Forwards & Futures

Futures

  • Futures use the same formulas as forwards
  • Futures are traded on exchanges while forwards are traded over the counter
  • Futures are cash settled, and deal with margins
  • Some language
    • Notional Value = Index * Size
    • Margin = %age of notional value that you have to keep in your margin account (in case you default)
    • Maintenance Margin = %age of your original margin balance that is the minimum allowed amount. You will get a marign call if you have less money than the maintenance margin and have to deposit more money.
  • Ex: You enter into a futures contract to long the S&P 500. The size is $200, the index is 700, margin is 25%, maintenance margin is 80%, and risk free rate is .05.
    • Balances at t=0:
      • Notional Value = Index * Size = \(\$700(200) = \$140,000\)
      • Initial balance in the margin account = (Notional Value)(margin %) = \(\$140,000(.25) = \$35,000\)
      • Minimum balance before a margin call = \(\$35,000(.8) = \$28,000\)
    • 1 week passes. Index price rises to 730.
      • Margin account balance grows by rfr for 1 week: \(\$35,000e^{.05(\frac{1}{52})}=3.503367\times 10^{4}\)
      • Index price rose by \(730-700=30\), so margin balance increases by \(30*200=6,000\)
      • Margin balance at t = 1 week:
        • \(=\$35,000e^{.05(\frac{1}{52})} + 6,000=4.103367\times 10^{4}\)
    • Another week passes. t = 2 weeks. Index price falls to $650.
      • Margin grows by rfr, and margin balance decreases by \(730-650=\$80\) times the size of 200
      • Margin Balance at t = 2 weeks \(=\$4.103367\times 10^{4}e^{(\frac{.05}{52})} - (80)(200)=2.507314\times 10^{4}\)
      • So we would get a margin call to add more money to the margin account since the value dropped below the maintenance margin of $28,000

Options

General strategy for dealing with solo call/put payoffs

  • Look at the left and right tails
  • If the LEFT side goes up/down => Put
  • If the RIGHT side goes up/down => Call
  • If it goes UP => Long
  • If it goes DOWN => Put

Options Spreads

Option + Asset Combinations

Floor

Cap

Covered Call

Covered Put

Spreads

  • Combination of calls and puts

Bull Spread

  • 2 Calls OR 2 Puts.
  • Buy LOW & Sell HIGH

Bear Spread

  • 2 Calls OR 2 Puts

  • Buy HIGH & Sell LOW

  • 2 Calls OR 2 Puts

Box Spread

  • Rare to see on IFM
  • LONG a synthetic forward at \(K_1\), and SHORT a synthetic forward at \(K_2\). (\(K_1 < K_2\))
    • This creates a BEAR Spread
  • SHORT a synthetic forward at \(K_1\) and LONG a synthetic forward at \(K_2\)
    • This creates a BULL Spread

Ratio Spread

  • Combination of buying/selling m options at one strike price, and n options at a different strike price.

Collars

Purchased Collar

  • BUY Put low, WRITE Call High.
  • So it’s flat in the middle
  • Collar Width: Distance between the Put and the Call
  • Weird that a Purchased collar looks like a bearish position

Written Collar

  • WRITE Put Low, LONG Call High

Zero Cost Collar

  • A Collar Spread (either purchased or written) that you break even (0 profit) in the middle

Long Straddle

  • Buy a call and a put at the same strike price
  • Betting on high volatility

Written Straddle

  • Writting both a call and a put at the same strike price
  • Betting on low volatility

Strangle

  • Long both a call and a put at different strike prices (it doesn’t matter which one is low/high)

Written Strangle

  • Write both a call and a put at different strike prices

Butterfly Spread

  • Combination of 4 option positions at 3 different strike prices (\(K_1 < K_2 < K_3\))

  • For a purchased Butterfly Spread, there are several ways to construct.
  • Thinking in terms of just individual options:
    • Buy Put at \(K_1\)
    • Sell Put at \(K_2\)
    • Sell Call at \(K_2\)
    • Buy Call at \(K_3\)
  • OR:
    • Write a Straddle at \(K_2\)
    • Buy a Strangle with strikes \(K_1\) & \(K_3\)
    • When created this way it is technically called an “Iron Butterfly”
  • OR:
    • Bull spread: buy put at \(K_1\) & sell put at \(K_2\)
    • Bear Spread: sell put at \(K_2\) & buy put at \(K_3\)
    • (yes, you can construct a butterfly w/ all calls or all puts)
    • Technically, a true butterfly spread is constructed using all calls or all puts.

Each of these positions create this identical plot:

  • Written Butterfly
    • Switch long/short positions from above

Asymetric Butterfly

  • Still work w/ 3 strike prices \(K_1 < K_2 < K_3\)
    • But this time \(K_2\) isn’t exactly in the middle
    • Working with 3 options this time (instead of 4)

To construct an asymetric butterfly w/ all calls:

  • You have some \(\lambda\) where:

    • \(\lambda = \frac{K_3 - K_2}{K_3 - K_1}\)
    • Solving for \(K_2 = \lambda K_1 + (1- \lambda)K_3\)
    • So for every \(K_2\) Call you write, you must:
      • Buy \(\lambda K_1\) Calls
      • Buy \((1-\lambda) K_3\) Calls

Ex: Suppose you wish to create an asymetric butterfly with strikes 100, 110, and 115. You wish to buy/sell 12 options total. How many of each option do you buy/sell?

  • \(\lambda = \frac{K_3 - K_2}{K_3 - K_1} = \frac{115 - 110}{115 - 100} = \frac{1}{3}\)

  • So our ratios are as follows:

    • 1 \(K_2\) Call sold
    • 1/3 \(K_1\) calls bought
    • 2/3 \(K_3\) calls bought

  • Want 12 options total:

    • \(1x + \frac{1}{3}x + \frac{2}{3}x = 12\)
    • \(x = 6\)

  • Multiply the above ratios by 6 each to see how many of each to buy

Iron Condor

Like a butterfly, but with space between the options at the point. So it has 4 strikes \(K_1 < K_2 < K_3 < K_4\) - Not even sure if this is tested on IFM but it’s easy

Synthetic Forward

  • Buying a call and Selling a put at the same strike price w/ the same expiration mimics a forward contract.

Put Call Parity

Option Pricing

Price using Put Call Parity

  • \(C-P = Se^{-\delta t}-Ke^{-rt}\)
    • Given one value solve for the other

Price using 1 period Binomial Tree

  • Assume stock can either rise from \(S_0\) to \(S_u\), or go down to \(S_d\) at time t.

  • Option Premium = \(\Delta S + B\)
    • \(\Delta = e^{-\delta t} \frac{C_u - C_d}{S(u - d)}\)
    • \(B=e^{-rt} \frac{u C_d - d C_u}{u-d}\)
      • \(C_d\) or \(C_u\) = call payoff if stock drops/rises
      • \(S\) = \(S_0\) = Stock price at t=0
      • \(u\) & \(d\) = Ratio of increase or decrease
        • \(u = \frac{S_1}{S_0}\)
        • \(d = \frac{S_0}{S_1}\)
    • This formula holds for both calls and puts, just substitute the put payoff \(P_x\) for \(C_x\) in the formula

    • For calls we’ll get a \(+\Delta\) & \(-B\). For puts we’ll get a \(-\Delta\) and \(+B\)

    • Synthetic position
      • Can use this formula to create an equivalent synthetic position
      • Call payoff = purchasing \(+\Delta\) shares of stock, and borrowing \(-B\) to do so
      • Put payoff = shorting \(-\Delta\) shares of stock, and lending \(+B\)

  • Arbitrage

    • If the Option Prem \(\neq\) \(\Delta S + B\), then we buy what’s lower, and short what’s higher
    • Profit = \(|C^* - C|e^{rt}\)
      • Difference between what premium actually is and what it should be
      • Profit is calculated at time t=t

Risk Neutral pricing w Binomial Trees

  • This time we incorporate probabily that the stock rises/decreases between now and the next time interval

  • \(Prem = e^{-rt}[p(C_u)+(1-p)C_d]\)

    • \(p = \frac{e^{(r-\delta)t}-d}{u-d}\)
      • \(u=e^{(r-\delta)t+\sigma \sqrt{t}}\)
      • \(d=e^{(r-\delta)t-\sigma \sqrt{t}}\)
    • Can switch \(C_u\) <==> \(P_u\)

  • Volatility (\(\sigma\)) is calculated in 3 steps

    1. Find log ratios
      • \(log(\frac{S_t}{S_{t-h}})\)
    2. Find unbiased estimator of the variance of these ratios
      • \(S^2 = \frac{1}{n-1}\sum(x_i-\bar{x})^2\)
      • Or just plug values into multiview
    3. Multiply by 1/h and square root to get SD
      • \(\sigma = \sqrt{S^2/h}\)
        • (For weekly values, h = 1/52, so we’d multiply it by 52)

Price using multiple period tree

European Pricing

  • Start with very right big side of the tree and find all payoffs there
  • Then we sum up all of these payoff possibilities, times their probabilities (w combinatorics factor), and discount
  • \(Prem = e^{-rt}[\sum \binom{m}{k} (prob_i)(payoff_i)]\)
    • Combinatorics factor is the total number of periods (4 in this pic), choose either of the probability exponents
    • \(p = \frac{e^{(r-\delta)h-d}}{u-d}\)
      • \(u=e^{(r-\delta)h + \sigma \sqrt{t}}\)
      • \(d=e^{(r-\delta)h - \sigma \sqrt{t}}\)
        • We replace t w/ h which is our step size

American Pricing

  • American -> can exercise option at any time
  • Key is to work backwards
  1. Find all payoffs at end periosd
  2. Work backwards step by step
    • For each step back you take the max between:{
      • Immediate payoff if exercised immediately
      • \(e^{-rh}[pC_u + (1-p)C_d]\)}
  3. Repeat until you work your way back to the first step

Black Scholes

  • This is how modern options are priced today. Based on the lognormal distribution.
  • \(C = Se^{-\delta t}N(d_1) - Ke^{-rt}N(d_2)\)
  • \(P = Ke^{-rt}N(-d_2) - Se^{-\delta t}N(-d_1)\)
    • \(d_1 = \frac{log(\frac{S}{K}) + (r - \delta + .5 \sigma^2)t}{\sigma \sqrt{t}}\)
    • \(d_2 = d_1 - \sigma \sqrt{t}\)
  • When dealing w/ discrete dividends we can use prepaid fwd equations (these formulas also work w/ continuous divs)
  • \(C = F^p(S)N(d_1) - F^p(K)N(d_2)\)
  • \(P = F^p(K)N(-d_2) - F^p(S)N(-d_1)\)
    • \(d_1 = \frac{log(\frac{F^p(S)}{F^p(K)}) + .5 \sigma^2 t}{\sigma \sqrt{t}}\)
    • \(d_2 = d_1 - \sigma \sqrt{t}\)
    • \(F^p(S) = S_0 - \sum div_i (e^{-rt_i} = S_0 e^{-\delta t}\)
      • Subtract the PV of each dividend from the stock price, or discount by \(\delta\)
    • \(F^p(K) = Se^{-rt}\)
      • Just discount \(S_0\) by the rfr

The GREEKS

GREEK Formula from IFM Tables Explanation Values/Magnitude
\(\Delta\) Call = \(e^{-\delta(T-t)}N(d_1)\)
Put = \(-e^{-\delta(T-t)}N(-d_1)\)		 Change in Option price as stock increases by $1 Higher Magnitude when the profit is higher. + for Calls, - for Puts
\(\Gamma\) Call & Put = \(\frac{e^{-\delta(T-t)}N'(d_1)}{S\delta\sqrt{T-t}}\) Change in Delta as the stock increases by $1 Always Positive for both calls and puts
Vega or \(\kappa\) Call & Put = \(Se^{-\delta(T-t)}N'(d_1)\sqrt{T-t}\) Change in option price when volatility increases by 1% High when stock and strike prices are close. Usually +
\(\theta\) big equation Change in option price when time moves forward one day closer to maturity Usually - b/c options usually lose value w/ smaller time frame, but can be + if deep in the money
\(\rho\) big equation Change in option price when interest rate increases by 1% Higher mag when in the money & T is longer. Calls = +, Puts -
\(\psi\) big equation Change in option price when dividend yield \(\delta\) increases by 1% Higher mag when in the money & T is longer. Calls = -, Puts = +

Greeks of multiple positions

  • \(\Delta = \sum_{i=1}^{N} (n_i)(greek_i)\)

    • For a portfolio of options, you just sum up all the \(\Delta\)’s or \(\Gamma\)’s or any Greek of each option, to get the net Greek value.
    • When long, \(\Delta\) => 1. When Short => \(\Delta\) => -1.

Other Greek-ish formulas to memorize

  • Option Elasticity (\(\Omega\))

    • \(\Omega = \frac{S\Delta}{V}\)
      • V = Option premium for Call or Put
      • \(\Omega\) is the 3rd derivative of the options price, or derivative of \(\Gamma\)
      • Represents: If the stock changes by 1%, by what % does the option change?

  • Option Volatility (\(\sigma_{option}\))

    • \(\sigma_{option} = \sigma_{stock} |\Omega|\)
      • Option volatility is higher than stock volatility

  • Risk Premium

    • Risk Premium on a Stock = \(\alpha - r\)
      • \(\alpha\) = Expected rate of return on a stock
      • \(r\) = rfr (t-bill)
    • Risk Premium of an Option = \(\gamma - r = (\alpha - r)\Omega\)
      • \(\gamma = \Omega\alpha + (1-\Omega)r\) = Expected return of an option

  • Sharpe Ratio

    • Sharpe Ratio \(=\frac{\alpha - r}{\sigma}\)
    • Ratio between risk premium and volatility
    • Same equation for Stocks and Options (but for Options I think we use \(\gamma - r\) for the numerator b/c risk premium)
    • We want this value to be high b/c we want high return for low risk

Delta Hedging

About

  • Delta hedging might be the only thing actuaries actually use from IFM material
  • Delta hedging is offsetting an options position by buying/selling \(\delta\) shares of stock
  • Market Maker is the person who sells the options contract

Delta Gamma Theta (\(\Delta\Gamma\theta\)) approximations

  • \(C_{t+h}(S+\epsilon)\approx C_t(S) + \Delta\epsilon + .5\Gamma\epsilon^2 + \theta h\)
    • A way to approximate the change in options price given the change in a stock price
    • Only use part of the formula for however much they want you to approximate

Marking to Market

  • Market maker adjusts his portfolio every day to stay hedged against market swings
  • Best demonstrated with an example: Customer buys a t=91/365 day call option, so market maker sells the call option. Suppose S = $40, K = $40, \(\sigma\) = .3, \(r\) = .08, \(\delta\) = 0, and call contract is for 100 shares.
    • Using Black Scholes framework and Greek Formulas we obtain these values in respect to the Call buyer:
      • C(40) = 2.7804, \(\Delta\) = .5824, \(\Gamma\) = .0652, \(\theta\) = -.0173
      • For the market maker, multiply each of these by -1 b/c we are selling the call
        • C(40) = -2.7804, \(\Delta\) = -.5824, \(\Gamma\) = -.0652, \(\theta\) = .0173
    • To hedge, the market maker buys .5824 shares of stock for every call
    • Day 0:
      • Market maker sells option contract so receives premium
        • 2.7804*100 = 278.04
      • Market maker buys \(\Delta\) shares for every option to hedge and spends:
        • -.5824(100)(40) = -2329.6
      • Market maker net position at t=0
        • 278.04 - 2329.6 = -2051.56
        • B/c it’s negative, we must borrow this money and will pay interest on it. If it were +, we would be lending this money and gaining interest
    • Day 1: Stock rises to $40.50. Using Black Scholes w/ t=90/365, C(40) => $3.0621
      • Market maker profit on shares he owns
        • (40.50 - 40)(58.24) = 29.12
      • Market maker profit on change in option value
        • (2.7804 - 3.0621)(100) = -28.17
      • Interest expense market maker must pay. We take previous period balance and compound by rfr
        • \(-2051.56(e^{.08(\frac{1}{365})}-1) = -0.45\)
      • Market Maker’s overnight profit:
        • \(29.12 + -28.17 + -0.45 = 0.5\)
      • Market maker must rebalance his portfolio
        • Use Black Scholes to calculate new \(\Delta\) to be => .6142. So we must buy additional shares:
          • (.6142 - .5824)(100) = 3.18
        • So we must buy 3.18 more shares at new $40.50 price to rebalance portfolio: = 3.18(40.5) = 128.79
        • This isn’t used in the profit equation for the day though
      • Market maker’s new net position at Day 1:
        • (3.0621)(100 options we sold) - (40.5)(61.42 shares we paid for) = -2181.3
    • Day 2: Stock falls to $39.25, t=89/365, using Black Scholes C(40) => 2.3282
      • Market maker profit on 61.42 shares he now owns
        • (39.25 - 40.5)(61.42) = -76.775
      • Market maker profit on option value change
        • (3.0621 - 2.3282)(100) = 73.39
      • Market maker pays interest on previous loan balance
        • \(-2181.3(e^{.08/365}-1) = -0.48\)
      • Market maker’s overnight profit:
        • \(-76.775 + 73.39 + -0.48 = -3.865\)
    • Here’s a summary table of what happens each day:

  • Steps in Marking to Market
    • Find market maker’s net Delta position, and offset by buying/selling that many shares
    • Find market maker’s net position from shares and option premiums
    • For the next day
      • Use new Stock price and t in the Black Scholes framework to find new Delta and Call premium
      • Find the profits for your stock and option premium positions
      • Find profit for how much you pay/receive in interest from previous position
      • Add/subtract shares from portfolio to mark to market (rebalance portfolio) w/ new Delta
      • Find new Net position
    • Remember, interest profit never gets added to the net portfolio value

Gamma Hedging

  • To stay even more hedged, not only do we make our net \(\Delta\) of our position = 0, but we also make our net \(\Gamma\) of our position = 0.
  • We do this by buying/selling shares of another call option to get a net \(\Gamma\) of 0, but then have to buy/sell stock shares to get a net \(\Delta\) of 0
  • We have 2 options w/ strikes: \(K_1\) and \(K_2\), each w/ their own \(\Delta\)’s and \(\Gamma\)’s
    • If we sell the \(C(K_1)\) option, we \(\Gamma\) hedge by buying \(\frac{\Gamma_1}{\Gamma_2}\) \(K_2\) Call options to make our net \(\Gamma\) 0
    • Our combined \(\Delta\) becomes:
      • \(\Delta_{combined} = -\Delta_1+\frac{\Gamma_1}{\Gamma_2}\Delta_2\)
      • You can logic your way to this equation if you just remember to buy \(\frac{\Gamma_1}{\Gamma_2}\) contracts of option 2
    • So if this is negative we purchase that many shares to offset our position

Exotic Options

Asian Options

  • Deals with the average price of the stock
  • 2 different kinds of averages
    • Arithmetic Average A(T) = just regular mean
    • Geometric Mean = \((S_1*S_2*S_3*S_4*...)^{1/N}\)
  • Recall that payoffs are calculated:
    • Call payoff = Max{0, S-K}
    • Put payoff = Max{0, K-S}
  • For Asian options, we replace either S, or K using either A(T) or G(T)
    • Between Call/Put, S/K, A(T)/G(T) there are 8 diff combinations
    • Prob will say “avg price” for replacing S, or “avg strike” for replacing K

Barrier Options

  • Knock-in Option (“Up and In”) = Can only exercise option if at any point between time 0 and T, the stock price crosses the barrier at least once
  • Knock-out Option (“Down and Out”) = Option goes out of existence if at any point between time 0 and T, the stock price crosses the barrier once.
  • These are priced using computer simulations
  • Parity relationship for option premiums
    • Knock-in + Knock-out = Ordinary option

Compound Options

  • Option to buy an Option
  • Have 2 strikes and 2 expirations
  • \(t_0\) is current time. At \(t_1\), you have the option to buy (for a price of \(\$x\)) a European option with a strike price of \(\$K\). At time \(T\), the underlying European option expires.
  • \(S^*\) is the critical value that if the stock rises above this value, then it makes sense to exercise the compound option.

  • Compound Option Parity

    • CallonCall - PutonCall = Call - \(xe^{-rt_1}\)
    • CallonPut - PutonPut = Put - \(xe^{-rt_1}\)

Gap Options

  • Gap options have a regular strike price \(K_1\) that you buy/sell the asset for, but you can only do so if the stock is more/less than the “trigger” price \(K_2\) at expiration
  • Use the same Black Scholes formulas except we change the \(d_1\) formula to have \(K_2\) in place of \(K_1\)
  • \(C = Se^{-\delta t}N(d_1)-K_1e^{-rt}N(d_2)\)
  • \(P = K_1e^{-rt}N(-d_2) - Se^{-\delta t}N(-d_1)\)
    • \(d_1 = \frac{log(\frac{Se^{-\delta t}}{K_2e^{-rt}}) + .5\sigma^2t}{\sigma\sqrt{t}}\)
    • \(d_2 = d_1 - \sigma\sqrt{t}\)

Exchange Options

  • AKA Outperformance Options
  • Basically an option to exchange an asset with another
  • S is the price of risky asset 1, K is the price of risky asset 2
  • We are given dividend yields (\(\delta_S\) & \(\delta_K\)), volatilites (\(\sigma_S\) & \(\sigma_K\)), and correlation between the 2 continuously compounded dividend yields (\(\rho\))
  • Call option is giving asset K (2), and receiving asset S (1). (Paying strike K, receiving stock S) (Same as normal)
  • Call Prem = \(Se^{-\delta t}N(d_1) - Ke^{-rt}N(d_2)\)
    • \(d_1 = \frac{log(\frac{Se^{-\delta_S t}}{Ke^{-\delta_K t}}) + .5\sigma^2t}{\sigma\sqrt{t}}\)
    • \(d_2 = d_1 - \sigma\sqrt{t}\)
    • \(\sigma = \sqrt{\sigma^2_S + \sigma^2_K -2\rho\sigma_S \sigma_K}\)

Properties of Lognormal

Expected value of the stock at time t

  • \(E[S_t]=S_0e^{(\alpha-\delta)t}\)

Median value of the stock at time t

  • \(Med[S_t]=S_0e^{(\alpha-\delta-.5\sigma^2)t}=E[S_t]e^{-.5\sigma^2 t}\)

Expected value of the stock given it’s greater/less than some value K

  • \(E[S_t|S_t>K]=S_0e^{(\alpha-\delta)t}\frac{N(d_1)}{N(d_2)}\)
  • \(E[S_t|S_t<K]=S_0e^{(\alpha-\delta)t}\frac{N(-d_1)}{N(-d_2)}\)
    • \(d_1 = \frac{log(\frac{S_0}{K})+(r-\delta+.5\sigma^2)t}{\sigma\sqrt{t}}\)
      • Assuming no arbitrage \(\alpha = r\)
    • \(d_2 = d_1 - \sigma\sqrt{t} = \frac{log(\frac{S_0}{K})+(r-\delta-.5\sigma^2)t}{\sigma\sqrt{t}}\)
      • Only diff is we subtract \(.5\sigma^2\) from the numerator

Probability the stock is > or < K at time t

  • \(P(S<K)=N(-d_2)\)
  • \(P(S>K)=N(d_2)\)
    • \(d_1 = \frac{log(\frac{S_0}{K})+(\alpha-\delta+.5\sigma^2)t}{\sigma\sqrt{t}}\)
      • Only diff is we use \(\alpha\) instead of \(r\) here b/c we assume no arbitrage
    • \(d_2=d_1-\sigma\sqrt{t}\)

Variance of the stock at time t

  • \(Var(S_t)=S_0^2 Var(X)\)
    • \(Var(X)=e^{2m+v^2}(e^{v^2}-1)\)
      • \(m=(\alpha-\delta-.5\sigma^2)t\)
      • \(v^2=\sigma^2 t\)
  • \(Var(S_t)=S_0^2 e^{2t(\alpha - \delta -.5\sigma^2)+\sigma^2 t}(e^{\sigma^2 t}-1)\)

Corporate Finance

Chap 10

Historical Return

  • \(R_{t+1}=\frac{Div_{t+1}+P_{t+1}}{P_t}-1\)
    • \(R\): Return
    • \(Div_t\): dividend amount received at t=t
    • \(P_{t}\) Price at t=t
    • Basically this is just how much you have at the end, divided by how much you started w/, -1 to get the return.

Quarterly -> Annual returns

  • \(1+R_{annual}=(1+R_{Q1})(1+R_{Q2})(1+R_{Q3})(1+R_{Q4})\)
    • These can be any time interval, as long as they all add up to 1 year. Can even combine weeks and months

Risk

  • Common Risk: Systematic Risk, Undiversifiable Risk, Market Risk
    • Ex: Terrorist attack
  • Independent Risk: Firm specific risk, idiosyncratic risk, unique risk, unsystematic risk, diversifiable risk
    • Ex: scandal at 1 firm

Sensitivity to systematic risk (\(\beta\))

  • Expected % change of the excess return of a security, for a 1% change in the excess return of a mkt portfolio.
  • \(\beta = \frac{Range of returns for a stock}{Range of returns of market}\)
  • Or calculate \(\beta\) through regression techniques

Market Risk Premium

  • =\(E[R_{mkt}]-r_f\)
    • How much higher your market return is than the rfr

CAPM Capital Asset Pricing Model

  • \(E[R]=r_f +\beta(E[R_{mkt}]-r_f)\)
    • Used to find expected return on a stock

Chapter 11

Return on Portfolio

  • Actual return in portfolio
    • \(R_p=\sum x_i R_i\)
      • \(x_i\) = portfolio weight of asset i
      • \(R_i\) = return from asset i
  • Expected return in portfolio
    • \(E[R_p]=\sum x_i E[R_i]\)

Volatility of portfolio

  • Main Equation for the variance of a portfolio of returns
    • \(Var(R_p)=\sum \sum x_i x_j Cov(R_i, R_j)\)
      • You do this for every combination of returns, w/ combinatorics factors
      • For ex, this is for a portfolio of 3 stocks
        • \(Var = 2x_1x_2Cov(R_1, R_2) + 2x_1x_3Cov(R_1, R_3) + 2x_2x_3Cov(R_2, R_3) +\) \(x_1^2Cov(R_1, R_1) + x_2^2Cov(R_2, R_2) + x_3^2Cov(R_3, R_3)\)
          • And recall that the Cov() of a return w/ itself is just the variance
  • Covariance and Correlation formulae
    • \(Cov(R_i, R_j)=E[(R_i-E[R_i])(R_j-E[R_j])] = \frac{1}{T-1} \sum (R_{i,t} - \bar{R_i})(R_{j,t} - \bar{R_j})\)
    • \(Corr(R_i, R_j)=\frac{Cov(R_i, R_j)}{SD(R_i)SD(R_j)}\)

Efficient Portfolio

  • A portfolio is efficient if you have the lowest amount of volatility for a given return

Adding T Bills

  • Do this to reduce some risk
  • Expected return with adding T bills to portfolio
    • \(E[R_{xp}] = xE[R_p] + (1-x)r_f\)
      • x = %age of portfolio that consissts of stocks
      • \(r_f\) = rfr from investing in bonds
  • Standard Deviation
    • \(SD(R_{xp}) = xSD(R_p)\)
      • Standard deviation of a portfolio with t bills, is just the %age of stocks in portfolio, times the SD of the stock portfolio. B/c there is no standard deviation w/ T bills.

Sharpe Ratio again

  • Sharpe Ratio = \(\frac{E[R_p]-r_f}{SD(R_p)}\)

Portfolio improvement

  • \(\beta\) of 2 diff assets, one w/ respect to the other. How much the movement of 1 can be described by the movement of another
  • \(\beta_i^P = \frac{SD(R_i)Corr(R_i, R_P)}{SD(R_P)}\)
    • Beta of stock i, w/ portfolio P (market portfolio)
  • Beta of a portfolio (just weighted avg of all the \(\beta\)’s)
    • \(\beta_P = \sum x_i \beta_i\)

Required Rate of return

  • \(r_{rrr}=r_f+\beta_i^P(E[R_P]-r_f)\)
  • Subjective, diff for each person. What they want to receive in return
  • This should be what you require to return for adding a new stock, i, to your portfolio. Calculate the required return using the above, then if the individual stock gives you a higher return than the required rate of return, then it makes sense to add the stock to your portfolio.

Chap 12 Cost of Capital

Equity Cost of Capital

  • Use CAPM to calculate cost of capital. (Return on equity)
    • \(r_E=E(R_i)=r_i + \beta(E[R_{mkt}]-r_i)\)

Debt Cost of Capital

  • Method 1: Use expected return of a bond
    • \(r_d=y-PL\)
      • \(r_d\) = return on debt
      • \(y\) = yield to maturity of bond
      • \(P\) = probability of default
      • \(L\) = Expected loss rate. Expected loss per $1 of debt.
  • Method 2: Use market risk premium (CAPM style)
    • \(r_d = r_f + \beta(E[R_{mkt}]-r_f)\)
      • \(\beta\) = The \(\beta\) used here is diff than our equity \(\beta\). Risker bonds have higher \(\beta\), so we look it up on a table based on it’s rating (AAA, A, BBB, BB, …)

Asset (unlevered) Cost of Capital

  • Expected return required by investors to hold the firm’s underlying assets
  • Weighted avg of the firm’s cost of equity and cost of debt
    • \(r_U = \frac{E}{E+D}r_E + \frac{D}{E+D}r_D\)
      • \(r_U\) = unlevered return
      • \(r_E\) = return on equity
      • \(r_D\) = return on debt
      • \(E\) = Amt of equity
      • \(D\) = Net Debt = Debt - Cash
  • Asset (Unlevered) Beta
    • \(\beta_U = \frac{E}{E+D}\beta_E + \frac{D}{E+D}\beta_D\)
      • This is the underlying beta of the business enterprise
  • Random formula
    • Enterprise Value = Mkt Cap + Net Debt
      • Mkt Cap is equity