QUANTITATIVE METHODS IN FINANCE ASSIGNMENT

QUESTION 1 - Using the following financial models as examples:

• Write down their equations,
• Explain what type of equations they are, and
• Briefly describe the key properties of each type of equation.

(a) The Market Model Characteristic Line.

(b) The Capital Market Line (CML), which is part of the Capital Asset Pricing Model (CAPM).

(c) The security market line (SML), also part of CAPM.

(d) An equation used in the arbitrage pricing theory (APT) model.

QUESTION 2 - Logarithms are usually either to the base 10 or e, but this need not be the case. Find:

(a) log₄(64) = ?

(b) log_0.5(0.125) = ?

(c) log₉(9²) = ?

(d) log₂(1) = ?

QUESTION 3 - For the following functions:

0 = -50x² + 3x + 1000

0 = 20x² -2x + 50

0 = 16x² + 15x + 25

(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, excel instructions follow).

(d) Use excel to graph of these functions.

To use the solver:

• Construct the formula for the equation 0 = -50x² + 3 - 1000; in B1 SUM(C1: E1), C1 =-50*C2^2, D1 = 3*C2, E1 = +1000, 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.

QUESTION 4 - In the Excel file ECON1095 Data Sem 1 2018.xls on Canvas you will find the monthly share market indexes from 2007 to 2017 for a number of different countries.

(a) Calculate the continuous month returns for all of these indexes for the whole period and then find the average continuous returns as well as the standard deviations of these returns.

(b) Graph the continuous returns for the highest, the lowest and the riskiest indexes shares over this period. Please use separate graphs.

(Use Excel to do this question).

QUESTION 5 - Find the derivative with respect to x for these functions:

(a) y = 20x + 5x^0.5 + x^-2

(b) y = (2x+ x⁵)(3x + x²)

(c) y = x^-2 - (e^-2x/2x) + e^x + ln(2x²)

Find the definite integral of the following function for values of x from 2 to 3:

(d) f(x) = 2x + 8x⁹, that is find ₂∫³2x+8x⁹ dx

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.

(c) If you were told that for this 5 year bond F = 1,500, C = 25 and the YTM = 0.05; what will be the price (P) and Macauley Duration (MD) of this bond? This must be done both by hand and using excel.

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 (d²P/dYTM²) (1/P)

QUESTION 8 - Using the information from Questions 6 and 7, 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:

ΔP = MMD P [ΔYTM] + Convexity P [ΔYTM]² [MMD is the modified Macauley Duration]

QUESTION 9 - (use excel for this question)

Assume you have four different bonds

• B1 - A two year bond with a nominal rate of 3% per annum
• B2 - A five year bond with a nominal rate of 4 % per annum
• B3 - A ten year bond with a nominal rate of 5 % per annum
• B4 - A twenty-five year bond with a nominal rate of 6 % per annum

All these bonds have six monthly coupons and a face value of $3,000. Calculate their present values, Macauly durations and convexities using a YTM of 5% (YTM = 0.05).

QUESTION 10 - (use excel for this question)

Suppose a fund manager is committed to making payments of $40,000 every 6 months for the next 15 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 9. Assume that she is trying to minimize transaction costs; use the figures in Question 9 to write the equations that would need to satisfy to immunize the annuity described in this question. Note that the fund manager is concerned that the application of these conditions could result in them only holding one or two different types of bonds. 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.

(c) Using the methods described in the Topic 10 course notes; 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 Convexity payments > Convexity receipts. 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.

Attachment:- Assignment Files.rar