Basic Statistics for Finance
Tutorial 4
1. (a) Let z ~ N(0,1). Find (i) P( - 1 < z ); (ii) P( z < - 1.2 ).
(b) Let X ~ N(3,9). Find (i) P( X > 7.5 ); (ii) P( - 6 < X < 0 ); (iii) P( 2X < 12 ).
2. The NPV of an investment project (measured in £ million) is believed to have the following
probability distribution: N( 16, 64). Find the probability that the NPV will be
(i) less than zero; (ii) greater than 20.
3. Consider an investment project which would cost £ 7.5 million and have a 3 year life. The
net annual cash flows (in £ million), CFi , i = 1,2,3, are random variables with probability
distributions N(2, 1), N(4, 1.44) and N(8, 2.25) respectively.
The correlation (ρij) between cash flow i and cash flow j is as follows:
ρ12 = 0.4, ρ13 = 0.2, ρ23 = 0.6. The appropriate discount rate is 6%.
(i) Calculate the expected NPV for the project and its standard deviation.
(ii) Find the probability that the project's NPV is less than £ 2 million.
4. The vectors and matrices used in the questions below are as follows:
a' = b' = A = B = C =
(a) Find (i) a' + b' (ii) A + B (iii) B – A.
(b) Write down (i) a (ii) A' (iii) (A + B)'
(c) Evaluate (i) a'b (ii) ab'
(d) Evaluate (i) a'C (ii) AB .
(e) Show that (AB)' = B'A'.
(f) Find A-1.