2. Two stocks A and B have expected returns, and a variance-covariance matrix of returns given in Table 1.

a) What is the correlation coefficient between the returns on stock A and stock B?

b) What is the expected return and standard deviation of portfolio S which is invested 80% in stock A and 20% in stock B?

c) If you combine portfolio S with a risk free asset paying a return of 4%, what would be the expected return on a new portfolio V if you desire a standard deviation of 27.9%?

d) Plot in mean-standard deviation space the efficiency frontier between Stock A and Stock B, and identify portfolios S and V.

A particular single-machine workstation has a capacity of 1,000 units per day and variability is moderate, such that V = (SCV of arrivals + SCV of effective process time)/2 = 1. Demand is currently 900 units per day. Suppose management has decided that cycle times should be no longer than 1.5 times the average process time.

A) What is the current cycle time in multiples of the process time? (i.e. if the current cycle time was 2 times longer than the process time, put 2 in the answer box)

B) If variability is not changed, what would the daily capacity have to be in order to meet the requirement that average cycle time be no longer than 1.5 times process time?

C) If capacity is not changed, what value would be needed for V in order to meet the requirement that average cycle time be no longer than 1.5 times process time?

THERMODYNAMIC

The specifications of a typical reciprocating internal combustion engine coupled with a
generator are given in Table Q2. With the aid of a P-v diagram, determine the following
engine performance characteristics by using constant specific heat at room temperature:
i) the total mass contained in the cylinder per cycle,
ii) the mass of fuel burned per cycle,
iii) the mean effective pressure,
iv) the engine power in kW, and
v) the specific fuel consumption in g/kWh .

Table Q2
Item Specification
Cycle 4-stroke
Fuel Type Diesel
Fuel Calorific Value 43 MJ/kg
Combustion Efficiency(%) 95
Compression Ratio 19
No. of Cylinder 4
Engine Capacity (cc) 2000
Intake Pressure (kPa) 95 kPa
Intake Temperature (°C) 30
AFR 28:1
Engine Speed (rpm) 1800

Both a call and a put currently are traded on stock XYZ; both have strike prices of $50 and expirations of 6 months.

a. What will be the profit to an investor who buys the call for $4.8 in the following scenarios for stock prices in 6 months? (i) $40; (ii) $45; (iii) $50; (iv) $55; (v) $60. (Leave no cells blank - be certain to enter "0" wherever required. Negative amounts should be indicated by a minus sign. Round your answers to 1 decimal place.)

b. What will be the profit to an investor who buys the put for $7.5 in the following scenarios for stock prices in 6 months? (i) $40; (ii) $45; (iii) $50; (iv) $55; (v) $60. (Leave no cells blank - be certain to enter "0" wherever required. Negative amounts should be indicated by a minus sign. Round your ans

. Let xj , j = 1, . . . n be n distinct values. Let yj be any n values. Let p(x) = c1 + c2x + c3x 2 + · · · + cn x ^n−1 be the unique polynomial that interpolates the data (xj , yj ), j = 1, . . . , n (Vandermonde approach).

(a) Remember that (xj , yj ), j = 1, . . . , n are given. Derive the n × n system Ac = b that determines the coefficients ck (as we did in class for n = 4).

(b) Write a MATLAB script that sets up the Vandermonde matrix V for any given vector x.

(c) Find the condition number (infinity norm) of the matrix V where x consists of n = 10, 20, 30, 40 equally spaced points spanning the interval [0, 1]. Report your results in a table of each n

Consider the following regression model: Yi = αXi + Ui , i = 1, .., n (2)

The error terms Ui are independently and identically distributed with E[Ui |X] = 0 and V[Ui |X] = σ^2 .

1. Write down the objective function of the method of least squares.

2. Write down the first order condition and derive the OLS estimator αˆ.

Suppose model (2) is estimated, although the (true) population regression model corresponds to: Yi = β0 + β1Xi + Ui , i = 1, .., n with β0 different to 0.

3. Derive the expectation of αˆ, E[ˆα], as a function of β0, β1 and Xi . Is αˆ an unbiased estimator for β1? [Hint: Derive first E[ˆα|X].]

4. Derive the conditional variance of αˆ, V[ˆα|X], as a function of σ^2 and Xi .

An electric vehicle has the following parameter values:

m_v = 800kg

C_D =0.2

A_f = 2.2 m²

ρ = 1.18 kg/m

f r = 0.008 + 0.6×10^-6×v² (v: vehicle speed in m/s)

The vehicle is on level road. It accelerates from 0 to 100 km/h in 10s such that its velocity profile is given by v (t) =0.29055t² for 0 < t < 10s. (The mass factor is assumed unit)

a. Define the traction force expression F_t

b. Sketch F_t versus time

c. Define the instantaneous traction power expression

d. Calculate the energy consumed during the acceleration (0 < t < 10s)

e. Calculate the energy lost for non-conservative forces (wind and rolling resistance)

f. Find the change in kinetic energy and the change in tractive energy during acceleration.

Partial Differential Equations

(a) Find the general solution to the given partial differential equation and (b) use it to find the solution satisfying the given initial data.

Exercise 1. 2∂u ∂x − ∂u ∂y = (x + y)u

u(x, x) = e −x 2

Exercise 2. ∂u ∂x = −(2x + y) ∂u ∂y

u(0, y) = 1 + y 2

Exercise 3. y ∂u ∂x + x ∂u ∂y = 0

u(x, 0) = x 4

Exercise 4. ∂u ∂x + 2y ∂u ∂y = e −x − u

u(0, y) = arctan y

Exercise 5. ∂u ∂x+v ∂u ∂y = −ru

(here r and v 6= 0 are constants) u(x, 0) = sin x x

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Discussion Questions:

How was America affected by the Cold War?

Why did the policy of “containment” develop and what were its goals?

How did the fear of Communism and the “red scare” affect American society during the 1950s?