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BUS315

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Feb 20, 2024

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A derivative is a financial instrument whose value, today and in the future, is derived from an underlying asset. Put differently: a contract that stipulates rights and/or obligations with respect to future cash flows, which depend on the underlying. Use of derivatives: take exposure on the underlying asset or reduce exposure to it, i.e. to reduce risk via hedging. Derivatives that we discuss in this course: Forwards and futures, Swaps, Options. Forward contract : obligation to buy or sell a security or commodity at a pre-specified price, i.e. the forward price, at some date in the future, i.e. the maturity date. timing of cash flows: only at the maturity date. long position: pay the forward price F 0 , receive the asset worth S T . short position: receive the forward price F 0 , deliver the asset worth S T . Payoffs: long position: S T -F 0 and short position: F 0 -S T Futures contract : a special type of forward contract, traded on organized exchanges. timing of cash flows: daily marking to market in margin account. Same payoffs. Swaps : A swap is an agreement between two counterparties to periodically exchange the cash flows of one security for the cash flows of another. The date of the last exchange determines the swap maturity. You can think about a swap as being a combination of multiple forward transactions. Interest rate swap : exchanges the cash flows of a fixed-rate bond for the cash flows of a floating-rate bond. Notional amount of the swap represents the size of the principal on which interest is calculated. Fixed-rate set such that the initial value of the swap is zero. Currency swap : exchange of cash flow streams in different currencies. Typically, par bonds. EU Options : The holder has the right to buy ( call option ) and/or sell ( put option ) at ( European style ) an agreed- upon expiry date , an agreed-upon quantity of the underlying at an agreed-upon price ( strike price ), from/to the writer (short position) of the option. AM Options : The holder has the right to buy ( call option ) and/or sell ( put option ) up until/at ( American style ) an agreed-upon expiry date , an agreed-upon quantity of the underlying at an agreed-upon price ( strike price ), from/to the writer (short position) of the option. Tracking portfolio and no-arbitrage : Idea: construct a portfolio that replicates the payoffs of the derivative. Under no-arbitrage , the law of one price implies that a derivative and its tracking portfolio must have the same price. Forward contract on stock (no dividends) : The no-arbitrage value of a forward contract on a non-dividend paying stock S 0 -((K)/(1+r f ) T ) . To construct a zero-cost forward contract at time zero, the forward price has to be F 0 =S 0 (1+r f ) T where S0 denotes the current stock price and rƒ is the riskfree rate. Valuing a currency forward contract : The no-arbitrage forward rate to buy foreign currency in T years is F 0 =S 0 ((1+r foreign )/(1+r domestic )) . Valuation with tracking portfolios : DS u +B(1+r f )=V u & DS d +B(1+r f )=V d where D=number of units of the underlying in the tracking portfolio, B= the amount invested in the risk-free asset, S u & S d = values of underlying in up- and down-state, respectively, & V u & V d = values of derivative in the up- and down-state, respectively. The current value of the tracking portfolio, and thus – due to the law of one price – the current value of the derivative is V=DS+B . Risk-neutral valuation : We can use the binomial process for the underlying to extract up and down probabilities for the purpose of valuation. These probabilities are risk-neutral . In the absence of arbitrage opportunities, the one-period forward price must be consistent with possible future spot prices: F=(P)S u +(1-P)S d . Option premium : the price paid by the holder to the writer of the option. Usually paid upfront. Intrinsic value : the (hypothetical) value of the option if it would be exercised right now. In/at/out of the money (ITM, ATM, OTM) : the strike relative to the current price is such that immediate exercise would yield a positive / zero / negative cashflow. Time value : = premium – intrinsic value. Positive if the market thinks that it’s better to postpone exercising. Call and put option payoffs: Long & Short Put-call parity for European options : C 0 -P 0 =S 0 -PV(K) . For dividend included: C 0 -P 0 =S 0 -PV(K)-PV(div) . PCP implications for American options : The price of an American option should exceed that of a corresponding European option by the value of its right to also exercise prior to the maturity date. (Without frictions,) American call options on stocks with no dividends should never be exercised before expiration, because with PV(K) < K, we have that S 0 -K<S 0 -PV(K) . Volatility and option prices : u=e sd({T/N & d=1/u Option price sensitivities : Sensitivity to changes in the underlying S , typically referred to as delta, and the moneyness of options: positive for calls, because increase in S increases moneyness. negative for puts, because increase in S decreases moneyness. Sensitivity to rƒ . For European options on stock w/o dividends: positive for calls, due to discounting of K. negative for puts, due to discounting of K. Sensitivity to T . For European options on stock w/o dividends: positive for calls, due to uncertainty and discounting of K. ambiguous for puts, due to uncertainty (+) and discounting of K (). FX derivatives : Exchange rates can be expressed as foreign currency units per one unit of home currency: S0 = #FC/1HC, ‘indirect quote’. home currency units per one unit of foreign currency: S0 = #HC/1FC, ‘direct quote’. With indirect quotes, Covered Interest Rate Parity implies: F 0 =S 0 ((1+r for )/(1+r dom )) , With direct quotes, Covered Interest Rate Parity implies: F 0 =S 0 ((1+r dom )/(1+r for )). Strategic option : possibility to alter a project that is underway or to enter new projects as a result of some other, earlier investment. Valuation of risky projects : Valuation based on discounting future cash flows Either, use expected future cash flows and a risk-adjusted discount rate OR use certainty-equivalent cash flows and the risk-free rate . Risk-adjustment based on CAPM or APT For the discount rate, we’ll also consider the dividend discount model. The risk-adjusted discount rate method : Ñ Discount expected cash flows at the project’s cost of capital . For this approach, we need: expected cash flows, expected return on market (and factor portfolios), CAPM beta (and factor betas) of project returns. To find the present value of the future cash flow, 1. compute the expected future cash flow next period, E(C˜), 2. compute the beta of the return of the project, 3. compute the expected return of the project by substituting the beta (from step 2) into the tangency portfolio risk-expected return equation 4. divide the expected future cash flow (from step 1) by one plus the expected return (from step 3) PV= E(C)/(1+r f +B(r T -r f ) where B =Beta and (r T -r f ) =MRP.

Risk Neutral Valuation, No Dividends: Given: market value of the underlying asset (a stock, above the node) and the cash flow generated by the derivative (below the node). No dividends. Using the risk-neutral valuation, calculate the present value of a derivative. The risk-free rate is 5% per period. 1. Solve the tree backwards, node by node using: 2. Find Vu and Vd. Find P and 1-P using S 0 *(1+r F )=(P)*Vu+(1-P)*Vd . 3. Find the values for U and D using EV=((P)*Pu+(1-P)*Pd)/(1+r f ) 4. At initial node, use PV=((P)*Pu+(1-P)*Pd)/(1+r f ) to get present value. EU Call Option Value : Find V given S 0 , S u and S d and rf. 1. Find P and 1-P using S 0 *(1+r F )=(P)*Vu+(1-P)*Vd 2. Figure out Pu and Pd as P=MAX(0, ST-K) where K is Strike Price and ST is Spot at T. 3. Find PV=((P)*Pu+(1-P)*Pd)/(1+r f ) EU Put Option Value : Find V given S 0 , S u and S d and rf. 1. Find P and 1-P using S 0 *(1+r F )=(P)*Vu+(1-P)*Vd 2. Figure out Pu and Pd as P=MAX(0, K-ST) where K is Strike Price and ST is Spot at T. 3. Find EV for each node. 4. Find PV=((P)*Pu+(1-P)*Pd)/(1+r f ) Hedging Using Tracking Portfolio and Delta : 1. We go from T=0 to T=T forward in time not backwards. 2. At T=0 you sell call option for X and you know the delta (delta can be found using D=(call price up-call price down)/(value undelying up-value underlying down) ), now D*Vunderlying-X=Y, so Y will be borrowed and to supplement X to be used to buy D of the stock for the tracking portfolio to hedge. 3. At T=1, Delta is new to this node so we have to adjust portfolio, D 0 -D 1 =Dextra we sell Dextra which is the difference of old and new delta (we would buy the diff of new and old Delta if the underlying goes up) to get income Dextra*VunderlyingT1=income, now Y*(1+rf)-income=current debt. 4. Keep doing for all nodes. At the end debt should equal 0. CAPM formula : ER=R f +B(ER m -R f ) where ER is the expected return, B is beta, R f is the risk-free rate, and ER m is the expected return of the market, making (ER m -R f ) the market risk premium. Beta formula : B=Covar(R i ,R m )/Var(R m ) where R i is the investment return and R m is the expected market return. Correlation : Correl=Covar/sd^2 Covariance : Covar=Corr*sd*sd and Covar=P1*((Ri-Ra)*(sdi-sda))+P2*((Ri-Ra)*(sdi-sda))… Valuing projects with the risk-adjusted discount rate method : Calculate the present value of a project which results in expected cash flows of 18,000 in one year, 21,000 in two years, and 23,000 in three years and zero thereafter. The expected return of the market portfolio is 7% p.a., the risk-free rate is 2.5% p.a., and the project has β = 1.25. Use PV=(18000/(1+rf+B(rm-rf)))+(21000/(1+rf+B(rm-rf))^2)+(23000/(1+rf+B(rm-rf))^3) Valuing projects with the Ratio Approach : You would like to value a (non-traded) firm by comparison with similar, traded firms. The company produces sweets in two business lines: cookies and donuts. In the table below you find information about the earnings forecast for the firm and the distribution of earnings across the two business lines. Expected earnings 4,000,000 Fraction of expected earnings from cookies 0.65 Fraction of expected earnings from donuts 0.35. The second table provides information about the relevant comparison firms for the cookies and donut industries. Cookies industry Donuts industry Price per share 52 58 Earnings per share 1.2 2.54. (a) What is the value of the firm? (b) The information above refers to the year 2019. Assume that the firm is financed with 70 million in debt. Suppose that the asset values of the firm increased by 10% from 2019 to 2020. Compute the percentage increase in the value of equity. (c) Compute the leverage ratio of this firm in 2019 and 2020. A) EF = $4000000Cookies: PPS = $52 EPS = $1.2 > % of EF = 65%, Donuts: PPS = 58 EPS = 2.54 > % of EF = 35%. Cookie Rev = $4000000*0.65 = $2600000 Cookie Shares = $2600000/$1.2 = 2166667 (rounded to nearest whole) Cookie Value = 2166667*$52 = $112666684. Donuts Rev = $4000000 - $2600000 = $1400000 Donuts Shares = $1400000/$2.54 = 551181 (rounded to nearest whole) Donuts Value = 551181*$58 = $31968498. Company Value = $144635182 B) B) 2019: Assets = $144635182 Liabilities = $70000000 SHE = $144635182 - $70000000 = $74635182. 2020: Assets = $144635182*1.1 = $159098700.20 Liabilities = $70000000 SHE = $159098700.20 - $70000000 = $89098700.20 %^ = ($89098700.20 - $74635182)/$74635182*100 = 19.38% (rounded) C) 2019: LR = $70000000/$74635182 = 0.9379 (rounded). 2020: LR = $70000000/$89098700.20 = 0.7856 (rounded) Valuing projects with lev/delev beta : In 1989, General Motors (GM) was evaluating the acquisition of Hughes. Recognizing that the appropriate discount rate for the projected cashflows of Hughes was different than its own cost of capital, GM assumed that Hughes had approximately the same risk as Lockheed or Northrop, which had low-risk defense contracts and products that were similar to Hughes. Specifically, assume the following inputs: GM βe 1.20 D/E 0.40, Lockheed βe 1.60 D/E 2, Northrop βe 0.85 D/E 0.7. Target D/E for Hughes’ acquisition = 1. Hughes’ expected cash flow next year = 300 million USD. Growth rate of Hughes’ cash flows = 5% per year. Appropriate discount rate on debt (riskless rate) = 8%. Expected return of the tangency portfolio = 14%. (a) Analyse the Hughes acquisition (which took place) by first computing the betas of the comparison firms, Lockheed and Northrop, as if they were all equity financed. Assume there are no taxes. (b) Compute the beta of the assets of the Hughes acquisition, assuming there are no taxes, by taking the average of the asset betas of Lockheed and Northrop. (c) Compute the cost of capital for the Hughes acquisition, assuming there are no taxes. (d) Compute the value of Hughes with the cost of capital estimated in exercise (c). (a) When a firm is financed only with equity, then βA=βE. Thus, we can interpret this question as asking us to compute βA of the comparison firms. Since we know βE for both comparison firms,D/E for both comparison firms, and thatβA=(E/(D+E))*βE, we can solve the general expression for βA: βA=(1/((D/E)+1))*βE and using the appropriate values we get that for GM βGMA= 0.86, for Lockheed βLA= 0.53and for Northrop βNA= 0.50. (b) β=1/2β+1/2β yields β= 0.52. (c) Using the CAPM, we can calculate the cost of capital knowing that rf= 8% and the expected return of tangency portfolio is 14%. E= rf+β(E(RT)−rf) = 8% + 0.52 ×(14% −8%) = 11.10%. d) Working in millions, and denoting the growth rate of Hughes’ cash flow as gH we have: PV=(E(C1))/(E(r)−gH)=300/(11.10% −5%) = 4′918. Valuing projects with the certainty equivalent method : Each share of BA, has a cash flow beta of $5.125 when computed against the tangency portfolio. One year from now, this subsidiary has a 0.9 probability of being worth $5 per share and 0.1 probability of being worth $4 per share. The rf is 6% pa. The tangency portfolio has an expected return of 14% pa. What is the PV of BA, assuming no dividend payments to the parent firm in the coming year? PV=(CE(C))/(1+rf) and CE(C)=E(C)- B(Rt-rf) , so E(c)=0.9*5+0.1*4=4.9, then CE(C)=4.9-5.125(0.14-0.06)=4.49, back to PV=4.49/(1+0.06)=4.24 Valuing American options : Assume that, in each period, the non-dividend-paying equity of BBVA, a major Spanish bank, can either double or halve in value: that is, u 2, d 0.5. If the initial share price is $20 per share and the risk-free rate is 25 per cent per period (this is incredibly high, but we need it to make our example work!), what is the value of an American put expiring two periods from now with a strike price of $27.50? S0=20, u=2, d=0.5, rf=0.25. Find P and 1-P. At node U, 3>0 so we do not exercise the option. At Node D, 17.5>12, so we exercise the option immediately if we arrive at D. When calculating V, we use 17.5 instead of 12. Because V(8.2)>7.5 intrinsic value, we don’t exercise the option here, instead waiting.

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