AFM 272

AFM 272 - Global Capital Markets (Valuation)
MW 1:00PM - 2:20PM
HH 2104
Daniel Sang Kim

Office Hours: 2:30-4:30pm Wed HH 286F
10% clicker | 40% midterm | 50% final (handwritten)

1 | Time Value of Money

TVOM Formulas

One-time payment

$$PV=\frac{C}{(1+r{)}^n}$$

Regular Perpetuity

$$PV=\frac{C_1}{r}$$

Growing Perpetuity

$$PV=\frac{C_1}{r-g}$$

Regular Annuity

$$PV=\frac{C_1}{r}\times [1-\frac{1}{(1+r{)}^n}]$$

Growing Annuity

$$PV=\frac{C_1}{r-g}\times [1-(\frac{1+g}{1+r}{)}^n]$$

EAR

$$EAR=(1+\frac{ARR}{k}{)}^k-1$$

General Streams

Daily Article (09/03)

Article: Stocks sink on Bay and Wall Streets, but Loonie rises following new tariffs

US sets tariffs on Canada

• Analogy: would you rather have $50oracoinflipfor$100?
• Cash Flow (CF) Story: $Profit=Revenue-Expenses$
  • Does it change the $50?
• Discount Rate (DR) Story: 50% chance for $0,50$100
  • Does it change the risk? Who knows what Trump will do tomorrow?

Cash Flows and NPV Ex: Friend offers you $5000nowinexchangefor$1700 per month for the next 4 months

$$PV=5000-\frac{1700}{1+r}-\frac{1700}{(1+r{)}^2}-\frac{1700}{(1+r{)}^3}$$

General Stream of Cash Flows

$$P{V}_0=\sum _{t=0}^n\frac{C_t}{(1+r{)}^n}$$

Definition: NPV

$$\text{NPV = PV(benefts) - PV(costs)}$$

Note: the formula produces the same results if you discount forward or backward

Perpetuities and Annuities

Cyclical

• Pro-cyclical: moves with the market (most securities)
• Counter-cyclical: moves against the market (gold, insurance)

$$\text{r = risk free rate + risk premium}$$

What happens if you want to spend the same amount every year while investing? $t(0)=10000-x$ $t(1)=1.05(10000-x)-x$ $t(2)=1.05(1.05(10000-x)-x)=0$

$$1.05^2\times 10000-1.05^{2}x-1.05x-x=0$$ $$x=10000\times \frac{1.05^2}{1+1.05+1.05^2}$$

PV of a cash flow stream:

$$P{V}_0=C_0+\frac{C_1}{(1+r)}+\frac{C_2}{(1+r{)}^2}+\cdots +\frac{C_n}{(1+r{)}^n}=\sum _{t=0}^n\frac{C_t}{(1+r{)}^t}$$

Present Value of Perpetuities

$$P{V}_0=\sum _{t=1}^{\infty }\frac{c_1(1+g{)}^{t-1}}{(1+r{)}^t}$$

Deriving the Regular Perpetuity:

$$\begin{array}{rl}PV& =\frac{C}{1+r}+\frac{C}{(1+r{)}^2}+\cdots \\ PV(1+r)& =C+\frac{C}{1+r}+\frac{C}{(1+r{)}^2}+\cdots \\ PV(1+r)-C& =PV\\ PV+rPV-C& =PV\\ PV(GP)& =\frac{C}{r-g}\end{array}$$

Present Value of Annuities

$$P{V}_0\text{(growing)}=\frac{c_1}{r-g}[1-(\frac{1+g}{1+r}{)}^n]$$ $$P{V}_0\text{(regular)}=\frac{c}{r}[1-(\frac{1}{1+r}{)}^n]$$

• c = Payment
• r = Interest Rate
• g = Growth Rate
• n = Duration

Future Value of Annuities

$$P{V}_0=C\times A_r^n=C\times \frac{1}{r}[1-\frac{1}{(1+r{)}^n}]$$ $$F{V}_n=P{V}_0\times (1+r{)}^n=C\times \frac{1}{r}[(1+r{)}^n-1]$$

Examples

Example: if Duane deposits $1,000 at the end of each year for the next 15 years in an account paying 2% interest per year, how much money will be in his account after 15 years?

$$S_n=a_1(\frac{1-r^n}{1-r})=1000(\frac{1-1.02^{15}}{1-1.02})=FV=\frac{1000}{0.02}(1.02^{15}-1)=1729.34$$

Example: Fred wants to save money at the end of each of the next 10 years. He plans to save $1,000 at the end of the first year, and to have this increase by 4% each year. If he can earn 3% interest each year on his savings, how much money will he have saved 10 years from today?

$$FV=(1.03{)}^{10}\times \frac{1000}{0.03-0.04}[1-(\frac{1+0.04}{1+0.03}{)}^{10}]$$ $$FV=1000\cdot 1.03^9\cdot \frac{1-(\frac{1.04}{1.03}{)}^4}{1-\frac{1.04}{1.03}}$$

IRR

IRR (regular)

$$IRR=(\frac{FV}{PV}{)}^{1/n}-1$$

• PV = investment today
• FV = return that period
• n = number of periods

IRR (perpetuity)

$$IRR=\frac{C_1}{PV}+g$$

• $C_1$ = return per period
• $g$ = growth rate per period

Derivation:

$$\begin{array}{rl}5000& =200+\frac{200(1.015)}{1+r}+\frac{200(1.015{)}^2}{(1+r{)}^2}+\cdots \\ \frac{1.015}{1+r}5000& =200\frac{1.015}{1+r}+\frac{200(1.015{)}^2}{(1+r{)}^2}+\frac{200(1.015{)}^3}{(1+r{)}^3}+\cdots \\ [1-\frac{1.015}{1+r}]5000& =200\\ 1-\frac{1.015}{1+r}& =\frac{200}{5000}\\ \frac{1.015}{1+r}& =\frac{4800}{5000}\\ r& =\frac{5000}{4800}(1.015)-1\end{array}$$

Solving for the Number of Periods

$$\ln [(1+r{)}^n]=n\ln (1+r)$$ $$n=\frac{\ln [(1+r{)}^n]}{\ln (1+r)}$$

2 | Interest Rates

Quotes and Adjustments

Effective Annual Rate (EAR) = total amount of compound interest earned over a year

$$\text{Equivalent n-Period Effective Rate}=(1+r{)}^n-1$$

Example: Step 1: Find $r$

$$(1+r{)}^4=1.05⟹r=0.012$$

Step 2: Calculate FV

$$\begin{array}{rl}FV& =1000(1+r{)}^{20}+1000(1+r{)}^{19}+\cdots +1000(1+r)\\ \frac{1}{1+r}& =1000(1+r{)}^{19}+1000(1+r{)}^{18}+\cdots +1000\\ [1-\frac{1}{1+r}]FV& =[1000(1+r{)}^{20}-1]\\ FV& =1+\frac{1+r}{r}1000[(1+r{)}^{20}-1]\\ & =1+\frac{1.012}{0.012}1000(1.012{)}^{20}-1\\ & =22,789\end{array}$$

Example:

$$EAR=1.05⟹r=\sqrt[4]{1.05}=1.0122$$ $$\begin{array}{rl}F{V}_n& =P{V}_0\times (1+r{)}^n\\ & =\frac{c_1}{r-g}[1-(\frac{1+g}{1+r}{)}^n](1+r{)}^n\\ & =\frac{1000}{0.0122-0.4}[1-(\frac{1.04}{1.012}{)}^{20}](1.0122{)}^4\\ & =33,399\end{array}$$

Annual Percentage Rate (APR) = total amount of simple interest earned over a year

Given an APR with $k$ compounding periods per year, the implied effective rate $r$ earned each compounding period is $r=\frac{APR}{k}$ To convert an APR to an EAR:

$$1+EAR=(1+r{)}^k=(1+\frac{APR}{k}{)}^k$$ $$EAR=(1+\frac{APR}{k}{)}^k-1$$

Continuous Compounding

An infinite number of compounding periods:

$$EAR=\underset{k\to \infty }{\lim }(1+\frac{APR}{k}{)}^k-1=e^{APR}-1$$

If we know the EAR, we can rearrange this to solve for the APR with continuous compounding

$$APR=\ln (1+EAR)$$

In the case of a single cash flow:

• If $C_T$ will be received $T$ years from now, then

$$P{V}_0=C_T{e}^{-r_{CC}T}$$

• If $C_0$ will be invested today, then

$$F{V}_T=C_0{e}^{-r_{CC}T}$$

Continuous Cash Flows Suppose that cash flows start immediately at an initial rate of $C$ per year, and assume that this rate increases continuously at the rate $g_{cc}$

• If these cash flows are perpetual, then

$$P{V}_0={\int }_0^{\infty }C\cdot \exp (g_{cc}t)\cdot \exp (-r_{cc}t)dt=\frac{c}{r_{cc}-g_{cc}}$$

• If these cash flows stop after $T$ years, then

$$\begin{array}{rl}P{V}_0& ={\int }_0^{T}C\cdot \exp (g_{cc}t)\cdot \exp (-r_{cc}t)dt\\ & =\frac{c}{r_{cc}-g_{cc}}\cdot [1-\exp (-(r_{cc}-g_{cc})T)]\end{array}$$

• $r_{cc}=\ln (r),g_{cc}=\ln (g)$

Loan Calculations

Determinants of Interest Rates

The Yield Curve

Risk, Tasks, Opportunity Cost of Capital

3 | Bond Valuation

Introduction

Coupon payment is determined from bond’s coupon rate and face value

$$CPN=\frac{\text{(couopn rate) x (face value)}}{\text{(number of coupon payments per year)}}$$

Zero-Coupon Bonds

$$P=\frac{FV}{(1+YTM{)}^n}⟹YTM=(\frac{FV}{P}{)}^{1/n}-1$$

• YTM = yield to maturity
• P = price
• FV = face value
• n = years to maturity

Coupon Bonds

Price vs Face Value

• (P = FV), at a discount (P < FV), or at a premium (P > FV)

$$P=(\frac{CPN}{y})[1-\frac{1}{(1+y{)}^n}]+\frac{FV}{(1+y{)}^n}$$

• CPN = coupon payment
• n = number of periods
• P = bond price
• y = interest rate
• FV = face value

If a fraction $f$ of the current coupon period has elapsed, then the price of a bond (dirty price) with $n$ remaining coupon payments is

$$P=(1+y{)}^f\times (\frac{CPN}{y})[1-\frac{1}{(1+y{)}^n}]+\frac{FV}{(1+y{)}^n}$$

Dynamic Behaviour of Bond Prices

$$\text{clean price = dirty price - accured interest}$$

The Yield Curve and Bond Arbitrage

PV of a risk-free cash flow $C_n$ received in $n$ years, where $r_n$ is the risk-free effective annual rate (spot rate)

$$P{V}_0=\frac{C_n}{(1+r_n{)}^n}$$

PV of a risk-free cash flow stream

$$P{V}_0=\sum _{t=0}^n\frac{C_t}{(1+r_t{)}^t}$$

The shape of the yield curve is strongly influenced by interest rate expectations

• Upward-sloping = higher expected future interest
• Downward-sloping = lower expected future interest

Corporate Bonds and Sovereign Bonds

Corporate Bonds

• Default risk (credit risk) implies that corporate bonds are not certain
• Corporate bonds will have lower prices because they carry higher risk
• Credit rating firms give ratings such as AA, BBB

Sovereign Bonds

• Have lower default risk

Forward Interest Rates and the Term Structure

Let $s_1,s_2$ be the spot rates

$$(1+r_2{)}^2=s_1\times s_2$$

Forward rate = interest rate guaranteed today

$$(1+r_n{)}^n=(1+r_{n-1}{)}^{n-1}(1+r_n)⟹f_n=\frac{(1+r_n{)}^n}{(1+r_{n-1}{)}^{n-1}}$$

4 | Stock Valuation

Dividend-Discount Model

$P_0$ = price of 1 share of stock today $P_1$ = price of 1 share of stock 1 year from today $Di{v}_1$ = dividends (per share) paid to owners over the year

0 years: $-P_0$ 1 year: $Di{v}_1+P_1$

$r_E$ = equity cost of capital

$$P_0=\frac{Di{v}_1+P_1}{1+r_E}$$

Dividend yield and Capital gain rate are separate for tax purposes

$$\text{Dividend yield =}\frac{Di{v}_1}{P_0},\text{Capital gain rate}=\frac{P_1-P_0}{P_0}$$

But some investors will have horizons longer than 1 year:

Ex: $Di{v}_1=1.8,g=0.03,r_E=0.08$

$$\frac{1.8}{0.8-0.3}=36$$

What about in one year (after one dividend was paid)?

$$\frac{1.8\times 1.03}{0.8-0.3}=37$$

Dividend Payout Rate

$$Di{v}_t=\frac{{\text{earnings}}_t}{{\text{shares outstanding}}_t}\times {\text{(dividend payout rate)}}_t$$ $$EP{S}_t=\frac{{\text{earnings}}_t}{{\text{shares outstanding}}_t}$$

Ex: $EP{S}_1=6,P_0=60,g=0$, payout rate = 75%, return = 8%

$$P_0=\frac{Di{v}_1}{r-g}=\frac{6\times 0.75}{0.10-(0.08\times 0.75)}=56.25$$

Total Payout Model

$$\text{value of share}=\frac{\text{total equity}}{\text{shares outstanding}}$$

Discounted Free Cash Flow Model

$$P_0=\frac{V_0+{\text{Cash}}_0-{\text{Debt}}_0}{(\text{Shares Outstanding}{)}_0}$$

Comparables

P/E ratio (price per share / equity per share) EBITDA

Market Efficiency

5 | Capital Markets and Risk

Introductions

S&P > S&PTSX > Canada Bonds > USA Bonds > CPI > Holding

Common Measures of Risk and Return

Variance and Standard Deviation

• Are good but don’t distinguish between upside and downside

Historical Returns

$$R_{t+1}=\frac{Di{v}_{t+1}+P_{t+1}}{P_t}-1=\frac{Di{v}_{t+1}}{P_t}+\frac{P_{t+1}-P_t}{P_t}$$

Confidence Intervals

Arithmetic Average vs Geometric Average

• $\overline{R}$ (arithmetic) for estimating expected return over a future horizon
  • how much money after investing in S&P for 25 years
• $\overline{R_g}$ (geometric) for long-run historical performance
  • how much money after investing in stocks for 1 year

Historical Tradeoff between Risk and Return

Inversely correlated

Common risk = affects all securities (can’t be diversified) Independent risk = affects a specific security (can be diversified)

Beta = how correlated you are with the market (covariance/variance)

Capital Asset Pricing Model (CAPM)

$$E[R_i]-r_f={\beta }_i[E(R_{mkt})-r_f]$$

• Market Risk Premium = $E[R_{mkt}-r_f]$
• Risk Premium for security $i$ = $E[R_i]-r_f$
• When $i=mkt,\beta =1$

Diversification

Systemic Risk

6 | Portfolio Theory and the CAPM

Portfolio Return and Volatility

Portfolio weight for $i$-th investment is deterministic (price-agnostic)

$$x_i=\frac{\text{Value of investment i}}{\text{Total value of portfolio}}$$

Realized return for a portfolio

$$R_p=x_1{R}_1+x_2{R}_2+\cdots +x_n{R}_n=\sum _i{x}_i{R}_i$$

• $x=(x_1,x_2,\dots ,x_n)$
• $R=(R_1,R_2,\dots R_n{)}^{\prime }$

Expected return for a portfolio

$$E[R_p]=x_{1}E(R_1)+\cdots +x_{N}E(R_n)=\sum _i{x}_{i}E[R_i]$$

Covariance (TESTABLE)

$$\begin{array}{rl}Cov(R_1,R_2)& =E[(R_1-E[R_1])(R_2-E[R_2])]\\ & =E[R_1{R}_2-R(E(R_2)-E(R_1)R_2+E(R_1)E(R_2))]\\ & =E(R_1{R}_2)-E[R_{1}E(R_2)]-E[E(R_1)R_2]+E[E(R_1)E(R_2)]\\ & =E[R_1{R}_2]-E[R_1]E[R_2]-E[R_1]E[R_2]+E[R_1]E[R_2]\\ & =E[R_1{R}_2]-E[R_1]E[R_2]\end{array}$$

Estimating covariance

$$\begin{array}{rl}Cov(R_i,R_j)& =\frac{1}{T-1}\sum _t(R_{i,t}-\overline{R_i})(R_{j,t}-\overline{R_j})\\ & =\frac{1}{T-1}[\sum _r(R_{i,t}\times R_{j,t})-T\overline{R_i}\overline{R_j}]\end{array}$$

Correlation

$$Corr(R_i,R_j)=\frac{Cov(R_i,R_j)}{SD(R_i)SD(R_j)}$$

Volatility of a portfolio with two stocks

$$\begin{array}{rl}Var(R_p)& =(x_1{)}^{2}Var(R_1)+(x_2{)}^{2}Var(R_2)+2x_1{x}_{2}Corr(R_1,R_2)\sqrt{Var(R_1)Var(R_2)}\end{array}$$ $$Volatility=SD(R_p)=\sqrt{Var(R_p)}$$

Efficient Portfolios of Risky Assets

A portfolio $P$ is inefficient if there exists another portfolio $Q$ such that either there’s both lower risk and lower reward

1. $E[R_Q]>E[R_P]$ AND $SD(R_Q)\le SD(R_P)$
2. $E[R_Q]\le E[R_P]$ AND $SD(R_Q)<SD(R_P)$

Shorting ${\sum }_i{x}_i=1$ but it’s not necessary that $0\le x_i\le 1$

Volatility of a portfolio with $n$

$$Var(R_p)=\sum _{i=1}^n\sum _{j=1}^n{x}_i{x}_{j}Cov(R_i,R_j)Cov(R_i,R_j)=x^{\prime }\Omega x$$ $$Var(R_p)=Cov(R_p,R_p)=Cov(\sum _i{x}_i{R}_i,R_p)$$

Rearranging

$$\begin{array}{rl}Var(R_p)& =\sum _{i}Cov(R_i,R_p)\\ & =\sum _i\sum _j{x}_i{x}_{j}Cov(R_i,R_j)\\ & =\sum _i{x}_i^{2}Var(R_i)+\sum _i\sum _{j,j\ne i}{x}_i{x}_{j}Cov(R_i,R_j)\end{array}$$

Volatility of an equally weighted portfolio with $n$ securities:

$$\begin{array}{rl}Var(R_p)& =(\frac{1}{n})\times \text{Avg variance of individual securities}\\ & +(1-\frac{1}{n})\times \text{Avg covaraince between the securities}\end{array}$$

Risk-Free Saving and Borrowing

Buying on Margin

• Setting $x>1$ such that the fraction invested in the risk free asset $(1-x)<0$

Sharpe Ratio and Tangent Portfolio

Sharpe Ratio: Slope of the line through portfolio

$$Sharpe=\frac{E[R_P]-r_f}{SD(R_P)}$$

• $E[R_P]$ = Expected return
• $r_f$ = Risk free rate
• $SD(R_p)$ = Volatility

Expected Returns and Efficient Portfolios

$$E[R_i]=r_f+{\beta }_i^Q\times (E[R_{Mkt}-r_f])$$ $$\beta =\frac{Cov(R_i,R_{Mkt})}{Var(R_{Mkt})}=SD(R_i)\times \frac{Corr(R_i,R_{Mkt})}{SD(R_{Mkt})}$$

Capital Asset Pricing Model

Beta of a portfolio

$$E[R_p]=\sum _i{x}_{i}E[R_i]=x^{\prime }\mu$$

Volatility of a portfolio is not a weighted average

$$SD(R_p)=\sqrt{\sum _i\sum _j{x}_i{x}_{j}Cov(R_i,R_j)}=\sqrt{x^{\prime }\Omega x}$$

Beta of a portfolio is a weighted average

$${\beta }_p=\frac{Cov(R_p,R_{Mkt})}{Var(R_{Mkt})}=\sum _i{x}_i{\beta }_i=x^{\prime }\beta$$

where $\beta =({\beta }_1,{\beta }_2,\dots ,{\beta }_n)$

7 | Financial Options

Background

Moneyness notation

• Expiration time: $T$
• Stock price at time: $S_t$
• Strike: $K$
• Value of call option at time $t:C_t$
• Value of put option at time $t:P_t$

At the money / in the money / out of the money

Option Payoffs and Profits At Expiration

Can’t lose more than the premium

Combinations

Straddle = long call and long put with the same $T,K$

Butterfly spread:

• Consider 3 calls on the same stock with the same $T$ and evenly-spaced strikes $K_1<K_2<K_3$
• Butterfly spread = long $K_1$ call, one $K_3$ call, short two $K_2$ calls

Protective put

• something about hedging

Put-Call Parity

European option = can only be exercised at expiration American option = can be exercised anytime

Suppose you buy one share, one EU put, one EU Call (same $K,T$). Assume no dividends. What’s the payout?

Case 1: stock doesn’t move:

• Stock: $K$
• Put: $0$
• Call: $0$

Case 2: stock goes up:

• Stock: $S_T$
• Put: $0$
• Call: $K-S_T$

Case 3: stock goes down

• Stock: $S_T$
• Put: $K-S_T$
• Call: $0$

Total in all cases is $K$ - (price of 2 options). In summary, everything cancels out

Put-call Parity

$$C_t=S_t+P_t-P{V}_t(K)$$

If the underlying asset pays dividends before $T$

$$C_t=S_t-P{V}_t(Div)+P_t-P{V}_t(K)$$

• $P{V}_t(Div)$ = PV of current time t dividends paid between $t,T$

Example: $S_t=30,t=2,P_t=3,r=0.05$, what is $C_t$?

$$C_t=25+3-\frac{30}{1.05^2}$$

Factors Affecting Option Values

Intrinsic value = value if it would be immediately exercised Time value = current value - intrinsic value

Early Exercise

Discount on strike price:

$$dis(K)=K-P{V}_t(K)$$

Early Exercise Call option, no dividends

$$\begin{array}{rl}{C}_t& =(S_t-K)+(dis(K)+P_t)\\ & =\text{(intrinsic value) - (time value)}\end{array}$$

Put option, no dividends

$$P_t=(K-S_t)+(C_t-dis(K))$$

Call option, dividends

$$C_t=(S_t-K)+(dis(K)+P_t-P{V}_t(Div))$$

Put option, dividends

$$P_T=(K-S_t)+(C_t-dis(K)+P{V}_t(Div))$$

8 | Option Valuation

Two State Option Pricing

Ex: If $r=0.05,K=100,C_1^{+}=25,C_1^{-}=0,C_0=?$

Form a portfolio $V$ (called the replicating portfolio) with $\Delta$ (delta) shares of stock and $B$ cash $V_1^{+}=125\Delta +1.05B+C_1^{+}=25$ $V_1^{-}=80\Delta +1.05B+C_1^{-}=0$ $\Delta =0.5556,B=-42.3280$

$$V_0=100(.5556)-42.3280=13.3375$$

Delta Hedging: Make portfolio insensitive to small price changes in the underlying stock

Risk-Neutral Valuation

Under risk-neutral probabilities, option values are expected future payoffs discounted at the risk-free rate

Risk-neutral probability:

$$p\ast =\frac{S_0(1+r_f)-S_1^{-}}{S_1^{+}-S_1^{-}}=\frac{1+r_f-d}{u-d}$$

The Binomial Model

$$p\ast =\frac{\sqrt{1+r_f}-d}{u-d}$$

Convergence of the Binomial Model

• $nT$ periods of expiration $T$, each of length $h=\frac{1}{n}$ years
• $\sigma$ is the standard deviation of the annual rate of return of the underlying stock, with $u=e^{\sigma \sqrt{h}},d=\frac{1}{u}$
• The risk-free rate $r$ is assumed to be continuously compounded
• The discount factor over any period is $e^{-rh}$ and the risk-neutral probability of an up move in any period is

$$p^{\ast }+\frac{e^{rh}-d}{u-d}$$

• After $n$ periods there are $n+1$ possible stock prices $S_0{u}^j{d}^{n-j}$ for $j=0,1,\dots ,n$ with associated risk-neutral probabilities

$$\left(\genfrac{}{}{0ex}{}{n}{j}\right)(p^{\ast }{)}^j(1-p^{\ast }{)}^{n-j}$$

The Black-Scholes Model

Can be viewed as the limit of the binomial model as $n\to \infty$

• The basic version applies to European options on stocks which don’t pay dividends before $T$
• Assumes changes in underlying stock prices have a lognormal distribution (changes in the natural log of the price are normally distributed)

Notation

• Current stock price = $S_0$
• Expiration time (years) = $T$
• Strike = $K$
• Risk-free interest rate (continuously compounded) = $r$
• Standard deviation (volatility) = $\sigma$

Black-Scholes Formula

$$\begin{array}{rl}{C}_0& =S_{0}N(d_1)-K{e}^{-rT}N(d_2)\\ P_0& =K{e}^{-rT}N(-d_2)-S_{0}N(-d_1)\\ d_1& =\frac{\ln [\frac{S_0}{PV(K))}]}{\sigma \sqrt{T}}+\frac{\sigma \sqrt{T}}{2}\\ d_2& =d_1-\sigma \sqrt{T}\end{array}$$

Risk and Expected Return for Options

Option betas can be calculated using the replicating portfolio since the option value is $S_0\Delta +B$

$$\begin{array}{rl}{\beta }_{option}& =[\frac{S_0\Delta }{S_0\Delta +B}]\times {\beta }_S+[\frac{S_0\Delta }{S_0\Delta +B}]\times {\beta }_B\\ & =[\frac{S_0\Delta }{S_0\Delta +B}]\times {\beta }_S\end{array}$$ $$[\frac{S_0\Delta }{S_0\Delta +B}]=\text{leverage ratio}$$

Under Black-Scholes,

• Call: $\Delta =N(D_1),B=-K{e}^{-rT}N(d_2)$
• Put: $\Delta =-N(-d_1),B=K{e}^{-rT}N(-d_2)$

Leverage ratio

• For a call option, $\Delta >0,B<0$, so the leverage ratio $>1$ and ${\beta }_{option}>{\beta }_s$
• For a put option, $\Delta <),B>0$, so the leverage ratio $<0$ and ${\beta }_{option}<0$
• The magnitude of the leverage ratio is higher for out-of-money options

9 | Financial Risk Management

Forwards

Forward = an agreement to buy or sell an asset at a specified future delivery date at a price set today called the forward price (a.k.a. delivery price)

• like an option without the option
• the buying party is long, the selling party is short

How are forward prices determined?

• For financial assets (stocks, bond, currencies) and commodities held by a lot of investors (gold), price is determined by no-arbitrage
• For commodities held by firms for production (copper), price is determined by supply and demand

Buy now and hold VS buy forward for later?

• Depends on whether the asset pays income or if it incurs storage costs (oil in 2020)

Futures

Future = forwards that are exchange-traded

How to reduce default risk

• Margin = investors depositing collateral
• Marking to market = calculating price changes at EoD and removing from margin accounts. If balances get too low, they get margin called

Interest rate hedging

• Can be used to hedge against interest rates
• Forward rate = interest rate guaranteed today for a loan/investment in the future

Duration-based hedging

• A firm with assets and liabilities having different durations has a duration mismatch
• Since equity = assets - liabilities, the value of equity

Example: Oil producers

Exchange rate risk

• FX forward contracts used to hedge against that

Swaps

Swap = private agreement to exchange future cash flows in case something happens

• event contracts?
• Credit Default Swap (CDS) = if someone defaults, they have to give up cash flows (insurance against defaulting)