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ECON 1095 QUANTITATIVE METHODS IN FINANCE
Assignment 1 is due Sunday 21st April and contributes 25% to the assessment of this course.
INSTRUCTIONS
Please up load one (and one only) either word or pdf file. For the excel sections just take screen shots of your work to show some of your workings then cut and paste into your document. For the maths sections (if you prefer) you can hand write scan and then add to word doc.
QUESTION 1
(a)What is the intercept, slope and elasticity for the following function?
y = a + bx +cx2
(b)If t (time) is the independent variable, what is the growth rate for this function?
y = a + b.t + c.t2
(c)If t (time) is the independent variable, what is the growth rate for this function?
y = eat
(d)For the function y = xa, prove that the elasticity equals“a”.
(e)Suppose the price movements of an asset can be modeled as a continuous return. That is, pt = pt-1.eR. Define R in terms of pt and pt-1.
(2.5 marks)
QUESTION 2
(a)Simplify the following mathematical functions:
(b)Logarithms are usually to the base 10 or e, but this need not be the case. Find:
(c)Below is a table showing a series of numbers as well as their logarithms[Note that the logarithms have been rounded to two decimal places.] and the changes in these logarithms, with some numbers removed and marked with an X:
Number Log Change in Log
50.0 1.70
60.0 1.78 0.08
X X 0.08
X 1.94 X
Please copy this table then fill in the missing numbers. What is the constant percentage change between the numbers?
(d) The general form of the exponential (or semi-log) function is written as: Yt = ?0et
In its current form this is a non-linear function and so cannot be estimated using linear regression. Use logarithms to convert this function so that it can be estimated using linear regression. Once the function has been estimated, what does 1 represent?
(e) The following function relates the quantity of good Y sold (Qy) to its own price (Py), the price of another good X (Px) and income levels (INC).
Qy = 0Py 1.Px2.INC3
Again, use logarithms to convert this function into a form that can be estimated using linear regression. Once estimated, what do the values of 1, 2 and 3 represent?
(2.5 marks)
QUESTION 3
For the following functions:
0 = -5×2 + 3x – 20
0 = x2 – 3x +5
0 = 1.5×2 -2.5x – 2.5
(a)What are their discriminates?
(b)What is the significance of these discriminates?
(c)Complete the square and thereby solve for x for these equations (please do this both by hand and using the excel solver, see excel instructions below).
(d)Use excel to graph of these functions.
To use the solver:
construct the formula for the equation 0 = -5×2 + 3 – 20; in B1 SUM (C1: E1), C1=-5C2^2, D1 = 3C2, E1 = -20, C2 the starting value for x.
go to solver (tools or formula), Set Cell $B$1, Equal To 0, By Changing C2, Solve.
keep trying different starting values of x until you have found all solutions.
if you have trouble go to help.
(2 marks)
QUESTION 4 (use Excel to do this question)
In the Excel file ECON 1095 Data Sem 1 2019 you will find daily share market and interest rate data for the last twelve months.
(a)Why do the share prices remain the same between 24th and 25th April 2018?
(b)Calculated the continuous returns for the market index (S&P/ASX 200 – PRICE INDEX), as well as for all the stocks (use the unadjusted share prices) and then find the average continuous returns for each for the whole period.
(c)Which stock has the highest and which stock has the lowest average returns over this period?
(d)Calculate the standard deviations for the continuous returns for the market and for each stock for the entire period. Which is the riskiest stock?
(e)Graph the highest returning stock against the market index for the twelve-month period, then on a separate graph do the same for the lowest returning stock, then on a third graph do this for the riskiest stock.
(f)Assume that the Beta (?) coefficient for the highest returning stock was 1.00, and that you believe that the last twelve months will give you a good indicator of what is likely to happen in the future, would you invest in this stock? Briefly explain.
(3 marks)
QUESTION 5
Find the derivative of y with respect to x for these functions:
(e)Find the indefinite integral of the following function
(2.5 marks)
QUESTION 6
Consider a 5-year bond from which you receive 5 coupon payments (C), one at the end of every year. The face value of the bond F is received at the end of year 5.
(a)Write the formula for the price P of this bond, where C is the coupon payment, F is the face value of the bond and YTM is the Yield to Maturity.
(b)Obtain the first derivative of the price of this bond with respect to the YTM (dP/dYTM). Write this derivative using the Macauley Duration (MD). The formula for the MD when there are n periods is shown below.
MD = (c)If you were told that for this 5-year bond F = 1,750, C = 125 and the YTM = 0.05; what will be the price (P) and Macauley Duration (MD) of the bond? This must be done both by hand and using excel.
(2 marks)
QUESTION 7
Consider the 5-year bond in QUESTION 6
(a) Find the second derivative of P w.r.t. the YTM (d2P / dYTM2). That is, using the Chain Rule (showing each step) differentiate the first derivative to obtain the second derivative.
(b) Use your answer to part (a), and the following formula to find the Convexity of this bond.
Convexity = 0.5 (c) Using the information from parts (a) and (b), find both the actual change in P when the YTM changes from 0.05 to 0.04 and the quadratic approximation to this change using the following formula. Briefly comment on the quality of this approximation:
P= MMD P [YTM] + Convexity P [YTM]2 [MMD is the modified Macauley Duration.]
(3 marks)
QUESTION 8(use Excel to do this question)
Assume you have four different bonds
B1 – A two-year bond with a nominal rate of 2 % per annum
B2 – A five-year bond with a nominal rate of 3 % per annum
B3 – A ten-year bond with a nominal rate of 4 % per annum
B4 – A twenty-five-year bond with a nominal rate of 5 % per annum
All these bonds have six monthly coupons and a face value of $2,500. Calculate their present values, Macauly durations and convexities using a YTM of 5% (YTM = 0.05).
(2 marks)
QUESTION 9 (use Excel to do this question)
Suppose a fund manager is committed to making payments of $35,000 every 6 months for the next 20 years (an annuity). The fund manager uses a discount rate of 0.05 or 5 % pa.
(a)What is the present value of these payments?
(b)To fund these payments the fund manager must invest in the four bonds described in Question 8. Assume that she is trying to minimize transaction costs; use the figures in Question 8 to write the equations that would need to be satisfied to immunize the annuity described in this question. Note that the fund manager is concerned that the application of these conditions could result in only one or two different types of bonds being held. As this is considered risky she introduces a diversification condition whereby she must hold a minimum of five of each of B1, B2, B3 and B4. Note; these conditions will need to be considered in your equations.
(2 marks)
QUESTION 10 (use Excel to do this question)
For the portfolio described in QUESTIONS 8 and 9 and the methods outined in the course notes on Mathematical Programming; that is, using the solver in excel, find the portfolio of bonds that the fund manager must invest in to immunize the portfolio. Although you need to apply the diversification conditions, there is no need to apply the second order condition of Convexitypayments > Convexityreceipts. Therefore, all that is required is for the two streams of payments and receipts to have the same present value, their Macaulay Durations must be equal and for the diversification conditions need to be satisfied. Submit you answer and sensitivity reports and write a brief paragraph explaining how much of each bond the fund manager should buy.
(3.5 marks)
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