FM430
MT
Topic 1 Present Values
Topic 2 Fixed-Income Securities
Topic 3 Stock Valuation
Topic 4 Portfolio Theory
Topic 5 CAPM
Topic 6 Market Efficiency
Topic 7 Option
Topic 8 Forwards & Futures
LT
Topic 9 Capital Budgeting
Topic 10 Capital Structure
Topic 11 Mergers & Acquisitions
Topic 12 IPOs
,Topic 1 Present Values
Present Value
- $1 today =/= $1 in the future
- PV is sensitive to the interest rate
' '-
- 𝑃𝑉# = 𝐶# + ( + ⋯ + -
)*+( ()*+- )
- 𝐹𝑉1 = 𝑃𝑉# ∗ 1 + 𝑟 1 = 𝐶# (1 + 𝑟)1 + 𝐶) (1 + 𝑟)15) + ⋯ + 𝐶1
- Interest rates don’t have to be constant, depends on term structure (R as a function of T)
- Risky cashflows have to be discounted using risky rate, not risk-free rate
Attitude Towards Risk
- People care about reference points
- People overweigh small probabilities
APR vs EAR
- APR is a quote rate, no compounding effect
- EAR = Monthly (compounding) APR is a true rate, with compounding effect
- EAR always > APR
9:; 1 9:; 1
- 1 + 𝐸𝐴𝑅 = (1 + ) => 𝐸𝐴𝑅 = (1 + ) −1
1 1
Real vs Nominal Dollars
- Nominal $ = regular $ Real $ = $ adjusted for inflation
)*BCDEBFG +FHI
- 1 + 𝑟𝑒𝑎𝑙 𝑟𝑎𝑡𝑒 =
)*EBJGFHECB +FHI
- Discount real (nominal) CF using real (nominal) rate
Perpetuities
- Stream of constant cash flows, starting one period from today, lasting forever
'
- 𝑃𝑉# = ( , where r > g
+5K
- First cash flow comes one period from today
- If C1 is annual CF, use annual rate r etc…
Annuities
- Stream of constant cash flows, starting one period from today, lasting for T periods
' ) ' ) ) ' )*K
- 𝑃𝑉# = − -
=𝐶[ − -
] With growth: 𝑃𝑉# = [ 1 − ( )1 ]
+ )*+ + + + )*+ +5K )*+
- First cash flow comes one period from today
- If C is annual CF, use annual rate r etc…
Internal Rate of Return (IRR)
- Discount rate in the PV calculation that makes the PV equal to 0, or project specific rate
- Decreasing function (PV against r)
- IRR = r: Breakeven IRR > r: Accept IRR < r: Reject
2
,Topic 2 Fixed-Income Securities
Type of Bond
Corporate Bond (Involves default risk)
- Zero-Coupon Bond: Promises a single cash flow and face value at maturity, always issue at par
- Coupon Bond: Promises a periodic cash flow (coupon) and face value at maturity
Treasury Bond (Risk-free)
- Treasury bills (1yr, no C), Treasury note (1-10yrs, semi C), Treasury bond (>10 yrs, semi C)
Term Structure of Spot Rates (Yield Curve)
- T-year spot rate is the annual interest rate for year t
- Yield curve represents the spot rates as a function of maturity
- Can have many shapes, generally slopes up (longer maturities have higher spot rates)
- Spots rates, for both short & long maturities, move substantially over time (daily news)
Forward Rates
- Spot rates = Today’s rate
- Forward rates = Rates guaranteed today for investing in the future
- Forward rates can be derived from spot rates 1 + 𝑟H H (1 + H𝑓1 )1 = (1 + 𝑟H*1 )H*1
Synthetic Replication
- Absence of arbitrage (no making money out of nothing), Law of one price
- A portfolio of zero-coupon bond synthetically creates a replicating portfolio for coupon bond
- A portfolio of coupon bond synthetically creates a replicating portfolio for zero-coupon bond
Yield to Maturity (YTM) & Return
- A single discount rate that equates the PV of the bond’s CF to the bond’s price (i.e. bond’s IRR)
- A complicated average of the spot rates
- If C is annual, YTM is quoted as annual APR etc…
- For zero coupon bond: YTM is the return of investing in the bond & holding until maturity
- For coupon bond: YTM is the return of investing in the bond & hold until maturity & reinvesting coupon
at a rate equal to YTM
YTM & Coupon Rate
- Coupon Rate > YTM Price > Par Value Premium
- Coupon Rate = YTM Price = Par Value Par
- Coupon Rate < YTM Price < Par Value Discount
YTM & Default Risk
- Bond with default risk has lower price, higher YTM than an identical bond with no default risk
- Default spread = Difference in YTM of the same risky & risk-free bond
Corporate Bonds
- Involve default risk, corporation fails to pay the promised cash flows
- Valuation: Use expected cash flows, risk-adjusted discount rates
- Investment grade bonds: Aaa-Baa (Moody’s), AAA-BBB (S&P)
- Speculative grade (Junk) bonds: Ba (Moody’s), BB (S&P)
3
,Interest Rate & Bond Price
1. Bond prices are negatively related to interest rate (↑rate ↓PV / Investor sell bond when IR↑)
2. Interest rate sensitivity of bond prices ↑ with maturity (PV of long term bond is discounted heavier)
3. Bonds prices less sensitive to interest rate change with a higher coupon rate
Macaulay Duration & Modified Duration
R
- 𝐷 = 1HQ) 𝑤H 𝑡 𝐷∗ = where r is the current interest rate ∆𝑃 ≈ −𝑃𝐷∗ ∆𝑟
)*+
- Weighted average of the years where bond pays cash flows
- Effective maturity: Year around which the bond’s discounted CF are balanced
- Provides good approximation to actual change ONLY for a small shift in term structure (as Linear D)
1. Zero coupon bond with T year to maturity: D = T
2. ↑Time to Maturity → ↑D (Generally, not always)
3. ↑Coupon rate → ↓D
Convexity
- Convexity in addition to duration better approximate change in P
) 1 )
- ┌= V HQ) 𝑤H 𝑡 (𝑡 + 1) ∆𝑃 ≈ −𝑃𝐷∗ ∆𝑟 + 𝑃┌(∆𝑟)W
)*+ W
Immunization Using Duration (Making one insensitive to interest rate movement)
𝐶𝑜𝑠𝑡 𝑜𝑓 𝐼𝑚𝑚𝑢𝑛𝑖𝑧𝑎𝑡𝑖𝑜𝑛 = 𝐵𝑢𝑑𝑔𝑒𝑡 𝐶𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛 / 𝐸𝑥𝑐𝑒𝑠𝑠 𝐶𝑎𝑠ℎ
-
∆𝐴 = ∆𝐿 → −𝐴𝐷9∗ ∆𝑟 = −𝐿𝐷i∗ ∆𝑟 → 𝐴𝐷9∗ = 𝐿𝐷i∗
Topic 3 Stock Valuation
Stocks
- Common stocks, Non-voting stock, Preferred stock (Claim after debtholder, before common)
- Voting right, Rights to dividends, Limited liability
- Dividends vs Debt interest: Discretionary vs Mandatory, Non tax-deductible vs deductible
- Stock vs Bond: Stocks perform worse at bad time, Stock is risker -> higher return on avg
Stock Returns
- Return = Dividend payment + Capital gain
R *(: 5: )
- 𝑟= ( ( j
:j
Facts of Stocks
- Stocks with higher SD tend to have higher average return (i.e. High risk high return)
- Holding a diversified portfolio, can reduce risk without lowering expected return (i.e. Diversification)
Covariance & Correlation
u
nCo(p,q) tv((pt 5p)(qt 5q)
- 𝑐𝑜𝑟𝑟 𝑋, 𝑌 = 𝑐𝑜𝑣 𝑋, 𝑌 =
rR p ∗rR(q) w5)
- Cov: Positive if X&Y tend to be high at the same time, Negative if X is high Y is low or vice versa
- Corr: 1 (-1) if exact positive (negative) linear relationship btw X&Y
4
,Stock Valuation
- Stock price is PV of expected dividends discounted at the stock’s expected return
R( *:( R( R( ()*K) R( )*K -x( R(
1. Growing perpetuity: 𝑃# = = + + ⋯+ =
)*+ )*+ ()*+)V )*+ - +5K
Assume expected dividends grow at a constant rate g (Constant growth model)
:
2. Price-Earnings Ratio: Comparable firms have similar P/E 𝑃/𝐸 = j
y:r(
y:r(
3. PV of Growing Opp.: PVGO = Price – No-growth Price 𝑁𝐺𝑃 =
+
Valuation Input: Dividend d, Growth Rate g
- Historical growth: Sample average of historical dividend growth rates
- Forecasted growth: Sample average of forecasted dividend growth rates
- ROE (1 – Payout): Note that number of shares may not be constant, ROE may drop overtime
R
- Dividend Yield: 𝐷𝑌 = j ≈ 𝑟 − 𝑔
:j
Valuation Input: Expected Return r (By CAPM)
Topic 4 Portfolio Theory
Portfolio Return
- Weighted average of returns on individual stocks
- Weight sum to 1, with negative weights for stocks that are sold short
Expected Variance
- Variance of portfolio depends on variance of individuals stocks, and their covariances
- 𝜎(𝑅)W = 𝑤)W 𝜎(𝑅) )W + 𝑤WW 𝜎(𝑅W )W + 2𝑤) 𝑤W 𝜌(𝑅) , 𝑅W )𝜎(𝑅) )𝜎(𝑅W ) 2 Stocks
W W W
- 𝜎(𝑅) = BQ) 𝑤B 𝜎(𝑅B ) + 2 B•D 𝑤B 𝑤D 𝜌(𝑅B , 𝑅D )𝜎(𝑅B )𝜎(𝑅D ) Multiple stocks
Pros and Cons of Diversification
Pros
- Diversification can reduce risk substantially, compared with holding a pure individual stock
- Diversification does not necessarily reduce expected return
- Diversification outside group of assets is more effective in reducing risk
Cons
- Only idiosyncratic risk (particular company/industry) can be diversified but not systematic (mkt) risk
- When correlation between stocks in a portfolio is > 0, always limit to diversification
Mean-Variance Optimisation
Choosing Stock Portfolio
Step 1: Given expected return, choose the portfolio with minimum variance -> Portfolio frontier
Step 2: Choose the best portfolio on the PF by trading off risk & return
Portfolio Frontier
- 2 assets: Any portfolio is a frontier portfolio because no other portfolio having same E[R]
- Short sales: PF becomes hyperbola
- > 2 assets: Not all portfolios are frontier portfolios, adding assets shifts PF to the left
5
, Portfolio Frontier with a Riskless Assets
- 1 = Weight of Riskless Assets (w) + Weight of Risky Assets (1-w)
- Optimisation problem: Min variance (Cov of Rf asset is 0 with all risky assets)
- 𝜎(𝑅)W = 𝑤 W 𝜎 W (𝑅+E€•‚ ƒC+H ) 𝐸 𝑅 = 𝑤𝐸 𝑅+E€•‚ ƒC+H + (1 − 𝑤)𝑅J [Insert Graph pp35-36]
- PF links riskless assets with the tangent portfolio
- All frontier portfolios are combination of riskless asset & TP (with the highest sharpe ratio)
- Should choose portfolio on PF & trade off risk and return (assuming care only mean & variance)
∆y ; CJ ƒC+HJCGEC †EH‡ +J F€€IH & +E€•‚ ƒC+HJCGEC y ;Š 5;‹
- 𝐵𝑢𝑐𝑘 𝑓𝑜𝑟 𝑡ℎ𝑒 𝐵𝑎𝑛𝑔 𝑅𝑎𝑡𝑖𝑜: =
∆‰F+ ; CJ ƒC+HJCGEC †EH‡ +J F€€IH & +E€•‚ ƒC+HJCGEC W'Co(;Š ,;-Œ )
Topic 5 CAPM
Market Portfolio
D•H nFƒ CJ F nCDƒFB‚
- Value-weighted portfolio of N risky assets 𝑤𝑒𝑖𝑔ℎ𝑡 =
HCHFG D•H nFƒ CJ FGG nCDƒFBEI€
- Market risk premium = E[Rm] – Rf
- Beta of market portfolio = 1
𝐑𝐞𝐠𝐫𝐞𝐬𝐬𝐢𝐨𝐧 𝐨𝐧 𝐀𝐬𝐬𝐞𝐭 𝐑𝐞𝐭𝐮𝐫𝐧: 𝑹𝒏 − 𝑹𝒇 = 𝜶𝒏 + 𝜷𝒏 (𝑹𝒎 − 𝑹𝒇 ) + 𝝐𝒏
- Alpha: Measures asset’s attractiveness, CAPM says alpha = 0 (no under/overvalue, fairly priced)
'Co(;Š ,;Ÿ )
- Beta: Systematic risk; Measure asset’s sensitivity to market movements 𝐵B =
‰F+(;Ÿ )
- Sigma: Idiosyncratic risk; Standard deviation
CAPM Assumptions
1. There are N risky assets and 1 riskless asset
2. Trading (including shorting) of assets is costless
3. Investors care about mean & variance
4. Investors have the same info & beliefs
5. Investors have an one-period horizon
Market Equilibrium
- Investors care about mean & variance
- They choose portfolio (tangent portfolio + riskless assets)
- Very risk averse: P close to riskless A, Not very risk averse: P closer to Tangent P, even above TP
- Demand: Combination of tangent portfolio & riskless assets
- Supply: Total market cap of market portfolio + Market cap of the riskless asset
- At equilibrium, D=S →Tangent portfolio = Market portfolio
CAPM’s Key Insight
- Only systematic risk is priced in the market (i.e. relevant measure of A risk = beta, not Var)
- 0 beta: E[R] = Rf as idiosyncratic risk can be diversified → No contribution to portfolio risk
- + (-)beta: ↑(↓) E[R], asset that goes with the same (opp) direction as the market ↑(↓)portfolio risk
- Linearity: Asset’s expected excess return is a multiple of the market risk premium
- Alpha: If E[R] > E[R] given by CAPM → Positive 𝛼 → Underpriced → Invest
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