Given all the portfolios of n securities what criterion would an investor use to select a good portfolio?
An investor selects a good portfolio based on the trade-off between risk and return, guided by mathematical modeling. Using the mean-variance framework introduced by Markowitz, the investor evaluates __________ ____ ____ ___ _______ _______.
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Given all the portfolios of n securities what criterion would an investor use to select a good portfolio?
Suppose that the previous forecast was 2090 and the actual value of the variable of interest for the period was 1985 and the oldest value of interest was 1955. Using the moving average technique based upon the most recent four observations find new forecast for the next period.
A monopolist sets a price ‘p’ per unit and the quantity demanded ‘q’ is given by the following relation:
Let there be a fixed cost of Rs.9 and a marginal cost of Rs.1 per unit.
i) Write the profit function of monopolist.
ii) For maximum profit, find the number ‘x’ of units produced. Also find the maximum profit.
iii) A potential entrant enters into the business of the monopolist. He believes that the monopolist will go on making ‘x’ units. Write the profit function of the entrant.
iv) For maximum profit of entrant, find the number ‘z’ of units produced.
v) Find the maximum profit of the entrant. Explain whether he should enter into the business or not.
vi) Find the profit made by the monopolist after the entrant has entered into business.
Characterise the following as discrete or continuous giving reasons for your answers.
i) Effects of radiation treatment on a tumour when applied for short period of time but at regular intervals.
ii) Effects of chemotherapy drugs on a tumour when introduced into a patient for a given duration of time.
Consider the group of individuals born in a given year )0 t( = and let )t(n be the number of these individuals surviving t year later. Let )t(x be the number of members of this group who have not had smallpox by year t and are therefore still susceptible. Let β be the rate at which susceptibles contract smallpox and let v be the rate at which people who contract smallpox die from the disease. Finally, let )t( µ be the death rate from all causes other than smallpox. If dx dt/ and dn dt/ are, respectively the rates at which the number of susceptibles and entire population decline due to contraction from smallpox and also due to death from all causes then
i) Formulate the above problem by writing equations for dx dt/ and dn dt/ .
ii) Taking z = x/n , show that z satisfies the initial value problem
iii) Find )t(z at any time t.
iv) Bernoulli estimated that Using these values, determine the proportion of 20 years old who have not had smallpox.
Consider a one-dimensional growth c(x, t) of phytoplankton in a water mass. Formulate the model describing the dynamics of growth taking into account the following: D , its diffusion coefficient, r its rate of growth, R its mortality rate due to sinking. Fixing the area of interest as 0 ≤ x ≤ 2 and the initial concentration of phytoplankton as 30moles/ 3 cm , find the concentration distribution of phytoplankton in 0 ≤ x ≤ 2 at any time t .
A particle of mass m moves on a straight line towards the centre of attraction, starting from rest at a distance a from the centre. Its velocity at a distance x from the centre varies as . Find the law of force.