Investment Management Easter 2018 Eric Dyke Question 1a Stop buy order definition Stop loss order definition Both are for exiting positions / limiting losses 3 3 1 1a. Explain the difference between a stop buy order and a stop loss order. (7 marks) Market orders Buy or sell at the best current price Condition Limit orders specify limits Price above Price below on the maximum buy or the limit the limit minimum sell price Action Buy Sell LimitBuy Order StopBuy Order StopLoss Order LimitSell Order Stop loss: Contingent order for someone wishing to protect a profitable long position, or limit its losses, if it drops to a given price. It then becomes a market order; it does not guarantee sale price. Stop buy order Contingent order for someone wishing to protect a profitable short position, or limit its losses, if it rises above a given price. It then becomes a market order; it does not guarantee purchase price Question 1b Implicit rf for 90 (Or NPV comparison) 7.14% (0.933) 3 Implicit rf for 110 15.79% (0.864) 3 Opportunity for arbitrage? 3 You are given the following option prices on a stock, which is currently trading at 100. A 1-year call with exercise price 90 is valued at 26 and a 1-year put with exercise price 90 is valued at 10; and a 1-year call with exercise price 110 is valued at 18 and a 1-year put with exercise 110 is valued at 13. Demonstrate whether there are arbitrage opportunities? (9 marks) 𝑋 𝑋 = 𝑆 + 𝑃 − 𝐶 → 𝑟𝑓 = −1 1 + 𝑟𝑓 𝑆+𝑃−𝐶 Arbitrage free, require PV(X) = S+P-C Call Put S X PV(X) rf 84 90 84.00 7.14% 𝐷𝐹 = 90 = 0.9333 100 110 95.00 15.79% 𝐷𝐹 = 110 = 0.8636 26 10 100 18 13 Borrow at low rate and lend at higher rate 95 Future profit on trading present sum of 100 is 15.79 -7.14 = 8.647 𝑃𝑉 = 8.647 1.1579 = 7.4675 Scaled to 93.33 (7.4675)0.9333 6.97 Buy future money (lend) at low price Present profit on trading future sum of 100 is 93.33-86.36 = 6 .97 and sell (borrow) at high price Assuming rf = 0% and arbitraging both sides of put-call parity equation is incorrect but was worth 2 marks X 90 110 C+X/(1+rf) S+P 116.00 > 110.00 128.00 > 113.00 Question 1c Alpha 3% Idiosyncratic risk 6.24% W* = 0.74 and passive 0.26 9 marks for this 3 3 3 You are given the following information about a portfolio, denoted A, and the market portfolio. The risk free rate is 3%. Portfolio A 10% 12% 0.8 Expected return Variance Beta Market portfolio 8% 9% 1 According to the Treynor-Black model, the optimal mix of the A and market portfolios for variance-averse investors is given by the formula 𝑤∗ = 𝛼𝐴 𝛼𝐴 1−𝛽𝐴 +(𝑟𝑚 −𝑟𝑓 ) 𝑉𝑎𝑟(𝑒𝐴) 𝜎2 𝑚 In this formula, w* is the weight on portfolio A, αA is Jensen’s alpha of portfolio A, βA is the beta of portfolio A, rm is the expected return on the market portfolio, rf is the risk free return, Var(eA) is the idiosyncratic risk of portfolio A, and σm2 is the variance of the market portfolio. Work out the optimal weight w*. (9 marks) α A rA- (rf β A (rm rf )) 10%- (3% 0.8(5%)) 3% 2 A2 A2 m2 12% 0.64(9%) 6.24% Var(eA) = eA 𝑤∗ = 𝛼𝐴 𝑉𝑎𝑟(𝑒𝐴) 𝛼𝐴 1−𝛽𝐴 +(𝑟𝑚 −𝑟𝑓 ) 𝜎2 𝑚 = 0.74 And hold 26% in the market portfolio. Question 2a Expectations hypothesis Segmentation theory Differences 3 3 1 How does the expectations hypothesis differ from the segmentation hypothesis as a theory of the term structure of interest rates? (7 marks) 5 Expectations Hypothesis Key Assumption: Investors are risk neutral Consider two £1 investments for a two year time horizon Two year (1 + 𝑠2 )2 Equating these 𝑠2 = One year rollovers 2 (1 + 𝑠1 )(1 + 𝑒 𝑠12 ) −1 𝑒 (1 + 𝑠1 )(1 + 𝑠12 ) 𝑒 𝑠1 + 𝑠12 ≈ 2 Long rate = average of expected short rates over life of bond. Implications: Expected return on bonds of different maturities are equal so they will be perfect substitutes. (1 + 𝑠2 )2 𝑒 𝑠12 = − 1 The expression on the right is the fwd rate. (1 + 𝑠1 )1 Thus fwd rate is an unbiased predictor of the expected future spot. An upward sloping term structure implies an expectation of rising spot rates with the expectations hypothesis. (1 + 𝑠2 )2 = (1 + 𝑠1 ) (1 + 𝑓12 ) 𝐼𝑓 𝑠2 > 𝑠1 then 𝑓12 > 𝑠1 Since the forward rate is the expected future spot an upward sloping term structure implies expectation of a rise in spot rates. 7 Segmented markets theory Key Assumption: Investors are completely risk averse Implication: Markets are totally segmented: interest rate at each maturity determined separately Borrowers will want to match two year investments with two year money and lenders will match their loans to their future money needs, so neither face interest rate risk. . Bonds of different maturities are not substitutes at all. The theory is consistent with any shaped yield curve. Question 2b Gross cash flows Margin cash flows IRR 3 3 3 You short 500 shares. The initial price of the shares is 100, and the shares pay a dividend at the end of the first year of 10 per share. The price of the shares (ex-dividend) at the end of the first year is 90, and you buy back 150 shares at this point. The dividend at the end of the second year is also 10 per share. The price of the shares (exdividend) at the end of the second year is 80, and you clear your position by buying back the remaining 350 shares at the end of the second year. The initial and maintenance margin is 40%. As a retail investor, you must leave the proceeds of the short sale in the margin account. Assuming you maintain the minimum margin requirements at all times, what is the 2-year return on your short sale transaction? (9 marks) Gross cash flow Margin cash flow £50,000 -£70,000 -£18,500 £25,900 -£31,500 £44,100 Net cash flow -£20,000 £7,400 £12,600 IRR 0% 3 marks 3 marks 3 marks Finding the IRR General form of a quadratic equation: a x2 b x c 0 −𝑏 ± 𝑏2 − 4𝑎𝑐 𝑥= 2𝑎 −20 + 7.4 1+𝑟 1 + 12.6 (1+𝑟)2 =0 −20(1 + 𝑟)2 + 7.4(1 + 𝑟)1 + 12.6 = 0 −7.4 ± 7.42 − 4(−20)(12.6) 𝑥= = −0.63 or 1 → 𝑟 = −163% 𝑜𝑟 0% 2(−20) We select the answer closest to zero, so r = 0% The money weighted return, IRR, is appropriate because it reflects the return on actual money invested. 11 Question 2c Market risk 8.1% 3 Ideosyncratic risk formula 3 Total risk 3 8.63% or 16.1% say Suppose you invest in an equally weighted portfolio of stocks with the same beta and the same level of unsystematic risk. Suppose the beta is 0.9, the variance of the market portfolio is 10%, and the idiosyncratic variance for each stock is 8%. What is the total variance of your portfolio if you include 15 stocks in your portfolio? (9 marks) If the unsystematic risks are uncorrelated then: 2 𝑛=15 1 2 0.08 = + 𝜎𝑒,𝑖 = 0.081 + = 8.63% 𝑛 15 𝑛=1 If the unsystematic risks are perfectly correlated then: 𝜎𝑝2 𝜎𝑃2 2 𝑏𝑝2 𝜎𝑚 = 8.63% + = 8.63% + 𝑖 12 𝑖 𝑗≠𝑖 𝑛 𝐶𝑜𝑣(𝑒𝑖 𝑒𝑗 ) 12 𝑗≠𝑖 𝑛 𝜌𝑒𝑖,𝑒𝑗 𝜎𝑒𝑖 𝜎𝑒𝑗 ) 2 1 = 8.63% + 152 − 15 1 15 0.08 = 8.63% + 7.47% = 16.1% Question 3a Formula for put Hedge portfolio, Delta, Hedge ratio 2 5 The Black-Scholes formula for call options is 𝑐 = 𝑆𝑁 𝑑1 − 𝑃𝑉 𝑋 𝑁(𝑑2 ) where c is the call price, S is the stock price, PV(X) is the discounted value of a risk free cash flow equal to the exercise price X at the maturity date of the option, N(.) is the cumulative standard normal distribution function, and d1 and d2 are parameters that are derived from the stock price, the exercise price, the time to maturity, the volatility of the stock, and the risk free rate of return. Using put-call parity, what is the formula for the value of a put option on the same stock with the same exercise price? Explain how you can use the call and the put options to create a volatility hedge that works well without rebalancing for small changes in the stock price. (7 marks) Straight from subject guide "Hedging Volatility" on page 116 contains the material for this question. Hedging volatility Black-Scholes model: 𝑐 = 𝑆𝑁 𝑑1 − 𝑋𝑒 −𝑟𝑡 𝑁(𝑑2 ) S 2 ln( ) ( r )t X 2 d1 t 𝑝 = −𝑆 + 𝑆𝑁 𝑑1 − 𝑋𝑒 −𝑟𝑡 𝑁 𝑑2 + 𝑋𝑒 −𝑟𝑡 d 2 d1 t = −𝑆 1 − 𝑁 𝑑1 + 𝑋𝑒 −𝑟𝑡 (1 − 𝑁 𝑑2 ) = −𝑆 1 − 𝑁 𝑑1 + 𝑋𝑒 −𝑟𝑡 (𝑁 −𝑑2 ) Option portfolio: xc + yp = 𝑥(𝑆 𝑁 𝑑1 − 𝑋𝑒 −𝑟𝑡 𝑁 𝑑2 𝑑 𝑂𝑝𝑡𝑖𝑜𝑛 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝐷𝑒𝑙𝑡𝑎 = 𝑑𝑆 𝑥 1−𝑁 𝑑1 0.496011 = = ≈1 𝑦 𝑁 𝑑1 0.503989 + y(−𝑆 1 − 𝑁 𝑑1 + 𝑋𝑒 −𝑟𝑡 𝑁 −𝑑2 ) = 𝑥𝑁 𝑑1 − 𝑦(1 − 𝑁 𝑑1 ) = 0 if d is small (eg = 0.01) Buy 1 call and 1 put to hedge a rise in volatility. 14 Question 3b Substitution in utility function Derivation result via optimisation Critical value lend / borrow 3 3 3 An investor has mean-variance preferences which can be expressed as the function U(µ,σ2) = µ - (ρ/2)σ2, where µ is the expected return on the investor’s portfolios, σ2 is the variance of the investor’s portfolio, and ρ > 0 is a parameter describing the investor’s variance aversion. Derive the optimal portfolio for the investor when all investors have mean-variance preferences and there exists a risk free asset. Also derive the critical value of ρ which determines the cut-off point between investors who are net lenders and net borrowers of the risk free asset. (9 marks) 𝜇 = 𝑤𝑟𝑚 +(1 − 𝑤)𝑟𝑓 Two fund separation 2 +(1 − 𝑤)2 𝜎 2 + 2𝑤(1 − 𝑤)𝜌 2 2 𝜎𝑝2 = 𝑤 2 𝜎𝑚 𝑚,𝑟𝑓 𝜎𝑚 𝜎𝑟𝑓 =𝑤 𝜎𝑚 𝑟𝑓 If investors have a CARA utility function and portfolio returns are normal, we can write the expected utility of holding a portfolio as : 𝐸 𝑢 𝑥 𝜌 𝜌 2 = 𝜇 − 2 𝜎𝑝2 = 𝑤𝑟𝑚 + 1 − 𝑤 𝑟𝑓 − 2 𝑤 2 𝜎𝑚 𝑑𝑈 𝜌 2 =0 = 𝑟𝑚 − 𝑟𝑓 − 2𝑤𝜎𝑚 𝑑𝑤 2 ∴𝑤= 1 𝑟𝑚 − 𝑟𝑓 2 𝜌 𝜎𝑚 1 𝑟𝑚 −𝑟𝑓 2 𝜎𝑚 ∴𝜌=𝑤 = 𝑟𝑚 −𝑟𝑓 2 𝜎𝑚 for w = 1 The more risk averse an individual is the less the weight on the market portfolio ∴𝜌> 𝑟𝑚 −𝑟𝑓 2 𝜎𝑚 for w < 1 𝑎𝑛𝑑 𝜌 < 𝑟𝑚 −𝑟𝑓 2 𝜎𝑚 for w >1 Question 3c Log transformation (standard normal variable) VaR calculation 4 5 If the value of a portfolio follows a geometric Brownian motion with drift rate 6% and volatility 20%, then the log return of the portfolio from time 𝑡 to time 𝑇 is normally distributed with mean 6% − 0.5 ∗ (20%)^2 (𝑇 – 𝑡) and variance 0.04 ∗ (𝑇 – 𝑡). What is the 10-day, 1% VaR of the portfolio? You should give your answer in terms of logreturns. You are also given the following number: For a standard normal random variable with zero mean and unit variance, the probability that z is less than or equal to -2.33 is approximately 1%. (9 marks) Answer to actual question Answer to intended question (A bracket was put in the wrong 𝑉 place!) 𝑙𝑛 𝑉𝑇 −𝜇10 𝑑𝑎𝑦 𝑍 = 𝜎𝑡 = -2.33 10 𝑑𝑎𝑦 𝜇10 𝑑𝑎𝑦 = (𝜇 𝜎10 𝑑𝑎𝑦 = 𝜎 𝑉 𝑙𝑛 𝑉𝑇 𝑡 = 0.0011 - 2.33 (0.0331) = -0.076 𝑉𝑎𝑅 = 𝑉𝑡 − 𝑉𝑇 = (1 − 𝑒 −0.076 )𝑉𝑡 = (1 − 0.9268)𝑉𝑡 = 7.32% of 𝑉𝑡 𝜎 2 10 − 2 ) 365 = 0.0011 10 = 0.0331 365 ↔ 𝑉𝑇 𝑉𝑡 = 𝑒 −0.076 𝜇10 𝑑𝑎𝑦 = 𝜇 − 𝑙𝑛 𝑉𝑇 𝑉𝑡 𝜎 2 10 ( 2 ) 365 = -0.0177 ↔ = 0.05945 𝑉𝑇 𝑉𝑡 = 𝑒 −0.0177 𝑉𝑎𝑅 = 𝑉𝑡 − 𝑉𝑇 = (1 − 𝑒 −0.0177 )𝑉𝑡 = 1.753% Question 4a What we mean by market microstructure Difference G-M and Kyle Which is the more general model 2 4 1 Explain what we mean by market microstructure. In intuitive terms, what is the essential difference between the Glosten-Milgrom model and the Kyle model of the microstructure of a market. Which model is the most general? (7 marks) Microstructure looks at the trading process facilitating trade in financial securities. The mix of insider and liquidity trader participants is a key factor determining trading costs through liquidity and the bid-ask spread. Glosten-Milgrom is about how the market maker determines the bid-ask spread when investors sequentially trade given quantities. Each step entails the MM and a single trader. MM prices are set using the nature of the order to ascertain the probability that the fundamental price is high or low. The Kyle model is a wider strategic game between a market maker and many insider traders. It focuses on the insider’s optimal order size, to maximise profits, as well as the market maker’s optimal price (not the spread). The MM adjusts price according to the size of the net order flow, with which it is correlated. An insider trader will try to hide the size of their trades amongst the orders given by noise traders. Question 4b Duration 4.2859 (or modified 4.1611) 3 Convexity 22.9128 3 Predicted price 126.714 (or -4.047% or -5.495) 3 Consider a 5-year bond with a coupon of 10% and standing on a yield of 3%. What are the duration and the convexity of this bond? Using these numbers only, predict the bond’s new price if its yield moved to 4%. [9 marks] 10 10 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 132.058 10 + 1.03 10 10 110 𝑃 = 1.03 + (1.03)2 + (1.03)3 + (1.03)4 + (1.03)5 = 132.058 1 10 𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1 10 10 110 2 (1.03)2 + 3 (1.03)3 + 4 (1.03)4 + 5 (1.03)5 = 4.2859 10 4.2859 = 4.1611 1.03 10 10 10 110 𝐶𝑜𝑛𝑣𝑒𝑥𝑖𝑡𝑦 = 132.058(1.03)2 2 1.03 + 6 (1.03)2 + 12 (1.03)3 + 20 (1.03)4 + 30 (1.03)5 = 22.9128 Study Guide answer of 11.456 is fully acceptable. 𝑑𝑝 1 ∗ = −𝐷 𝑑𝑟 + 𝐶𝑑𝑟 2 𝑝 2 -4.1611(0.01) ∴ 𝑃∗ =132.058 (1- 0.04161) = 126.714 22.9128 (0.01) 2 4.047% 2 18 Question 4c Yields 4%, 5.95%, 7.91% 3 Spots Borrowing profitable 4%, 6%, NPV 9,626 sqrt(1.2597)-1 8% f13=10.06% 3 3 Consider the data below on prices and coupon rates: Bond price per £100 nominal Coupon rate Time to maturity £102.88 7% 1 year £98.26 5% 2 years £87.32 3% 3 years Calculate the yields y1 and y2, of the one and two year bonds and provide an expression for solving the three year bond yield y3. Calculate the one and two year spot rates s1 and s2 and provide an expression for solving the three year spot rate s3. You are to assume that coupons are paid annually, with the next payment being in 12 months’ time. You get the opportunity to borrow £100,000 at the end of year 1, against promising to repay £109,000 at the end of year 3. Should you take this borrowing opportunity? Explain. Yields: 107 102.88 = 1+𝑦 1 5 → 𝑦1 = 4% 105 98.26 = 1+𝑦 + (1+𝑦 2) 2 87.32 = Spots: 3 1+𝑦3 3 + (1+𝑦 )2 3 107 102.88 = 1+𝑠 1 5 + 103 (1+𝑦3 )2 → 𝑦3 = 7.91% 105 2) 2 3 3 + 1.04 (1.06)2 + → 𝑠2 = 6% 103 (1+𝑠3 )3 → 𝑠3 = 8% 103 81.7654 = (1 + 𝑠3 )3 (1 + 𝑠3 )3 = But you are not required to work this out! → 𝑠1 = 4% 98.26 = 1.04 + (1+𝑠 87.32 = → 𝑦2 = 5.95% 2 103 = 1.259702 81.7654 But you are not required to work this out! Sell 100,000 due in year 1 for 100,000 96,154 Buy 109,000 in year 3 for Liability covered by cash from loan in year 1. Proceeds cover loan repayment. No future liability. 109,000 109,000 = = 86,528 (1 + 𝑠3 )3 1.259712 1.04 £100,000 𝑠3 = 8% 96,154 LOAN: 0 1 𝑠1 = 4% 2 86,528 3 Current profit 96,154 - 86,528 = 9,626 £109,000 This is profitable and zero risk, so repeat many times. Alternatively show that the forward rate for the loan period exceeds the cost of the loan. 3 (1 + 𝑠3 ) = (1 + 𝑠1 ) (1 + 𝑓13 )2 𝑓13 = (1 + 𝑠3 )3 − 1 = 10.06% (1 + 𝑠1 ) The annual cost of the loan is (1.09)0.5-1 = 4.4% The annual loan rate is much cheaper than the market. Question 5a True in isolation Pricing of derivatives Allows reduction of deadweight costs (financial distress), agency + info costs + tax efficient 3 4 “Hedge transactions involving the trading of derivatives have zero net present value, so will never increase the value of the corporation.” Discuss this statement, and explain why hedging of corporate risk nonetheless can add value to corporations. (7 marks) Investors care mainly about the trade off between risk and return (risk aversion). Risk management for investors concerns measuring this and aligning investments to the risk appetite of the investor. Corporations are risk neutral and care about shortfall of capital as this could lead to financial distress. Derivative positions have zero NPV and do not advance profits when bankruptcy is far from a question. Corporate risk management programmes focus on measuring the likelihood of a critical shortfall of capital and rectifying it (value-at-risk based programmes, or cash-flow-at risk programmes, for instance). Question 5b Pension liability Pension duration Position in zero coupon bond 500 51 364.29 3 3 3 Your company has a pension liability which leads to projected annual payments of $10m forever. The term structure is flat at 2%. You seek to immunise your equity holders against the risk of parallel fluctuations in the term structure, and consider trading a zero coupon bond with a maturity of 70 years. Explain how you work out the optimal dollar investment in the bond. (9 marks) 𝐹𝑜𝑟 0 < 𝑥 < 1, 𝑥 + 𝑥 2 +𝑥 3 + … = ∞ 𝑡 𝑃𝑒𝑛𝑠𝑖𝑜𝑛 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑡=1 51 Total £mn 𝑎𝑛𝑑 𝑥 + 2𝑥 2 +3𝑥 3 + … = 𝑃𝑒𝑛𝑠𝑖𝑜𝑛 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 Pension liability: Duration Assets 70 Bond A 0 Cash / Bond B 𝑥 1−𝑥 10 (1 + 𝑟)𝑡 𝑃𝑡 = 10 10 + 1.02 (1.02)2 𝑥+2𝑥 2 +3𝑥 3 + 𝑥 1−𝑥 … 𝑥 𝑤ℎ𝑒𝑟𝑒 (1−𝑥)2 10 + (1.02)3 + ⋯ = = $mn $x $(500-x) Liabilities Pension liability $mn $500 Duration 51 $500.00 Total £mn $500 51 𝑥 (1−𝑥)2 𝑥 1−𝑥 1 = 1−𝑥 = 10 0.02 1+𝑟 𝑟 𝑥 70 500 + ∴𝑥= 500 51 70 1 𝑥 = 1+𝑟 = 500 1.02 = 0.02 = 51 (500−𝑥) 0 500 500 = 500 51 = 364.29 Note that this is only covers part of liabilities. A sum of $135.71 must be held as cash. Question 5c Market return 20.66% Returns 33.33%,25,3.64 Weights 4.23%,1.45, -5.67 ($7,448, 2,551,-10,000) 1 2 6 An investor wants to implement a returns-based momentum strategy. Using the method in the Study Guide, what money position should be held in each of the three stocks, given below, if the total long position is to be $10,000? Explain your reasoning. [9 marks] Security A B C Price (pence) Yesterday Today 27 36 52 65 55 57 24 Return based trading strategies Invest on the basis of yesterday’s returns Security Period 1 Period 2 % Return, t-1 A 27 36 33.33% B 52 65 25% C 55 57 3.64% 20.66% Mean 𝑤𝑖,𝑡 1 = ± 𝑅𝑖, 𝑡−1 − 𝑅𝑚, 𝑡−1 𝑁 𝑖 𝑤𝑖, 𝑡 𝑤𝐴,𝑡 = i.e. 1 3 = 1 𝑖 𝑁 𝑅𝑖, 𝑡−1 − 𝑅𝑚, 𝑡−1 = 0 in A Costless longshort position Extreme divergence merits a large weight + for momentum strategy -ve contrarian Ideally buy ‘winners’ using funds from the sale of ‘losers’, so that the total money spent will sum up to zero. i.e. Market return is equally weighted index 33.33% − 20.66% = 4.226% 4.226 =74.48% 4.226+1.448 Guide Weights, t weights $7,448 4.226% $2,551 1.448% -$10,000 -5.673% $0 0.00% …and weights constitute the dollar amount to be invested in each stock. For every $4.226 in A, put $1.448 in B. Question 6a ETF definition – Traded as stock Track other assets eg index trackers Cheap + Diversification + flex benefits 2 2 3 Explain what we mean by exchange traded funds. What benefits do these funds offer to investors? (7 marks) ETFs are exchange traded funds, which are funds that can be traded as individual stocks on the exchange, but are ‘tracking’ the value of other assets such as the stock market index. They are used to achieve diversification benefits without requiring to make an investment across many stocks, a job done for them by a financial institution making the market. Question 6b Derivation alone Cov is -ve and c = 0.01P/2 Calculation Cov = - 0.25 5 2 2 Consider an asset which has a fundamental price mt with increments (mt – mt-1) between time t-1 and time t. The increments are identically and independently distributed over small time increments with mean value zero and constant variance. The transaction price depends on mt and the market spread, so that if the transaction is a buy transaction, the price is pt = mt + c, and if it is a sell transaction, the price is pt = mt – c, where c is the “half-spread”. Assume that buy and sell transactions are equally likely. Derive the covariance term. If the fundamental price mt is currently 100 and the total spread is 1% of this price, what is the approximate covariance of the next successive price changes (pt+1 – pt) and (pt+2 – pt+1)? (9 marks) 𝑚𝑡 = 𝑚𝑡−1 + 𝑢𝑡 mt is the EMH price at time t. Ask 𝑝𝑡 = 𝑚𝑡 + 𝑐 Bid 𝑝𝑡 = 𝑚𝑡 − 𝑐 pt is the transaction price at time t. Generally 𝑝𝑡 = 𝑚𝑡 + 𝑞𝑡 𝑐 𝑢~𝑁(0, 𝜎𝑢2 ) where 𝑞𝑡 = +1 𝑜𝑟 − 1, ∴ 𝐸 𝑞 = 0 𝑉𝑎𝑟 𝑞 = 𝐸(𝑞 − 0)2 = 𝐸 𝑞 2 = 1 ∆𝑝𝑡 = 𝑚𝑡 − 𝑚𝑡−1 + 𝑞𝑡 𝑐 − 𝑞𝑡−1 𝑐 = 𝑢𝑡 + 𝑞𝑡 − 𝑞𝑡−1 𝑐 𝐶𝑜𝑣(∆𝑝𝑡−1 , ∆𝑝𝑡 ) =E(∆𝑝𝑡−1 ∆𝑝𝑡 ) = E 𝑢𝑡−1 + 𝑞𝑡−1 − 𝑞𝑡−2 𝑐 2 = −𝑐 2 E(𝑞𝑡−1 ) = −𝑐 2 𝐶𝑜𝑣(∆𝑝𝑡−1 , ∆𝑝𝑡 ) = −𝑐 2 =− 1%𝑃 2 = 2 - 0.25 𝑢𝑡 + 𝑞𝑡 − 𝑞𝑡−1 𝑐 Question 6c Invest risk free / calls 93.46 /6.54 Max value of X 81.78% Porttfolio return 8.178% or 108.18 4 3 2 You are designing a savings scheme by which customers can invest a fixed amount (say 100) at the beginning of the year, and receive at the end of the year max(100,100*(1+X*rM)), which is the maximum of 100 (the amount invested), and 100 plus a return on 100 equal to X times the return on the stock market index over the year rM. So, for instance, if you have invested 100 and the stock market index has increased by 15% over the year, you will get back 100(1+0.15*X) at the end of the year. Suppose the risk free return is 7% and a one-year call option on the stock market index with exercise price equal to the current index value is trading at 8% of the index value. What is the maximum value of x you can promise your savings customers? What then would portfolio return be if rm=10% (9 marks) 𝐼𝑛𝑣𝑒𝑠𝑡 𝑟𝑖𝑠𝑘 𝑓𝑟𝑒𝑒 6.54 𝑖𝑛 𝑐𝑎𝑙𝑙s → 100 1.07 6.54 8 = 93.46 → 6.54 𝑡𝑜 𝑖𝑛𝑣𝑒𝑠𝑡 𝑖𝑛 𝑐𝑎𝑙𝑙 = 81.78% of a call 𝑃𝑎𝑦𝑜𝑓𝑓 𝑓𝑟𝑜𝑚 £100 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 = 𝑚𝑎𝑥 100,100 + 0.8178( 1 + 𝑟𝑚 100 − 100) = 𝑚𝑎𝑥 100,100(1 + 0.8178𝑟𝑚 ) Clients would find their portfolio worth 108.18 if the market return were 10%. Question 7a Protecting unsophisticated Fairness and transparency Limiting systemic risk 3 2 2 Give three reasons why financial markets are regulated. (7 marks) Regulation of financial markets encompasses : 1. regulation of the way investors and companies behave when they trade the company’s issued capital to give protection to small or unsophisticated investors, so they receive sound advice on investment opportunities and investment vehicles. 2. Requiring information releases from listed companies to be fair so all investors are treated equally.… and to ensure that sufficient information is released in order for investors to make a sound judgment of the company’s attractiveness as an investment. 3. regulation of financial institutions to make sure that we have sufficient confidence that our banks are safe from financial distress. Question 7b Dirty price 99.80 Pc (approx) 99.80-65(4.7)/365 = 98.96 AI = 0.84 (approx) OR Pc (exact) 99.18 and AI (exact) = 99.80 - 99.18 = 0.62 3 6 6 Explain what we mean by accrued interest for a bond that is traded in a market. Suppose a bond is trading at a yield of 5%, and that the bond has a coupon rate of 4.7%, paid annually. The bond has a maturity of 3 years and 300 days. Coupon payments are due in 300 days, 1 year and 300 days, 2 years and 300 days, and 3 years and 300 days. The discount factor (1/1.05)^x is equal to 0.96069192 when x = 300/365 and it is equal to 0.99134898 when x = (365-300)/365. What is the quoted price for the bond? (9 marks) 𝑃𝐷 = 4.7 300 + 1.05365 4.7 1.05 300 + 1+365 4.7 300 + 2+365 1.05 104.7 300 3+365 = 1.05 = 65 𝑃𝐶 ≈ 𝑃𝐷 − 3654.7 = £99.80 - £0.84 = £98.96 4.7 𝑃𝐶 = 1.051 + 4.7 1.052 104.7 + 1.053 = 99.18 1 4.7 300 1.05365 4.7 + 1.051 + 4.7+𝑐𝑙𝑒𝑎𝑛 𝑝𝑟𝑖𝑐𝑒 300 1.05365 4.7 1.052 104.7 + 1.053 = 99.80 But this is an approximation 𝐴𝐼 = 𝑃𝐷 − 𝑃𝐶 = 99.80 − 99.18 = 0.62 Question 7c X=1.6, 1-X = -0.6 Average return C =16%, β1=1.5 APT, λ1=7%, λ2=3% 3 3 3 Suppose stocks face a 2-factor structure. You are given the following data: Avg return Beta 1 Beta 2 Fund A 14.80% 1.2 0.8 Fund B 12.80% 0.7 1.3 Fund C 0.5 Funds A and B are well diversified. Fund C consists of a linear combination of A and B only. The risk free rate is 4%. What is your best estimate of the missing values for fund C? What is your best estimate of the risk premiums for factors 1 and 2? (9 marks) Factor 2: X(0.8)+(1-X)1.3 = 0.5. Therefore X = 1.60 Average return for fund C = 1.6*14.8+(-0.60)*12.8 =16% Factor 1 beta for fund C: 1.6*1.2+(-0.6)*0.7 = 1.5 𝑟𝐴 = 𝑟𝑓 + 𝑏𝐴1 λ1 + 𝑏𝐴2 λ2 = 4+ 1.2λ1 + 0.8λ2 𝑟𝐵 = 𝑟𝑓 + 𝑏𝐵1 λ1 + 𝑏𝐵2 λ2 = 4 + 0.7λ1 + 1.3λ2 𝑟𝐶 = 𝑟𝑓 + 𝑏𝐶1 λ1 + 𝑏𝐶2 λ2 = 4% +1.5(7%) + 0.5 3% = 16% Solving 1 7% and 2 3% Question 8a Definition of MM instruments Example 1 Exmple 2 3 2 2 Explain the defining characteristics of money market instruments, giving two examples of these types of instruments and stating clearly how they are issued and traded. (7 marks) Money market instruments are short maturity, liquid, fixed income (debt) instruments traded in large denominations and with relatively low risk. Examples: US T bills issued by the US Treasury (exchange traded – but often in low denominations atypical of money market instruments), free of default risk; certificate of deposits issued by private banks (tradeable rights to ownership of a bank deposit commercial paper issued by corporations (again tradeable), often protected by a bank credit line. Question 8b M2 formula Optimal weights, 1.75 (active) and -07.5 (mkt) Solving for r cutoff =9.29% 3 3 3 The expected return on the market index is 11%, with standard deviation 0.35, and the risk free return is 3%. You consider holding a portfolio that has at most standard deviation 0.2, subject to the constraint that the portfolio earns an M2 measure of 3%. What expected return is required to meet your investment objective if the portfolio has maximum risk? (9 marks) 𝑀2 𝜎𝑚 𝜎𝑚 = 1− 𝑟 + 𝑟 − 𝑟𝑚 𝜎𝑝 𝑓 𝜎𝑝 𝑝 35 35 3% = 1 − 20 3% + 20 𝑟𝑝 - 11% ∴ 3% = (−0.75)3% + 1.75𝑟𝑝 - 11% ∴ 𝑟𝑝 = 16.25% 1.75 = 9.29% Question 8c Tree P(H|Ask) 0.55 Ask 107.5 Bid 102.5 P(H|Bid) 0.45 3 3 3 A stock is currently valued at 105, but has a true value of either 130 or 80 with equal probabilities. There is a 90% chance the next trader is an uninformed trader who is equally likely to buy or sell one unit, and a 10% chance the next trader is an informed trader who trades on the basis of his or her (perfect) information about the true value of the stock. Work out the market maker’s bid and ask prices. (9 marks) P 𝐴𝑠𝑘 130 = 0.1 1 + 0.9 0.5 = 0.55 P 𝐵𝑖𝑑 80 = 1 − 0.45 = 0.55 P 𝐵𝑖𝑑 130 = 1 − 0.55 = 0.45 Consider the following tree diagram of conditional 1 probabilities: P 𝐴𝑠𝑘 80 = 0.1 0 + 0.9 0.5 = 0.45 Buy at Ask 0.1 I P 130 𝐴𝑠𝑘 = P 𝐴𝑠𝑘 130 𝑃130 P(𝐴𝑠𝑘) 130 0.5 0.9 Buy at Ask 0.45 N 0.5 0 .1 80 0.9 P 𝐴𝑠𝑘 130 𝑃130 P 𝐴𝑠𝑘 130 𝑃130 + P 𝐴𝑠𝑘 80 𝑃80 = 0.55+0.45 = 0.55 if P130 = P80 = 0.5 𝐴𝑠𝑘 𝑝𝑟𝑖𝑐𝑒 = 𝑃 130 𝐴𝑠𝑘 130 + 𝑃 80 𝐴𝑠𝑘 80 = (0.55)130 + (1-0.55)80 = 107.5 Sell at Bid 0.45 P 130 𝐵𝑖𝑑 = I = 0.55 0 .1 1 Application of Bayes is required for GM numericals Do not assume these concepts are the same 1 Sell at Bid 0.1 0.5 Buy at Ask 0.45 P = 𝐵𝑖𝑑 130 𝑃130 P(𝐵𝑖𝑑) 0.45 (0.5) 0.45(0.5)+0.55(0.5) = P 𝐵𝑖𝑑 130 𝑃130 P 𝐵𝑖𝑑 130 𝑃130 + P 𝐵𝑖𝑑 80 𝑃80 = 0.45 𝐵𝑖𝑑 𝑝𝑟𝑖𝑐𝑒 = 𝑃 130 𝐵𝑖𝑑 130 + 𝑃 80 𝐵𝑖𝑑 80 = (0.45)130 + (1-0.45)80 = 102.5 N 0.5 Sell at Bid 0.45 35
