Consider two types of investment: bonds and shares. Denote next year’s value of one unit of investment in bonds as X and one unit of investment in shares as Y. Suppose that the two types of investment are independent. The marginal density functions for X and Y are, respectively, given in table 1 and table 2:Table 1x242526P(X = x)0.20.30.5Table 2y48495051P(Y=y)0.10.20.40.3a) Determine the joint probability density function of X and Y.b) Suppose that you consider building a portfolio that mixes both bonds and shares. Also suppose that the portfolio’s return can be described by the following function: U =50X −30Y −15.What are the expected return and the variance of the portfolio?

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Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter2: Exponential, Logarithmic, And Trigonometric Functions

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Consider two types of investment: bonds and shares. Denote next year’s value of one unit of investment in bonds as X and one unit of investment in shares as Y. Suppose that the two types of investment are independent. The marginal density functions for X and Y are, respectively, given in table 1 and table 2:

Table 1

x	24	25	26
P(X = x)	0.2	0.3	0.5

Table 2

y	48	49	50	51
P(Y=y)	0.1	0.2	0.4	0.3

a) Determine the joint probability density function of X and Y.

b) Suppose that you consider building a portfolio that mixes both bonds and shares. Also suppose that the portfolio’s return can be described by the following function:

U =50X −30Y −15.
What are the expected return and the variance of the portfolio?

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