Suppose on 31-Dec-2019 you entered into a forward contract to buy one share of stock XYZ for delivery price = L = $50 with delivery date 31-Dec-2020.  You enter t and S(t) into your formula for V(S,t) to figure out today’s value of the contract.  Now suppose you learn to your surprise that interest rates are not constant; they can change in an uncertain way over the time period from today to 31-Dec-2020.

[A]  Assume that today’s stock price and the interest rates that prevail today have not changed.   Will the forward price that you compute today (for delivery date 31-Dec-2020) change because of your revised view that future interest rates are uncertain?

[B]  Will V(S,t), the value of the forward contract, change?  Why or why not?

An certain brand of upright freezer is available in three different rated capacities: 16 ft³, 18 ft³, and 20 ft³. Let X = the rated capacity of a freezer of this brand sold at a certain store. Suppose that X has the following pmf.

(a) Compute E(X), E(X²), and V(X).

(b) If the price of a freezer having capacity X is 60X − 650, what is the expected price paid by the next customer to buy a freezer?

(c) What is the variance of the price paid by the next customer?

(d) Suppose that although the rated capacity of a freezer is X, the actual capacity is h(X) = X − 0.009X². What is the expected actual capacity of the freezer purchased by the next customer?

Two isolated concentric conducting spherical shells have radii R1=0.5 m and R2=1 m,

uniform charges q1=+2.0 μC and q2=+1 μC, and negligible thickness. Assume that V=0 at infinity.​

(a) What is the magnitude of the electric field at a radial distance of r=4 m?
(b) What is the magnitude of the electric field at a radial distance of r=0.7 m?
(c) What is the magnitude of the electric field at a radial distance of r=0.2 m?
(d) What is the potential at r=4 m?
(e) What is the potential at r=1 m?
(f) What is the potential at r=0.7 m?
(g) What is the potential at r=0.5 m?
(h) What is the potential at r=0.2 m?
(i) What is the potential at r=0 m?
(j) sketch E(r) and V(r).​

Both a call and a put currently are traded on stock XYZ; both have strike prices of $35 and expirations of 6 months.

a. What will be the profit to an investor who buys the call for $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 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 answers to 1 decimal place.)

A rail gun uses electromagnetic forces to accelerate a projectile to very high velocities. The basic mechanism of acceleration is relatively simple and can be illustrated in the following example. A metal rod of mass 40.0 g and electrical resistance 0.200 Ωrests on parallel horizontal rails that have negligible electric resistance. The rails are a distance L = 10.0 cm apart. (Figure 1) The rails are also connected to a voltage source providing a voltage of V = 5.00 V .
The rod is placed in a vertical magnetic field. The rod begins to slide when the field reaches the value B = 7.84×10⁻² T . Assume that the rod has a slightly flattened bottom so that it slides instead of rolling. Use 9.80 m/s2 for the magnitude of the acceleration due to gravity.

Find μs, the coefficient of static friction between the rod and the rails.

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