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A random variable X has density function given by:
f(x)= {3e^-3x
x is greater than or equal to 0, otherwise. Find the first four moments.
Question three (9 marks)
ABC Company is being formed to make a 1 year investment in producing and marketing presidential campaign badges. The firm requires an investment of Sh 10,000,000 of which Sh 7,500,000 will be obtained by selling debt with a 10% interest rate and the other Sh 2,500,000 will be raised by selling common shares. All cash distribution to debt holders and shareholders will be made at the end of the one year. After this year is over the value of the firm will depend primarily on which candidates make it through the primary elections. The estimated probability of distribution of the firm is:
Probability Value `000'
0.7 20,000
0.2 5,000
0.1 0
Consider the shareholders value under the three states of nature and under the expected value.
Question two (8 marks)
The risk free rate is 10% and the expected return on the market portfolio is 15%. The expected returns for 4 securities are listed below together with their expected betas
SECURITY EXPECTED RETURN EXPECTED BETA
A 17.0% 1.3
B 14.5% 0.8
C 15.5% 1.1
D 18.0% 1.7
REQUIRED:
a. On the basis of these expectations, which securities are overvalued? Which are undervalued?
b. If the risk-free rate were to rise to 12% and the expected return on the market portfolio rose to 16%, which securities would be overvalued? which would be under-valued? (Assume the expected returns and the betas remain the same).
A cylindrical can is to be constructed to hold a fixed volume of liquid. The cost of the
material used for the top and bottom of the can is 3 cents per square inch, and the cost of the
material used for the curved side is 2 cents per square inch. Use calculus to derive a simple
relationship between the radius and height of the can that is the least costly to construct.
Discuss ways in which the current telephone numbering plan can be extended to accommodate the rapid demand for more telephone numbers. (See if you can find some of the proposals coming from the telecommunications industry.) For each new numbering plan you discuss, show how to find the number of different telephone numbers it supports.
Determine whether -104 is a quadratic residue or non residue of the
prime 997.
Determine those odd primes p for which -3p=1 and those for which -
3p=-1
.Solve𝑥 ≡ 7(𝑚𝑜𝑑 11) 𝑥 ≡ 6(𝑚𝑜𝑑 8) 𝑥 ≡ 10(𝑚𝑜𝑑 15) Also find the
smallest non-negative solutions
Solve the following congruences x≡1 mod3, x≡2mod4
x≡3mod5
Prove that n=2 mod4,and when n=p ,where p is a prime and p≡3 mod4
