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ECON3107 Economics of Finance GROUP ASSIGNMENT GROUPW18A01 Ruojing ZHANG (z5190800) Timothy UEN (z5165459) Haihang GAO (z5216951) Rohan AGRAWAL (z5257515) ECON3107 Assignment 2 (W18A01) 1 Question 1. Duration and Banking 1) We know, Bond Price = CP1 (1 + YTM)1 + CP2 (1 + YTM)2 + CP3 (1 + YTM)3 + CP4 (1 + YTM)4 + CP5 (1 + YTM)5 + FV (1 + YTM)5 Therefore: 95 = 3 (1 + YTM)1 + 3 (1 + YTM)2 + 3 (1 + YTM)3 + 3 (1 + YTM)4 + 3 (1 + YTM)5 + 100 (1 + YTM)5 Solving for YTM (Appendix Q1(1)): YTM=0.04127(5 d.p.) Using the Macaulay Duration Formula: Duration in Year = 1× Payment (1+Yield)1 +2× Payment (1+Yield)2 +3× Payment (1+Yield)3 +4× Payment (1+Yield)4 +5× Payment+Face Value (1+Yield)5 Current Price Duration = 1 × 3 1 + y + 2 × 3 (1 + y)2 + 3 × 3 (1 + y)3 + 4 × 3 (1 + y)4 + 5 × 103 (1 + y)5 95 Therefore as mentioned above we know y=4.127% and hence, Duration = 1 × 3 1 + 4.127% + 2 × 3 (1 + 4.127%)2 + 3 × 3 (1 + 4.127%)3 + 4 × 3 (1 + 4.127%)4 + 5 × 103 (1 + 4.127%)5 95 Therefore, Duration=4.70851 years 2) ECON3107 Assignment 2 (W18A01) 2 Modified Duration = Duration 1 + y Using y=4.127%, and Duration=4.70851 years = 4.70851 1 + 4.127% = . % = MD × 0.01 = 4.52189% × 0.01 = 0.0452% Therefore change in relative price is an increase by 0.0452% (y ∝ P (Hayes, 2021)1) 3) = 4 Assuming nominal face value (100) of Liabilities: 4 = 1 × CPN1 (1 + YTM)1 + 2 × CPN2 (1 + YTM)2 + 3 × CPN3 (1 + YTM)3 + 4 × (FV + CPN4) (1 + YTM)4 PLiabilities Since: coupon rate = 3% = YTM 4 = 1 × 3 (1 + 0.03)1 + 2 × 3 (1 + 0.03)2 + 3 × 3 (1 + 0.03)3 + 4 × (100 + 3) (1 + 0.03)4 PLiabilities = 95.71528 = 4 1.03% = 0.03883 () ≈ 0.03883 × 0.1 × 95.71528 × 60 100 = . () ≈ 0.04521 × 0.1 × 95 × 80 100 = . ≈ 0.3436 ( 0.2230) = . 1 Hayes, A., 2021. Duration. [online] investopedia.com. Available at: [Accessed 10 July 2021] ECON3107 Assignment 2 (W18A01) 3 Question 2. Option Pricing 1) Following the Put-Call Parity, when there is no arbitrage opportunity, the equation for call and put options is: C + PV(X) = P + S where PV(X) = Strike price 1 + r For semi-annual interest rate: r = (1 + _)2 1, r = 2.15%, semi_r = 1 + 2.15% 1 = 1.0693% = P + S PV(X) = 8.55 + 58.93 65 1 + 1.0693% = $. > $. The arbitrage free price of call option is greater than the market trading price, the call option was underpriced, this represents an arbitrage opportunity. 2) Equation of put-call parity: C + PV(X) = P + S We know: C = 1.46 ECON3107 Assignment 2 (W18A01) 4 PV(X) = 65 (1 + 2.15%) P = 8.55, S = 58.93 Therefore, we get: C+PV(X) = 65.77231 P+S = 67.48 Overall strategy: Buy: C+PV(X), which costs $ 65.77231 Sell: P+S, which earns $67.48 Then we can get a profit of: 67.48 – 65.77231 = $1.7077, by applying this arbitrage strategy ECON3107 Assignment 2 (W18A01) 5 Question 3. Real Option Note: = 290 = 1.2 = 0.8 πg = 0.65 1 πg = 0.35 1) c = 0 0 142 0 0 0 T = 1 1 0 0 0 0 ECON3107 Assignment 2 (W18A01) 6 = 1.06 1.2 1 1.06 0.8 0 0 0 1.06 1 0 0 0 1 1 1.2 0 0 0 0 1.06 0 0 0 0 0 0 0.8 0 0 0 1.06 1.2 0 1.06 0.8 T = × = 0.61320 0.33018 0.37602 0.20247 0.20247 0.10902 For the present value of developing the gold (Appendix Q3(1)): Period 1 (g): = × 70 0 0 0 0 0 = 42.92452 = × 0 0 0 0 0 0 = 0 ∴ Delay Period 1 (b): = × 0 50 0 0 0 0 = 16.50943 = × 0 0 0 0 0 0 = 0 ∴ Delay Period 2 (gg): = × 0 0 142 0 0 0 = . = × 0 0 0 0 0 0 = 0 ∴ Exercise Period 2 (gb): = × 0 0 0 2 0 0 = 0.40494 = × 0 0 0 0 0 0 = 0 ECON3107 Assignment 2 (W18A01) 7 ∴ Delay Period 2 (bg): = × 0 0 0 0 2 0 = 0.40494 = × 0 0 0 0 0 0 = 0 ∴ Delay Period 2 (bb): = × 70 0 0 0 0 98 = 10.68440 = × 0 0 0 0 0 0 = 0 ∴ Delay Overall, payoff tree represented by: / ____max{ , 0} = 53.39533 / __max{ , 0} = 0 __ ____max{ , 0} = 0 __max{ , 0} = 0 ___max{ , 0} = 0 __max{ , 0} = 0 Therefore, we choose to exercise the concession at the end of period 2: = × 0 0 142 0 0 0 = . Overall, the government would assume the market to be in perfect equilibrium. The NPV of the project at time 0, if they chose to develop is $10mil and if they chose to delay is $53.39533mil. Therefore, the government’s maximum fee that would be ECON3107 Assignment 2 (W18A01) 8 charged is $53.39533mil. We can only consider the fee based on the government’s perspective as at time 0 as all the future time periods are unknown and uncertain from the government perspective as they are just based on probability. However, the NPV at time 0 is known if they chose to develop or delay is known. Therefore, the maximum fee that the government should charge is $53.39533mil. 2) From part (1): = . = . = . = . = . = . = 0 0 142 0 0 0 = $. Valuing the project in terms of assets and real option value: Max {Value of the project} = max {Developed amount of gold payoff + n}, Since from above, we exercise the concession at the end of Period 2: max {Value of the project} = (432 290) 1.062 + 53.39533 max {Value of the project} = $179.77483 ECON3107 Assignment 2 (W18A01) 9 3） It is assumed that risk and return move in the same direction i.e higher risk infers higher level of returns. Risk averse refers to someone who avoids risk and hence they would settle for lower returns. Thus, if the company is risk averse, the valuation of the project will be lower as it has already been safeguarded from the risk of large losses. The return will be higher if the company is risk neutral as compared to risk averse, as a risk neutral company will still bear some level of risk of moderate losses. Overall, the atomic prices of a risk averse company will be higher as they will obtain relatively less returns than a risk neutral company. Since, for the risk neutral company having a higher risk of large losses will lead to higher returns, and thus their atomic prices will be lower compared to the risk averse company ECON3107 Assignment 2 (W18A01) 10 Question 4. Arrow-Debreu Economy 1) Let (1) denote the quantity of Arrow-Debreu Securities obtained at time 0 by consumer k for a given state 1 ∈ 1 , which pays 1 unit of consumption at time 1 at specific state and nothing otherwise. The budget constraints using the forms as follows: time 0: 0 (0) + 1∈1 (0, 1) (0, 1) = 0 (0) time 1: 1 (1) = 1 (0, 1) + 1 (1), 1 ∈ 1 = , , ∈ 1, 2, 3 The budget constraints at time 0: () + () () + () () + () () = () () + () () + () () + () () = () () + () () + () () + () () = () The budget constraints at time 1: () = () + () () = () + () () = () + () () = () + () () = () + () () = () + () ECON3107 Assignment 2 (W18A01) 11 () = () + () () = () + () () = () + () A Market Equilibrium in this economy is defined as an allocation (), () ∈ , (, ) ∈ (, ) ∈ : Give the prices (0, 1) 1∈1, the allocation solves the maximization problem ( ()) + ∈ ()( ()) , Subject to the budget constraints 0: 0 (0) + 1∈1 (0, 1) (0, 1) = 0 (0), time 1: 1 (1) = 1 (0, 1) + 1 (1), 1 ∈ 1 = , , Markets clear in every period and state = () = = (), = () = = (), ∈ Arrow-Debreu securities can be trade in this economy since each agent exhibits heterogeneity in endowments. 2) ECON3107 Assignment 2 (W18A01) 12 u(c) = c1 γ 1 γ max u (c0 k(s0)) + β 1∈1 π(s1)u(c1 k(s1)) u = u(c0 k(s0)) + β 1∈1 π(s1) c1 k(s1) 1 γ 1 γ , 1 ∈ 1 = , , Subject to the budget constraints and market clearing conditions The Lagrangian for the consumer’s optimization problem: = ( ()) + ∈ () () + ∈ (, ) (, ) () ∈ () () (, ) () , ∈ = , , ∈ , , Differentiate the Lagrangian with respect to 0 (0), 1 (1) 1∈1 , (0, 1) 1∈1 0 (0) = 1 0 k(0) = 0 1 (1) = (1)c1 k s1 γ 1 k(1) = 0, 1 ∈ 1 (1) = 0 k(0)(0, 1) + 1 (1) = 0, 1 ∈ 1 The prices q(0, 1) = 1 k(1) 0 k(0) , q(0, 1) = (1)c1 k s1 γ, ECON3107 Assignment 2 (W18A01) 13 for 1 ∈ 1 = 1, 2, 3 The price of each security is the same for all consumers, β and π(s1) are constant Therefore, 1 1(1) = 1 2(1) = 1 3(1) Under the market clearing condition (at time 1) =1 3 1 (1) = =1 3 1 (1), 1 ∈ 1 ∈ 1, 2, 3 Clearing conditions: 1 1() + 1 2() + 1 3() = 6 1 1() + 1 2() + 1 3() = 3.6 1 1() + 1 2() + 1 3() = 1.5 Equilibrium consumption: 1 1() = 1 2() = 1 3() = 1 1() = 1 2() = 1 3() = . 1 1() = 1 2() = 1 3() = . () = 0.98 × 0.4 × 2 0.2 = . () = 0.98 × 0.3 × 1.2 0.2 = . () = 0.98 × 0.3 × 0.5 0.2 = . Using the budget constraints at time 1 1 (0, 1) = 1 (1) 1 (1), 1 ∈ 1 = , , ∈ 1, 2, 3 ECON3107 Assignment 2 (W18A01) 14 Quantity of Arrow-Debreu securities () () () Consumer 1 2-3.2=-1.2 1.2-1.8=-0.6 0.5-0.9=-0.4 Consumer 2 2-1.6=0.4 1.2-1.2=0 0.5-0.4=0.1 Consumer 3 2-1.2=0.8 1.2-0.6=0.6 0.5-0.2=0.3 Using the budget constraints at time 0: 0 (0) = 0 (0) 1∈1 (0, 1) (0, 1), 1 ∈ 1 = , , ∈ 1, 2, 3 () = 0.4 0.3413 × ( 1.2) 0.2835 × ( 0.6) 0.3377 × ( 0.4) = . () = 1.2 0.3413 × 0.4 0.2835 × 0 0.3377 × 0.1 = . () = 2.0 0.3413 × 0.8 0.2835 × 0.6 0.3377 × 0.3 = . 3) The forward prices of the atomic (Arrow-Debreu) securities: s1 = (1) (1) = (1) 1∈1 (1) , where (1) = 1∈1 (1) = () + () + () = 0.3413 + 0.2835 + 0.3377 = 0.9625 () = 0.3413 0.9625 = . () = 0.2835 0.9625 = . ECON3107 Assignment 2 (W18A01) 15 () = 0.3377 0.9625 = . Risk premium: = , e(s1) = π(s1) for all 1 ∈ 1, = 0.4, 0.3, 0.3 () = 0.4 0.3546 = . > () = 0.3 0.2945 = . > () = 0.3 0.3509 = . < Forward value of security in the good state is less than the probability of this state occurring, agents care a good state payment less than probability. () = . < . = () But they care a bad state payment more than probability. () = . > . = () These results indicate that the consumers are risk averse. 4） Denote 2 ∈ 2= (Bond, Stock, Put option) as the line vector of the securities. Put option only exercises when price of stock lower than the exercise price. The payment would be (exercise price – stock price). Therefore, the payment matrix of the portfolios will be: ECON3107 Assignment 2 (W18A01) 16 G F B 1 2 0 1 1 0 1 0 1 We define the new constraint to be : : () + ∈ (, ) () = () : () = () + () + (), () = () + ()+ (), () = () + ( ) + () ∈ = , , , ∈ = , , ∈ , , 5) u(c) = c1 γ 1 γ max u (c0 k(s0)) + β 1∈1 π(s1)u(c1 k(s1)) u = u(c0 k(s0)) + β 1∈1 π(s1) c1 k(s1) 1 γ 1 γ , 1 ∈ 1 = , , Subject to the new budget constraints in previous question (4) and market clear conditions Define the Lagrangian for the consumer’s optimization problem as: = ( ()) + ∈ ECON3107 Assignment 2 (W18A01) 17 () () + ∈ (, ) () () () ( () () () ()) () ( () () () ()) () ( () () ( ) ()) Obtain the first order condition: 0 (0) = 1 0 k(0) = 0 1 (1) = (1)c1 k s1 γ 1 k(1) = 0, 1 ∈ 1 () = 0 k() + 1 k(G) + 1 k(F) + 1 k(B) = 0 () = 0 k() + 21 k(G) + 1 k(F) = 0 ( ) = 0 k( ) + 1 k(B) = 0 () = 1 k(G) + 1 k(F) + 1 k(B) 0 k () = 21 k(G) + 1 k(F) 0 k ( ) = 1 k(B) 0 k As we know: 1 1() = 1 2() = 1 3() = 2 1 1() = 1 2() = 1 3() = 1.2 1 1() = 1 2() = 1 3() = 0.5 1 () = () + 2() + 1 (), ECON3107 Assignment 2 (W18A01) 18 1 () = () + ()+1 (), 1 () = () + ( ) + 1 () ∈ 1, 2, 3 1 k(s1) = (1)c1 k s1 γ = 0.98(1)c1 k s1 0.2 () = . () = . () = . Therefore: () = 0.3413 + 0.2835 + 0.3377 = . () = 2 × 0.3413 + 0.2835 = . ( ) = . Calculate allocations using budget constraints at time 1 Quantity of securities () () ( ) Consumer 1 0 -0.6 -0.4 Consumer 2 -0.4 0.4 0.5 Consumer 3 0.4 0.2 -0.1 6) The payment matrix (Q) of securities is: ECON3107 Assignment 2 (W18A01) 19 G F B 1 2 0 1 1 0 1 0 1 We find the following prices for each security by using the atomic prices from question (5): q(s1) = 0.3413 0.2835 0.3377 q(s2) = 0.3413 0.2835 0.3377 × 1 2 0 1 1 0 1 0 1 = 0.9625 0.9661 0.3377 = () + () + () = . = 2 × () + () = . = () = . It can be concluded from question (5) and (6), the security prices calculated by solving optimization problems are the same as the result calculated by using atomic prices. By observing the composition of the price calculation equation, we can deduce that the price of any type of securities can be composed of certain combinations of Arrow-Debreu securities. The basic theory of Arrow-Debreu securities can also be used as support, that is “the current prices of securities can be decomposed into the prices of returns in different states in the future”. Therefore, the securities prices obtained by the two methods should be the same, and the market is an arbitrage-free market. ECON3107 Assignment 2 (W18A01) 20 Question 5. Investment 1) Sharpe Ratio = (rp rf) sdp Where: rp = return of portfolio rf = risk free rate sdp = standard deviation of portfolio return Then: = (0.1 0.03) 0.05 = 0.07 0.05 = . Since the Sharpe ratio of the market is based on a representative portfolio consisting of all the stocks in the market, hence themarket portfolio takes advantage of diversification benefits to minimise the unsystematic risk. While the systematic risk of the market cannot be reduced using diversification benefits, we note that no single stock in the market will consistently outperform other investments (The benefits of diversification Dixon Advisory, 2021)2, as large-scale investment gains on a typical growth asset are most usually associated with a relatively higher level of risk. Hence, in order to beat the Sharpe ratio of the market would likely need to obtain a diversified portfolio of at least one more stock in the market, to result in less risk 2 dixon.com.au. 2021. The benefits of diversification Dixon Advisory. [online] Available at: [Accessed 18 July 2021]. ECON3107 Assignment 2 (W18A01) 21 than the market portfolio for relatively greater excess returns, according to the equation: Sharpe Ratio = (rp rf) sdp . 2） () = 10% σ(rm) = 5% = 3% Consider the return of a portfolio of the riskless asset and a portfolio of stock, represented by the overall market portfolio: = × + (1 ) × 3. . . . . . . . . . (1) = 3 + × ( 3). . . . . . . . . . (2) Where: w is the % invested in the market portfolio Taking expectation: Using (1), E(rp) = E( × ) + E((1 ) × 3) = 10 × w+ 3 × (1 w) = 3 + 7. . . . . . . . . . (3) Taking variance: Using (2), 2() = 2 × 2( 3) () = × ( 3) () = × () = × 5. . . . . . . . . . (4) Using (3) and (4) in the investor’s Expected utility function: ECON3107 Assignment 2 (W18A01) 22 = 3 + 7 × (5 × )2 Assuming the investor is risk averse: Maximising Expected utility, = 7 (2 × × 25) 0 = 7 (50 × w) t Applying FOC 7 50 = w = 0.14t Calculating w for investor 1: = , = . From (3): = 3 + 7 × 0.14 = . % From (4): = 0.14 × 5 = . % Calculating w for investor 2: 2 = 0.5, 2 = 0.07 From (3): = 3 + 7 × 0.07 = . % From (4): = 0.07 × 5 = . % Therefore, the expected return for Investor 1 is 3.98% and the standard deviation is 0.7% with 86% invested in the risk-free security and 14% invested in the market ECON3107 Assignment 2 (W18A01) 23 portfolio. Investor 2 will have an expected return of 3.49% and a standard deviation of 0.35% with 93% of their portfolio in the risk-free security and 7% in the market portfolio. ECON3107 Assignment 2 (W18A01) 24 Appendix Question1 1) 2) 3) ECON3107 Assignment 2 (W18A01) 25 Question 3 1) ECON3107 Assignment 2 (W18A01) 26 2) Question 4 2) ECON3107 Assignment 2 (W18A01) 27 4) Question 5 1) 2) ECON3107 Assignment 2 (W18A01) 28