7.15 Channel equalization. We suppose that u1, . . . , um is a signal (time series) that is trans- mitted (for example by radio). A receiver receives the signal y = c ∗ u, where the n-vector c is called the channel impulse response. In most applications n is small, e.g., under 10, and m is much larger. An equalizer is a k-vector h that satisfies h∗c ≈ e1, the first unit vector of length n + k − 1. The receiver equalizes the received signal y by convolving it with the equalizer to obtain z = h ∗ y.

(a) How are z (the equalized received signal) and u (the original transmitted signal) related? Hint. Recall that h∗(c∗u) = (h∗c)∗u.

(b) Numerical example. Generate a signal u of length m = 50, with each entry a random value that is either −1 or +1. Plot u and y = c ∗ u, with c = (1,0.7,−0.3). Also plot the equalized signal z = h ∗ y, with

h = (0.9, −0.5, 0.5, −0.4, 0.3, −0.3, 0.2, −0.1).

1)

The risk free rate on treasury bills is 2.40%. Inflation is 1.3%. What is the real rate of return on treasury bills?

Meaning of what is the "real rate of return" ?

If investor saw calculated number two from above, would he be pleased or not?

2)

The nominal return on treasury bonds is 2.7%, the maturity risk premium is 0.2%. Real return can earn 0.3%.

What is the inflation premium?

Why is there maturity risk premium on US treasury bonds vs. bills?

1. Assume the rate of return given below are for two stocks listed on the Ghana

          Stock Exchange (GSE).

1. Calculate the arithmetic average return on the two stocks over the 5-year period.
2. Which of the two stocks has a greater dispersion around the mean?
3. Calculate the geometric average returns of each stock.

Assume that you set up a portfolio composed of Stock A and Stock B. You invested 40% of your capital on Stock A whereas 60% of your capital on Stock B. During the last 3 years, your portfolio showed the following performance.

What are your average portfolio return and risk (standard deviation or variance) during last three years? (20 points)

Assume that you set up a portfolio composed of Stock A and Stock B. You invested 40% of your capital on Stock A whereas 60% of your capital on Stock B. During the last 3 years, your portfolio showed the following performance.

What are your average portfolio return and risk (standard deviation or variance) during last three years? (20 points)

Suppose the probability that the probability of rain today in Vernon is 0.4 and the probability of rain today in Kelowna is 0.3. The probability that it will rain in both cities today is 0.2. A) What is the probability that it will rain in Kelowna or Vernon today? B) What is the probability that it will rain Kelowna today but not in Vernon? C) Suppose that it has started to rain in Kelowna today. What is the conditional probability that it will also start to rain in Vernon today? D) Is the event that it will rain today in Kelowna independent of the event that it will rain today in Vernon?

1. The price index from 2016 to 2018 is : 200, 210, 220. Derive a forecast for 2019 on the basis of adaptive expectations using 0.2, 0.3 and 0.5 as weights in 2016, 2017 and 2018 respectively.
2. The optimal forecast for 2019 is 225 using all available information. What is the expected price index in 2019 based on rational expectations?
3. The actual value in 2019 is 223. Based on this, Paul concluded that the expected value in (b) derived using rational expectations was wrong. Do you agree? Why?

Suppose that the index model for stocks A and B is estimated from excess returns with the following results: RA =0.03 + 0.7RM + eA RB = -0.02 + 1.2RM + eB σM = 0.2 R-squareA = 0.3; R-squareB = 0.25 Assume you create for portfolio Q with investment proportions of 0.50 in P, 0.30 in the market index, and 0.20 in T-bills, portfolio P is composed of 60% Stock A and 40% Stock B. What is the standard deviation of the portfolio Q?

0.4800

0.2556

0.1831

0.2766

Suppose that the index model for stocks A and B is estimated from excess returns with the following results:

R_A =0.03 + 0.7R_M + e_A

R_B = -0.02 + 1.2R_M + e_B

σ_M = 0.2

R-square_A = 0.3;

R-square_B = 0.25

Assume you create for portfolio Q with investment proportions of 0.50 in P, 0.30 in the market index, and 0.20 in T-bills, portfolio P is composed of 60% Stock A and 40% Stock B. What is the standard deviation of the portfolio Q?

0.2766

0.1831

0.4800

0.2556

.The following table displays the joint probability distribution of two discrete random variables X and Y.

1. What is P(X=1/Y=1)?   
2. What is the value of E(X/Y=1)?   
3. What is the value of VAR(X/Y = 1)?
4. What is the correlation between X and Y?
5. What is variance of W = 4X - 2Y.
6. What is covariance between X and W?