 200*13.7356% = $ 27.4712.