FM212 Highlights - LSE 2013

FM212 (2012-13 syllabus), [email protected] 1

FM212 HIGHLIGHTS (1, 2) Calculating present value Discount rates, discount factors, PV and NPV

Discount factor 𝐷𝐹 =1

1 + 𝐷𝑅 !

𝑃𝑉 = 𝐷𝐹!×𝐶𝐹! =𝐶𝐹!

1 + 𝐷𝑅 !

Perpetuity: An asset that pays CF in perpetuity

𝑃𝑉 Perpetuity =𝐶𝐹𝑟

𝑃𝑉 Perpetuity from 𝑡 + 1 =𝐶𝐹𝑟

11 + 𝑟 !

𝑃𝑉 Perpetuity with growth =

𝐶𝐹!𝑟 − 𝑔

Annuity: An asset that pays CF each year for a fixed number of years

𝑃𝑉 Annuity, year end =𝐶𝐹𝑟−

𝐶𝐹𝑟

11 + 𝑟 !

𝑃𝑉 Annuity, year start = 1 + 𝑟𝐶𝐹𝑟−

𝐶𝐹𝑟

11 + 𝑟 !

𝑃𝑉 Annuity with growth =𝐶𝐹!𝑟 − 𝑔

−1 + 𝑔1 + 𝑟

! 𝐶𝐹!𝑟 − 𝑔

Compound and simple interest Annual percentage/simple rate 𝐴𝑃𝑅 = 𝑟 =

𝑟𝑚×𝑚

Effective annual rate 𝐸𝐴𝑅 = 1 +

𝑟𝑚

!− 1

Effective monthly rate = 𝐸𝐴𝑅

!!" − 1

Continuous compounded 𝐸𝐴𝑅 = 𝑒! − 1

Nominal and real interest rates: Be consistent when discounting real/nominal CFs

1 + 𝑟 =1 + 𝑅1 + 𝑖

Fisher equation:𝑅! ≈ 𝑟!!! + 𝑖!!! (3) Value of bonds and stock Bonds YTM: Implicit constant interest rate based on future CF and current bond price

𝑃𝑉 Bond =𝐶𝐹!

1 + 𝑌𝑇𝑀 !

!

!!!

+Principal/par1 + 𝑌𝑇𝑀 !

𝐶𝐹 =Coupon rate × Principal/parNumber of payments per year

𝑌𝑇𝑀Per period = 1 + 𝑌𝑇𝑀

No. of periods− 1

If coupon rate < 𝑌𝑇𝑀,𝑃𝑉 < Principal/par vice versa

Stocks

Expected return 𝑟 =𝐷𝐼𝑉!!!𝑃!

Dividend yield

+𝑃!!! − 𝑃!

𝑃!K appreciation

⟹ Capitalization rate 𝑟 =

𝐷𝐼𝑉!!!𝑃!

+ 𝑔

Valuing stocks: Dividend discount model

𝑃𝑉 Stock =𝐷𝐼𝑉!1 + 𝑟 !

!

!!!

+𝑃!

1 + 𝑟 !

𝑃𝑉 Stock in perpetuity =𝐷𝐼𝑉!𝑟

𝑃𝑉 Stock with growth in perpetuityAssume constant ROE and PBR

=𝐷𝐼𝑉!𝑟 − 𝑔

⟹ 𝑟 =

𝐷𝐼𝑉!𝑃𝑉

+ 𝑔 Valuing stocks: Dividend discount-perpetuity at t

𝑃𝑉 Stock =𝐷𝐼𝑉!1 + 𝑟 !

!!!

!!!

+1

1 + 𝑟 !!!×𝐷𝐼𝑉!𝑟

𝑃𝑉 Stock with 𝑔 =𝐷𝐼𝑉!1 + 𝑟 !

!!!

!!!

+1

1 + 𝑟 !!!×𝐷𝐼𝑉!𝑟 − 𝑔

Gordon growth multiple (g): Assume that ROE, PBR and g are constant

1 =𝐷𝐼𝑉𝐸𝑃𝑆

Payout ratio

+𝐸𝑃𝑆 − 𝐷𝐼𝑉

𝐸𝑃𝑆Plowback ratio

𝑔 =𝐸𝑃𝑆

Book equity per shareReturn on equity

×𝐸𝑃𝑆 − 𝐷𝐼𝑉

𝐸𝑃𝑆Plowback ratio

Present value of growth opportunities (PVGO)

𝑃! =𝐸𝑃𝑆𝑟

+ 𝑃𝑉𝐺𝑂⟹𝐷𝐼𝑉𝑟 − 𝑔

=𝐸𝑃𝑆𝑟

+ 𝑃𝑉𝐺𝑂

OR 𝐸𝑃𝑆𝑃!

= 𝑟 1 −𝑃𝑉𝐺𝑂𝑃!

=𝑃𝐸

!!


Price change with and without growth

𝑃Growth =𝐸𝑃𝑆(1 − Plowback ratio)

𝑟 − 𝑔

𝑃No growth =𝐷𝐼𝑉𝑟

=𝐸𝑃𝑆𝑟

∆ 𝑃Growth → 𝑃No growth =𝑃No growth − 𝑃Growth

𝑃Growth= ⋯

… =𝑔 − 𝑟×Plowback ratio𝑟 1 − Plowback ratio

↓

(4) Risk and return Variance: Measure of total risk of a security and is a measure of stand-alone risk. Total risk has both unique and market risk characteristics. Government and risk-free bonds have standard deviation of 0

𝜎! =𝑋! − 𝑋 !

𝑛 − 1

Portfolio risk

𝑥! = Share of portfolio in asset i (𝑥! < 1)

Expected portfolio return = 𝑥!𝑟!

!

!!!

Portfolio variance = 𝑥!𝑥!𝜌!"𝜎!𝜎!

!

!!!

!

!!!

Portfolio variance increases with higher financing margins (e.g. financing through borrowing). Portfolio standard deviation doubles when 50% of total amount can be borrowed

Stock 1 Stock 2

Stock 1 𝑥!!𝜎!! 𝑥!𝑥!𝜌!"𝜎!𝜎!

Stock 2 𝑥!𝑥!𝜌!"𝜎!𝜎! 𝑥!!𝜎!!

𝜌 =𝐶𝑜𝑣 𝑋,𝑌

𝑉𝑎𝑟 𝑋 𝑉𝑎𝑟 𝑌=

(𝑋! − 𝑋)(𝑌! − 𝑌)!!!!

𝑋! − 𝑋 !!!!! 𝑌! − 𝑌 !!

!!!

Beta: Measure of the volatility of the security’s returns to changes in market returns (measure of market risk). Diversification lowers idiosyncratic risks but does not affect market risk (ie. nondiversifiable risk)

𝐵! =Covariance with the marketVariance of the market

=𝐶𝑜𝑣 𝑖,𝑚𝑉𝑎𝑟 𝑚

Portfolio beta = 𝑥!𝐵!

!

!!!

𝐵! = 𝐺𝑟𝑎𝑑 𝑅Realized stock return,𝑅Realized market return

(5) Portfolio theory Markowitz portfolio theory: Combining stocks into portfolios can reduce SD below the level obtained from a simple weighted average calculation

Lending and borrowing: Lending and borrowing at the risk free rate enables one to attain all possible expected returns located on the line joining 𝑟! to the efficient portfolio Mean Variance Efficient portfolio: Combination of stocks that has the lowest risk for a given return expectation – the best possible portfolio Security market line: Linear relationship between risk (beta) and expected return that makes one indifferent to taking a long/short position on that asset. Assets above the SML are undervalued (long), whereas assets below the SML are overvalued (short) 𝐶𝐴𝑃𝑀: 𝑅

E(Return)= 𝑟! + 𝛽! 𝑟! − 𝑟!

Market price of risk ie. market risk premium

True SML vs CAPM: The true SML has a higher vertical intercept and is flatter than the SML modelled in the CAPM. This may be due to (1) risk-seeking investors who are leverage-constrained would demand high beta stocks, driving up their prices and driving down expected returns, (2) true beta measurements underestimate the market risk

Lend

Borrow

Long

Short

1

r_m


premium (ie. slope of the SML), (3) low beta stocks are often overlooked by investors and tend to be undervalued Sharpe ratio: The Sharpe ratio characterizes how well the return of an asset compensates the investor for the additional risk taken. Holding all else equal, an asset with a higher Sharpe ratio provides better returns for the same risk. The market portfolio has the highest Sharpe ratio

Sharpe ratio: Risk premium

Standard deviation=𝑟! − 𝑟!𝜎

Arbitrage Pricing Theory: An alternative to CAPM, relying on multiple betas (e.g. macroeconomic indicators) to measure sensitivity to multiple risk factors – not just the market factor

𝐴𝑃𝑇:𝑅 = 𝑟! + 𝛽!(𝑟Factor ! − 𝑟!)!

!!!

Comparing CAPM and APT Model Advantages Disadvantages CAPM Considers only

systematic risk, reflecting a reality in which most investors hold diversified portfolios Generates a theoretically-derived relationship between return and systematic risk

Difficult to estimate market return and beta World capital markets are not perfect, assets may be priced incorrectly and individual investors may not be able to borrow at the risk-free rate

APT Excludes the measurement of market efficient portfolios Allows for multiple sources of risk that affect stock returns

Demands that investors perceive and reasonably estimate factor sensitivities

(6) Market efficiency Random Walk Theory: Movement of stock prices from day to day do not reflect any pattern. Statistically, the movement of stock prices is random with a positive drift over the long term

Abnormal returns = 𝑟! − 𝑎! + 𝐵!𝑟! Forms of market efficiency

Form of efficiency Description Weak (ie. markets have no memory)

It is impossible to make consistently superior

profits by studying past returns. Prices follow a random walk

Semistrong (ie. there are no financial illusions)

Prices reflect all past information and current public information. Prices will adjust immediately to information as it becomes publically available

Strong (ie. trust market prices)

Prices reflect all information that can be acquired by the analysis of the company the economy – both public and private

(7) Put and call options Call option: The right to buy a security at a specified price within a specified time – exercise when P > Strike (call option is ‘in the money’)

IntrinsicCall = max(0,𝑃Stock − 𝑃Strike) Put option: The right to sell a security at a specified price within a specified time – exercise when P < Strike (put option is ‘in the money’)

IntrinsicPut = max(0,𝑃Strike − 𝑃Stock)

Long (Buy) Short (Sell)

Call option Right to buy Oligation to sell

Put option Right to sell Obligation to buy Black-Scholes variables: Any change in expected return has no effect on call option prices. Since the underlying prices are constant, a higher expected option payoff is discounted at an exactly offsetting higher rate

‡ 𝑃Call = 𝑁 𝑑! ×𝑃 + 𝑁 𝑑! ×𝑃𝑉 𝐸𝑋

‡ 𝑑! =log 𝑃

𝑃𝑉 𝐸𝑋𝜎 𝑡

+𝜎 𝑡2

‡ 𝑑! = 𝑑! − 𝜎 𝑡 =log 𝑃

𝑃𝑉 𝐸𝑋𝜎 𝑡

−𝜎 𝑡2

Variable (Reverse for )

Explanation for variable (holding all other factors constant) Call Put


𝑃Stock Higher price, higher intrinsic value

𝐸𝑋

𝑟! Lower PV of payment of strike price when exercised

Time to maturity, 𝑡

Delay in paying the exercise price (especially when interest rates are high) lowers PV

Volatility, 𝜎

Higher upside without higher downside – assuming that there is a positive probability of a positive payoff with a lower bound of zero

Closer to expiration

date

Option payoff (Option price = 0, kink at EX)

Call Put

Long

Shor

t

Option profit (Option price = p, kink at EX)

Call Put

Long

Shor

t

Put-call parity: If two security packages have identical payoffs in all states in the next period, they must have identical prices this period (assuming no arbitrage). Holding stock price and risk-free rate constant, anything that increases the call price must increase the put price by the same amount

𝑃Put + 𝑃Stock = 𝑃Call + 𝑃𝑉 𝐸𝑋 = 𝑃Call +𝐸𝑋1 + 𝑟

Invest EX risk free

Exploiting arbitrage opportunities

Put + Stock < Call + EX Put + Stock > Call + EX

Buy put and stock

Borrow PV(EX)

Sell call

Buy call

Lend PV(EX)

Sell put and stock

(8) Options pricing theory (1) Replicating strategy: Value options by constructing option equivalents. We calculate at each terminal stock price the portfolio of delta shares plus borrowing that has the same payoff as the option. We then set the price of the option to equal the replicating portfolio, working backwards until the starting date (d = option delta/hedge ratio, x = risk-free rate)

Option delta/hedge ratio: A measure of the sensitivity of changes in option price in relation to small changes in stock price. Delta tells us the fractional shares of

EXBE

EXBE

£50

£60 (£10)

60d - 1.01x = 10

£30 (0)

30d + 1.01x = 0


stock needed to hedge the risk of 1 option. A call will be exercised when delta is 1 (equivalent to buying the stock with a deferred payment – so a one-dollar change in the stock price matches a one-dollar change in the option price), not exercised when delta is 0 (option is essentially valueless, regardless of change in stock price)

𝛿Call =𝑃Option at high P − 𝑃Option at low P

𝑃Stock high − 𝑃Stock low> 0 (Buy to replicate)

𝑉Call (Period 1) = 𝛿𝑃! −𝛿𝑃Low (Period 1)

1 + 𝑟!(A): Borrow at !!

⟹ Buy 𝛿 shares, borrow (A)

𝛿Put =𝑃Option at high P − 𝑃Option at low P

𝑃Stock high − 𝑃Stock low< 0 (Sell to replicate)

𝑉Put (Period 1) = 𝛿𝑃! −𝛿𝑃High (Period 1)

1 + 𝑟!(B): Lend at !!

⟹ Sell 𝛿 shares, lend (B)

(2) Binomial risk-neutral probability: Value options by calculating risk-neutral probabilities – the hypothetical probabilities that upward and downward stock price movements will give an expected return that is equal to the risk-free return. We price the option by taking next-period prices and calculating expected values using risk-neutral probabilities, discounting backwards until the starting date

𝑝!×𝑢 + 𝑝!×𝑑 = 1 + 𝑟

⟹ 𝑝!×𝑢 + 1 − 𝑝! ×𝑑 = 1 + 𝑟

⟹ 𝑝! =1 + 𝑟 − 𝑑𝑢 − 𝑑

, 𝑝! = 1 − 𝑝!, where 𝑢 > 1,𝑑 < 1 Multi-period risk-neutral probability

𝑝! =1 + 𝑟 − 𝑑𝑢 − 𝑑

=1 + 𝑟 = 𝑒!"

𝑑 = 𝑒!! !

𝑢 = 𝑒! !

Binomial model Call options without dividends: In the absence of dividends, the value of a call option increases with time to maturity (ie. exercising early would reduce its value). Hence American and European calls have the same value

Call2 =𝑝! 𝑢!𝐴 − 𝐸𝑋 + 1 − 𝑝! [max 𝑢𝑑𝐴 − 𝐸𝑋 ]

1 + 𝑟

Call3 =

𝑝![𝑀𝑎𝑥 0,𝑢𝑑𝐴 − 𝐸𝑋 ] + (1 − 𝑝!)[max(0,𝑑!𝐴 − 𝐸𝑋)]1 + 𝑟

Call1 =𝑝! 𝐶𝑎𝑙𝑙! + (1 − 𝑝!)(𝐶𝑎𝑙𝑙!)

1 + 𝑟

Put options without dividends: American and European put options differ in price with or without dividends

𝐸𝑋!"# = £110

𝑟 = 0.01

𝑢 = 1.25 𝑑 = 0.8

𝑝! =715

𝑝! = 1 − 𝑝! =815

EU Put1 =715 5.28 + 1 − 7

15 (28.91)

1 + 0.01= £17.71

[1] £(A)

[2] £(uA)

[4] £(uuA) (uuA - EX)

[3] £(dA)

[5] £(udA) (udA - EX) or 0

[6] £(ddA) (ddA - EX) or 0

European Put £100

(£17.71)

£125 (£5.28)

£156.25 (0)

£80 (£28.91)

£100 (£10)

£64 (£46)

American Put £100

(£18.46)

£125 EX: -£15

No EX: £5.28

£156.25 (0)

£80 EX: £30

No EX: £28.91

£100 (£10)

£64 (£46)


AM Put1 =715 5.28 + 1 − 7

15 (30)

1 + 0.01= £18.46

Call options with dividends (same for put options)

𝐷𝐼𝑉 = 7

𝐸𝑋 = 90

Option can only be exercised at expiration

EU Call1 =715 28.78 + 1 − 7

15 (0.58)

1 + 0.01= £13.60

Make the intermediate choices that yield higher 𝐶𝑎𝑙𝑙! ie. Exercise early if the option is deep in the money

AM Call1 =715 35 + 1 − 7

15 (0.58)

1 + 0.01= £16.48

(9) Valuing government bonds Valuing a bond: The price of a bond is negatively related to yield/YTM. Coupon is negatively related to the length of the maturity period YTM: Implicit constant interest rate based on future CF and current bond price


1 + 𝑌𝑇𝑀 !

!

!!!

+Principal/par1 + 𝑌𝑇𝑀 !

𝐶𝐹 =Coupon rate × Principal/parNumber of payments per year

𝑌𝑇𝑀Per period = 1 + 𝑌𝑇𝑀

No. of periods− 1

If coupon rate < 𝑌𝑇𝑀,𝑃𝑉 < Principal/par vice versa

Bond duration: A weighted average time to maturity of all cash flow payments of the bond. It measures the true time length of the bond adjusted for the size of the cash flow and when it is received. A higher duration implies higher volatility

Duration =

𝐶𝐹!1 + 𝑌𝑇𝑀 !

𝑃𝑉×𝑖

!

!!!

⟹1×𝑃𝑉 𝐶𝐹!

𝑃𝑉+2×𝑃𝑉 𝐶𝐹!

𝑃𝑉+⋯+

𝑇×𝑃𝑉 𝐶𝐹!𝑃𝑉

Purpose of bond duration: Duration can be used to measure a portfolio’s interest rate risk exposure. Liabilities can also be matched with portfolios of similar durations so as to hedge risk from interest rate changes Modified duration/volatility: A measure of the sensitivity of changes in bond price in relation to a 1% change in interest rate (absolute, not % change)

Volatility/modified duration =Duration1 + 𝑌𝑇𝑀

∆𝑃 following ∆𝑌𝑇𝑀: ∆𝑃! = −Volatility (∆𝑌𝑇𝑀)(𝑃!)

𝑃! = 𝑃! + ∆𝑃!

Forward rate: The expected interest rate, fixed today, on a loan made in the future at a fixed time

1 + 𝑟! ! = 1 + 𝑟! 1 + 𝑓! … 1 + 𝑓!

1 + 𝑓! =1 + 𝑟! !

1 + 𝑟!!! !!!

Spot rate: The actual interest rate today for relevant maturity. The future rate refers to the spot rate in the future


1 + Spot! !

!

!!!

+Principal/par1 + Spot! !

𝑃𝑉 =𝐶𝐹!

1 + 𝑌𝑇𝑀 !

!

!!!

=𝐶𝐹!1 + 𝑓!

+𝐶𝐹!

1 + 𝑓! 1 + 𝑓!+⋯

Zero coupon/strip bonds: A method for solving for n-year spot rates (𝑟!)

−𝑃𝑉 +0

1 + 𝑟! ! +0

1 + 𝑟! ! +⋯+Principal/par

1 + 𝑟! ! = 0

European Call £100

(£13.60)

No EX: £118 (£28.78)

£147.25 (£57.25)

£94.4 (£4.4)

No EX: £73 (0.58)

£91.25 (£1.25)

£58.4 (0)

£100 (£16.48)

EX: £125 (£35)

No EX: £118 (£28.78)

£147.25 (£57.25)

£94.4 (£4.4)

EX: £80 (0)

No EX: £73 (0.58)

£91.25 (£1.25)

£58.4 (0)


⟹ 𝑃𝑉 =Principal/par

1 + 𝑟! !

Expectations theory and term structure: In equilibrium, investment in a series of short-maturity bonds must offer the same expected return as an investment in a single long-maturity bond (only then will investors be indifferent between holding both short and long-maturity bonds). The future spot rate is the forward rate. An upward sloping yield curve indicates that investors anticipate short term interest rates to rise in the future vice versa

1 + 𝑟! ! = 1 + 𝑟! 1 + 𝑓! Liquidity preference and term structure: Short-term rates tend to be lower than long rates due to the preferences of borrowers for shorter-term liabilities and lenders for longer-term assets. Short-term investors must be offered a positive risk premium to hold longer-term bonds (the future spot rate might be lower than predicted by expectations). Positive slope of the term structure does not necessarily mean that short-term rates are expected to increase Risk and term structure: In exchange for price and interest rate volatility, investors without long-term investment horizons will only hold long-term bonds if they offer higher returns – upward sloping yield curve Inflation and term structure: If rising inflation is an important risk for long-term investors (ie. future cash flows will be less valuable), borrowers must offer some extra incentives if they want investors to lend long – upward sloping yield curve Coupon is negatively related to yield: A bond with a higher (lower) coupon has a greater (smaller) proportion of its total payments coming earlier when interest rates are low. This explains a lower (higher) yield. This means that zero coupon bonds have the highest yields, whereas annuities (ie. fixed equal payments annually) have the lowest yields (10) Forwards and futures Spot contract: A contract for immediate sale and delivery of an asset. Spot prices are denoted as 𝑆! Forward contract: A contract for the delivery of an asset at a set price on a set date in the future Futures contract: Similar to a forward contract, but with a standardizing intermediary (e.g. clearing house) Futures vs forwards: Futures and forwards differ in the following aspects that may account for differences in prices

Difference Explanation Futures are exchange-traded whereas forwards are not

Reduces counterparty (credit) risk, making futures more desirable

and hence more expensive

Futures are marked to market, whereas forwards are not

Reduces counterparty (credit) risk, making futures more desirable and hence more expensive If interest rates are correlated with futures price, futures buyers can receive payments when interest rates are high and reinvest at a high rate. Futures are hence more expensive

Futures are written on standard underlying deliverables

Basis risk renders futures less perfect hedge than forwards. Futures may hence be less expensive

Marking to market: Resetting the contract at the end of each day to reflect changes in asset price so as to lower counterparty risk (ie. contract default)

Pricing financial futures: Assume that short-term dividend yield is risk-free, and dividends are paid just before the futures contract’s maturity date

𝑃!!! − 𝐹Buy future, sell asset

= 𝑃!!! − 𝑃! 1 + 𝑟! + 𝐷𝐼𝑉Repay loan+interest, receive DIV, sell asset

⟹ 𝐹 = 𝑃 1 + 𝑟! − 𝐷𝐼𝑉

⟹ 𝐹 = 𝑆! 1 + 𝑟! − 𝑦Dividend yield

!

Replicating a financial future

⟹ Borrow 𝑃! at 𝑟! and buy the asset today

⟹ Tomorrow, sell asset, receive DIV, repay loan 𝑃!!! − 𝑃! 1 + 𝑟! + 𝐷𝐼𝑉

⟹ Identical to the forward rate, otherwise arbitrage

Pricing commodity futures: Commodity futures differ from financial futures in the sense that yields are not observable and depend on storage costs and

Buyer(Long)

Seller(Short)

Asset price falls (-Y)Give seller Y

New contract price: F+X-Y

Asset price rises (+X)Give buyer X

New contract price: F+X


convenience. Net convenience yield is determined by commodity users’ desire to hold the commodity for production or consumption. NCY can vary over time due to inventory and seasonal factors. Excessive inventories can reduce convenience yield to zero, whereas commodity shortages can drive up convenience yields

𝑛𝑐𝑦: Net convenience yield from holding inventory

𝑃!!! − 𝐹Buy future, sell asset

= 𝑃!!! − 𝑃! 1 + 𝑟! − Net convenienceRepay loan+interest, sell asset, lose convenience

𝐹 = 𝑆! 1 + 𝑟! + Storage costs − Convenience yield-‐ Net convenience yield

!

⟹ 𝐹 = 𝑆! 1 + 𝑟! − 𝑛𝑐𝑦

! Replicating a commodity future

⟹ Borrow 𝑃! at 𝑟! and buy the commodity today ⟹ Tomorrow, sell commodity, pay storage, repay loan

𝑃!!! − 𝑃! 1 + 𝑟! − Convenience − Storage ⟹ Identical to the forward rate, otherwise arbitrage

Pricing foreign exchange futures: Let F denote the price of forex futures in home currency (e.g. $X/£). Let P denote the price of forex in home currency (e.g. $Y/£). Let r* denote the foreign risk-free interest rate and 𝑟! denote the domestic risk-free interest rate

𝑃!!! − 𝐹Buy future

= 𝑃!!! − 𝑃!1 + 𝑟∗

1 + 𝑟!

⟹ 𝐹 = 𝑃!1 + 𝑟∗

1 + 𝑟!

Reciprocal pricing: 𝑓 = 𝑝!1 + 𝑟!1 + 𝑟∗

Replicating a forex future: Suppose that £X is scheduled to be converted into $ in the next period, with exchange rates fixed today

⟹ Borrow £𝑋

1 + 𝑟! and convert today $

𝑃!𝑋1 + 𝑟!

⟹ Invest at 𝑟∗ to get $ 𝑃!𝑋1 + 𝑟∗

1 + 𝑟! in 𝑡 = 1

⟹ Identical to the forward rate, otherwise arbitrage

(11) Capital budgeting and the NPV rule NPV rule: Converting future FCFs into comparable risk-adjusted PVs that can be summed

𝑁𝑃𝑉! = 𝐹𝐶𝐹! +𝐸 𝐹𝐶𝐹!1 + 𝑟! !

!

!!!

⟹ Accept if 𝑁𝑃𝑉! > 0

Book rate of return (BRR): Average income divided by average book value over project life

BRR =Book incomeBook assets

Payback period: Number of years required for cumulative cash outflows to equal initial outlay

Choose project thatmin!𝐹𝐶𝐹! = 𝐹𝐶𝐹!

!

!!!

Internal rate of return (IRR): Discount rate that makes NPV equal to zero (higher IRR is preferred)

𝑁𝑃𝑉 = 𝐹𝐶𝐹! +𝐹𝐶𝐹!

1 + 𝐼𝑅𝑅 !

!

!!!

= 0⟹ Accept if 𝐼𝑅𝑅 > 𝑟!

IRR using incremental CFs

𝑁𝑃𝑉∆ = 𝐹𝐶𝐹!!,! − 𝐹𝐶𝐹!!,! +𝐹𝐶𝐹!!,! − 𝐹𝐶𝐹!!,!

1 + 𝐼𝑅𝑅∆ !

!

!!!

= 0

If 𝐼𝑅𝑅∆ > 𝑟! , choose Project 1

Sub 𝑟 < 𝐼𝑅𝑅∆ into NPV formula to yield similar results

Comparing investment appraisal methods Method Advantages Disadvantages NPV Recognizes the cost

of lending capital (e.g. time value and risk of money) Depends on forecasted CFs PVs are additive

Ignores flexibility of investment decisions (see Real options), potentially underestimating true value

BRR Market values and CFs not considered Average historic profitability is not the right hurdle for evaluating future investments

Payback FCFs after cutoff date are ignored

Interest rate differential Expected inflation rate difference

Forward and spot difference Expected change in spot rates

Interest rate parityExpectations theory

PPP

Equal real interest


FCFs before cutoff date are assigned equal weights

IRR Lending vs borrowing: Not all CFs decline with increasing DR (high returns for lenders and low returns for borrowers) Multiple rates of return: Certain CFs can generate NPV=0 at multiple DRs Mutually exclusive projects: Magnitude of project (in terms of NPV) may be ignored – use incremental CFs Term structure: DRs may not be stable over the project duration

Applying NPV: Profitability index 𝑃𝐼 =

𝑁𝑃𝑉Investment

⟹ Select projects with highest WAPI Free cash flows (FCF)

𝐹𝐶𝐹 = 1 − 𝑡 𝐸𝐵𝐼𝑇 + 𝐷𝑒𝑝 − ∆𝑁𝑊𝐶 − 𝐶𝐴𝑃𝐸𝑋

𝐹𝐶𝐹 = 1 − 𝑡 𝐸𝐵𝐼𝑇𝐷𝐴 + 𝑡𝐷𝑒𝑝 − ∆𝑁𝑊𝐶 − 𝐶𝐴𝑃𝐸𝑋

𝐹𝐶𝐹 = Operating CF − ∆𝑁𝑊𝐶 − 𝐶𝐴𝑃𝐸𝑋 Equivalent annual cost: An unbiased comparison of projects with different economic lives and NPVs

𝐸𝐴𝐶 =𝑁𝑃𝑉

Annuity factor

Timing: Delay project if deferred NPV is larger

Current NPV =𝐹𝑉!1 + 𝑟 !

(12) Real options Decision trees example: Call option to expand

𝑟! = 0.1

𝑃𝑉 𝐸𝐶𝐹Expand =0.6 0.8×800 + 0.2×100

1 + 0.1 ! = +327

𝑃𝑉 𝐸𝐶𝐹Not expand =0.6 0.8×410 + 0.2×180

1 + 0.1 ! = +180

𝑃𝑉 𝐸𝐶𝐹Expand > 𝑃𝑉 𝐸𝐶𝐹Not expand ⟹ Expand

𝑃𝑉 𝐸𝐶𝐹Down =0.4 0.4×220 + 0.6×100

1 + 0.1 ! = +49

𝑃𝑉 𝐸𝐶𝐹! =0.6× 100 − 150

Expand

+ 0.4×50

1 + 0.1= −9

𝐴𝑃𝑉! = −250

!!!

+ −9!"!!

+ 49!"!Down

+ 327!"!Expand

= +117

Decision trees example: No option to expand

𝑃𝑉 𝐸𝐶𝐹Up =0.6 0.8×410 + 0.2×180

1 + 0.1 ! = +180

𝑃𝑉 𝐸𝐶𝐹Down =0.4 0.4×220 + 0.6×100

1 + 0.1 ! = +49

𝑃𝑉 𝐸𝐶𝐹! =0.6×100 + 0.4×50

1 + 0.1= +73

𝑁𝑃𝑉! = −250

!!!

+ 73!"!!

+ 49!"!Down

+ 180!"!Up

= +52

Value of option to expand

Real option value = 117!"!!

− 52!"!!

= +65

CF_0 = -250 NPV = 117

CF_1a(0.6) = 100

Expand CF_1a = -150

CF_2a(0.8) = 800

CF_2b(0.2) = 100

Not expand

CF_2a(0.8) = 410

CF_2b(0.2) = 180

CF_1b(0.4) = 50

CF_2a(0.4) = 220

CF_2b(0.6) = 100

CF_0 = -250 NPV = 52

CF_1a(0.6) = 100

CF_2a(0.8) = 410

CF_2b(0.2) = 180

CF_1b(0.4) = 50

CF_2a(0.4) = 220

CF_2b(0.6) = 100


Decision tree example: Put option to abandon

𝑟! = 0.07

𝐴𝑃𝑉! =0.3805×18 + 0.6195×10

1 + 0.07= +12.19

Decision tree example: No option to abandon

𝑁𝑃𝑉! =0.3805×18 + 0.6195×8

1 + 0.07= +11.03

Value of option to abandon Abandonment option value = 12.19

!"!!

− 11.03!"!!

= +1.16

(13) Payout policy Dividend policy relevance vs irrelevance Modigliani-Miller (M&M) – Dividend policy irrelevance: Firm value and shareholders’ wealth do not change with dividend policy. There is merely a transfer of wealth between new and original shareholders. Original owners’ capital change exactly offsets change in cash dividends received

Lintner: Managers act as if dividend policy is relevant

Constant investment – but investments can be made when dividends are retained No transactions costs –

Firms have long-term target dividend payout ratios Dividend changes follow shifts in long-run

but investment banking costs are incurred Efficient capital markets – but information asymmetry and market mis-pricing exist Managers maximize shareholders’ wealth – but Principal-Agent problems may exist Homogeneous taxes – but tax rates on capital gains and dividends differ

sustainable earnings Managers are reluctant to make dividend changes that might have to be reversed Firms repurchase stock with excess cash/replace equity with debt

Earnings + Stock sales + Bond sales

Cash inflow

= 𝐼 + 𝐷𝐼𝑉 + 𝑅Cash outflow

Dividend decision 𝐷𝐼𝑉! − 𝐷𝐼𝑉! = Adj. rate × Target ratio×𝐸𝑃𝑆! − 𝐷𝐼𝑉!

Target change

M&M payout policy irrelevance: Issuing higher dividends via stock issue

𝑃! =𝑉!

NOSH! + NOSH!=

𝑉! + ∆𝐷𝐼𝑉NOSH! + NOSH!

… (1)

NOSH! =∆𝐷𝐼𝑉𝑃!

… 2

Sub 2 into 1 and solve for 𝑃! and NOSH!

Old owners are entitled toNOSH!

NOSH! + NOSH!future 𝐷𝐼𝑉

𝑃!NOSH!Old value

=𝐷𝐼𝑉! + ∆𝐷𝐼𝑉

1 + 𝑟+

NOSH!NOSH! + NOSH!

𝐷𝐼𝑉!1 + 𝑟 !

!

!!!New value

M&M payout policy irrelevance: Stock repurchases using excess cash (assuming all FCF are paid out as dividends)

𝑉! = Excess cash +𝐹𝐶𝐹𝑟

⟹ 𝑃! =Excess cash + 𝐹𝐶𝐹𝑟

NOSH!

NOSHRepurchased =Excess cash

𝑃!

𝐷𝐼𝑉! =𝐹𝐶𝐹

NOSH! − NOSHRepurchased⟹ 𝑃! =

𝐷𝐼𝑉!𝑟

= 𝑃!

Tax consequences: Examing the effects of dividend payout on share price

𝑃Cum DIV − 𝑃Ex DIV𝐷𝐼𝑉

=1 − 𝜏DIV

1 − 𝜏Capital gains

NPV(0) = 11.03

Good state (p = 0.3805)

Do not abandon NPV_1a = 18

Abandon NPV_1b = 10

Bad state (1 - p = 0.6195)


Abandon NPV_1b = 10

NPV(0) = 11.03

Good state (p = 0.3805)


Bad state (1 - p = 0.6195)



Views on payout policy

View Explanation Right: High dividend payout ratios are better than low ones. Dividend increases are followed by stock price increases of 0.36%, whereas dividend decreases are followed by stock price decline of -1.1% (Aharony and Swary)

Dividends are regarded as spendable income, whereas capital gains are merely additions to principal Regular dividends may relieve shareholders of transaction costs and inconvenience Shareholder discipline in spending only dividend income instead of “dipping into capital” Signaling mechanism. Paying out funds to shareholders prevents managers from misusing or wasting funds on negative NPV projects

Left: Firms should pay the lowest possible cash dividend – excess cash should be retained or used to repurchase shares when dividend taxes are higher than capital gain taxes

Financing high dividends via equity issue result in shareholders bearing tax and transaction costs Taxes on dividends need to be paid immediately whereas capital gains taxes can be deferred hence lowering PV of tax obligation

Middle: Firm value is not affected by dividend policy

Clientele effects result in firms having no incentive to change their payout policies, as there are already sufficient low and high payout firms Tax-exempt institutions are indifferent between holding low and high-payout stocks Dividend policy changes over the firm’s life cycle

(14) Capital structure M&M proposition I – Capital structure irrelevance: If capital markets are efficient, firms cannot increase their value by adjusting capital structure – firm value is independent of risk and amount of leverage undertaken, assuming that the standard M&M

assumptions apply AND there are no bankruptcy costs

𝑥𝑉Unlevered = 𝑥𝐸Unlevered → 𝑥𝜋 𝑥𝑉Levered = 𝑥 𝐷Levered + 𝐸Levered → 𝑥𝑅 + 𝑥 𝜋 − 𝑅 = 𝑥𝜋

Leverage boosts EPS

Operating income − InterestNOSHLevered

>Operating incomeNOSHUnlevered

Where NOSHLevered < NOSHUnlevered

𝑃𝐸 falls due to higher risk perceived

M&M capital structure irrelevance in the absence of bankruptcy costs (1 period example)

𝐶𝐹Bonds =Coupon if Coupon ≤ 𝐶𝐹𝐶𝐹 if 𝐶𝐹 ≤ Coupon

𝐶𝐹Shares = 𝐶𝐹 − 𝐶𝐹Bonds

𝑃𝑉 Bonds =𝐸 𝐶𝐹Bonds1 + 𝑟

𝑃𝑉 Stock =𝐸 𝐶𝐹Shares1 + 𝑟

𝑃𝑉 𝑉! =𝐸𝐶𝐹1 + 𝑟

=𝐸 𝐶𝐹Shares1 + 𝑟

+𝐸 𝐶𝐹Bonds1 + 𝑟

= 𝑃𝑉 𝑉! Exploiting arbitrage opportunities: Cost of owning a levered firm must equal that of an unlevered firm with equal and perfectly correlated cash flows. Otherwise one can short the overvalued stock while going long on the undervalued stock, earning positive income at zero risk

𝑥 𝑃!𝑁! = 𝑥 𝑃!𝑁! + 𝐷 Constructing zero-risk, zero-investment portfolios with constant positive income (when L is overvalued)

Short debt 𝐷! = +𝐷

𝑃!𝑁! + 𝐷×Cash

Short levered equity 𝐸! = +𝑃!𝑁!

𝑃!𝑁! + 𝐷×Cash

Long unlevered equity 𝐸∗ = −Cash

Net =𝐸∗

𝑃!𝑁!×𝐹𝐶𝐹!

!"!!

− 𝐷!×𝑟!Interest on D

−𝐸!

𝑃!𝑁!× 𝐹𝐶𝐹! − 𝐷×𝑟!

Repaying !"!!

Exchanging equal equity holdings with similar income stream and a one-off positive payoff (when L is overvalued)

Short current L equity holdings 𝐸! = +%×𝑃!𝑁!


Forego dividends 𝐷! : %× 𝐹𝐶𝐹! − 𝐷×𝑟! Long % U equity holdings 𝐸∗ = %×𝑃!𝑁! > %×𝑃!𝑁!

New dividends 𝐷∗ : %×𝐹𝐶𝐹!

Borrow 𝐵 =%×𝑃!𝑁! −%× 𝐹𝐶𝐹! − 𝐷×𝑟!

𝑟

Earn one-‐off positive payoff = 𝐸! + 𝐵 − 𝐸∗ > 0

Dividend income unchanged = 𝐷∗ − 𝑟𝐵 = 𝐷!

M&M proposition II: The expected return on equity of a levered firm increases in proportion to the D/E ratio. Any increase in expected return is offset by an increase in risk (hence leverage does not affect firm value) – WACC does not change

𝑟Unlevered equity = 𝑟! 𝛽Unlevered equity = 𝛽!

𝑊𝐴𝐶𝐶 or 𝑟! = 𝑟!×𝐷

𝐷 + 𝐸+ 𝑟!×

𝐸𝐷 + 𝐸

⟹ 𝑟! = 𝑟! + (𝑟! − 𝑟!)𝐷𝐸

𝛽! = 𝛽!×𝐷

𝐷 + 𝐸+ 𝛽!×

𝐸𝐷 + 𝐸

⟹ 𝛽! = 𝛽! +𝐷𝐸𝛽! − 𝛽!

After-tax WACC

𝑊𝐴𝐶𝐶 or 𝑟! = 𝑟!×𝐷

𝐷 + 𝐸× 1 − 𝜏Tax shield

+ 𝑟!×𝐸

𝐷 + 𝐸

Adjusted present value

𝐴𝑃𝑉 = 𝑁𝑃𝑉Use pre-‐tax WACC

+𝐷×𝜏×𝑟!

Pre-‐tax 𝑊𝐴𝐶𝐶!" Tax shield

−𝑝𝐶𝐹𝐷𝑟! + 𝑝!" !"#

𝐴𝑃𝑉 = 𝑁𝑃𝑉

Use post-‐tax WACC Assuming that CFD = 0

Traditional view on debt policy: Borrowing increases 𝑟! more slowly than M&M predicts but shoots up when excessive. WACC can be minimized at an optimum D/E ratio. This may be due to (1) investors fail to recognize the financial risk created by moderate borrowing and accept a lower rate of return than they should, (2) market imperfections result in firms being able to borrow more cheaply than individual investors, saving transaction costs and inconvenience (15, 16) Borrowing limits Refuting M&M (Taxes): Tax shield increases total distributed income, as equity capitalizes all future tax savings. Share price increases, and shareholders’ wealth increases accordingly

𝑃𝑉 Tax shield =𝐷×𝜏×𝑟!

𝑟!= 𝐷×𝜏

𝑃𝑉 Tax shield w assets =𝐷×𝜏×𝑟!

𝑟!

𝛽Tax shield = 𝛽!

Book values

Assets Equities and liabilities No change Debt ↑ 100%×𝐷 Equity ↓ 100%×𝐷

Market values Assets Equities and liabilities

Tax shield ↑ 𝜏×𝐷 Debt ↑ 100%×𝐷 Equity ↓ 1 − 𝜏 ×𝐷 Relative advantage of debt (RAD)

𝑅𝐴𝐷 =1 − 𝜏!

1 − 𝜏! 1 − 𝜏! Issue debt if 𝑅𝐴𝐷 > 1Issue equity if 𝑅𝐴𝐷 < 1

Refuting M&M (Costs of financial distress): Capital structure is based on a tradeoff between tax savings and costs of financial distress (empirically 2.5%) that increase with higher D/E ratios. This tradeoff determines optimal capital structure. Equity holders bear bankruptcy costs, as bondholders must be paid the risk-free rate to hold bonds

𝑉 = 𝑉All equity + 𝑃𝑉 Tax shield − 𝑃𝑉 𝐶𝐹𝐷 𝐶𝐹Bad with bankruptcy = 1 − 𝜏Bankruptcy 𝐶𝐹Bad no bankruptcy

𝐸𝐶𝐹 ↓,𝐸 ↓,𝑉 ↓, 𝑟! ↑

Risk shifting: Holding business risk constant, any increase in firm value is shared between shareholders and bondholders. Shareholders of levered firms gain when business risk increases

𝑃𝑉 StockRisky =𝐸𝐶𝐹S,Risky1 + 𝑟

>𝐸𝐶𝐹S,Safe1 + 𝑟

= 𝑃𝑉 StockSafe


𝑃𝑉 BondRisky =𝐸𝐶𝐹B,Risky1 + 𝑟

<𝐸𝐶𝐹B,Safe1 + 𝑟

= 𝑃𝑉 BondSafe Risky projects with negative NPV may be preferred, as shareholders' preferences are pursued over bondholders'

Action Debtors Owners Liquidation Win Lose

Take on more debt for zero NPV project

Lose Win

Issue stock for positive NPV project

Win Win

Extend debt maturity Lose Win

Refuting M&M (Constant investment): Due to asymmetric information, managers’ actions and capital structure serve as signals about expectations and future profitability

𝑃Project announcement =𝑃!×NOSH! + 𝑁𝑃𝑉

NOSH!

NOSH! = NOSH! +Cost of investment𝑃Project announcementNew shares issued, ∆NOSH

Old shareholders give up ∆NOSHNOSH!

of the firm

Equity issue with asymmetric information if perceived NPV is lower than true NPV results in new shareholders gaining at the old shareholders’ expense

𝑃!×NOSH! + 𝑁𝑃𝑉PerceivedNOSH!!Perceived

<𝑃!×NOSH! + 𝑁𝑃𝑉True

NOSH!!True

NOSHPerceived > NOSHTrue

𝑃!×NOSH! + 𝑁𝑃𝑉True + 𝑃Perceived×∆NOSH

NOSH!!Realized

< 𝑃True

𝑃Realized − 𝑃Perceived ∆NOSH

New shareholders' gain

= 𝑃True − 𝑃Realized NOSH!Old shareholders' loss

Debt issue increases share price due to gains from interest tax shield

𝑃!×NOSH! + 𝑁𝑃𝑉 + 𝐷 − Cost of investmentCancels out!!

+ 𝑃𝑉 𝑇𝑆NOSH!!True, debt

𝑃True, debt > 𝑃True, equity > 𝑃Realized > 𝑃Perceived

Asymmetric information problem: If managers strive to maximize original shareholders’ wealth, they will only issue additional equity (hence losing 1 − 𝜃 share of the

firm to new shareholders) and undertake the positive NPV project in the state that delivers a higher payoff to original shareholders. This is inefficient, because a project with positive NPV is forgone in some state of the world Good state Bad state Assets in place 𝑋 𝑌 Investment 𝐾 𝐾 NPV 𝐴 𝐵 Total 𝑋 + 𝐾 + 𝐴 𝑌 + 𝐾 + 𝐵 𝐸 𝐹𝑉 = 𝑃 Good × 𝑋 + 𝐾 + 𝐴 + 𝑃 Bad × 𝑌 + 𝐾 + 𝐵 Good state Bad state Invest 𝜃 𝑋 + 𝐾 + 𝐴 𝜃 𝑌 + 𝐾 + 𝐵 Do not invest 𝑋 𝑌 𝐸 Old, invest in both states > 𝐸 Old, invest in B only

Overcoming information asymmetry

Method Description Financing using retained earnings

Managers can consider investment projects on their own merit, rather than rely on expensive external financing. In doing so, managers are not forced to forgo positive NPV projects

Announcing the realized state

Managers can announce the realized state to the market. However, talk is cheap, and all managers will have an incentive to say that their equity is undervalued From a legal and competitive perspective, announcements of such a confidential nature may not be feasible

Bank debt Managers can reveal the confidential realized state to banks to lower the information asymmetry between lenders and borrowers

Rights issue Managers can issue new equity to existing shareholders. This eliminates the conflict between original and new shareholders, as they are the same people


Pecking order theory: Managers prefer retained earnings to external financing, because it allows them to consider projects on their own merit (rather than rely on market pricing). Financial slack is valuable, as retained earnings are cheaper than external financing. Equity financing may be perceived as existing equity being overvalued, and is hence used as a last resort. There is no well-defined target D/E ratio Tradeoff theory: Most companies have target debt ratios, and debt ratios are positively related to the percentage of tangible assets to total assets. D/E ratios are higher when firms have more taxable income to shield and are unlikely to incur the costs of financial distress. This could be due to (1) high profitability, (2) increasing marginal tax rates, (3) lower cost of financial distress, (4) less risky cash flows Pecking order vs tradeoff theory: (1) Profitable firms often rely on internally-generated funds and have low debt ratios, (2) large and mature firms that have access to bond markets seldom carry out equity financing, (3) new growth firms without tangible assets are likely to rely on equity issues Refuting M&M (Principal-agent problem/dark side to financial slack): Managers’ interests may not be aligned with that of the shareholders. In aligning incentives and ensuring that excess cash is directed to positive NPV projects, debt can serve as a bonding mechanism to discipline managers into staying efficient and generating sufficient CFs to meet debts (17) Types of debt Repayment provisions

Provision Description Sinking fund A fund established to

retire debt before maturity

Callable bond Bond with an option for issuer to buy back before maturity at a specified call price

Puttable bond Bond with an option for investor to demand repayment before maturity at a discount

Payment in kind Issuer can choose to pay interest in the form of cash or more bonds

Callable bonds: Call when market price equals call price. When market price exceeds call price, low yields mean that the firm should buy back existing debt and issue new debt in the market with lower coupon and same price

Convertible bonds: Financial instrument that starts life as a bond but may subsequently be converted into stock. The higher the conversion ratio, the more valuable the convertible vice versa. The higher the conversion price, the less valuable the convertible vice versa Conversion ratio: No. of shares converted into by 1 bond

Conversion price =Face value

Conversion ratio

Conversion value = 𝑃Stock×Conversion ratio

Convertible value = 𝑃𝑉 Straight bond

+𝑃𝑉 Option to acquire stock

Convertible value vs conversion value: Convertible values are higher due to a coversion premium. Convertibles are more secure and offer a higher interest payment than stock dividends. The value of the call option to convert as well as the difference between interest and dividend income reflects this Converting convertibles: As long as interest payments exceed dividends, conversion should be postponed. Exercise the option to convert early if the net dividend exceeds the difference between the option value and its intrinsic value

Terminal default:𝑉Firm < 𝐷 1 + 𝑟!

Convert when 𝑃!"×𝐶𝑅 ≥ 𝐷 1 + 𝑟! ⟹ 𝑃!" ≥𝐷 1 + 𝑟!

𝐶𝑅

⟹ OR Convert when 𝑉Firm ≥ 𝑃!"× NOSH! + 𝐶𝑅×𝐷

⟹ OR Convert when𝐶𝑅×𝐷

NOSH! + 𝐶𝑅×𝐷% shares held after converting

×𝑉Firm ≥ 𝐷

Stock dilution

𝑃!" =𝑉Firm

NOSH! + 𝐶𝑅×𝐷<𝑉Firm − 𝐷NOSH!

= 𝑃!"#

Dilution = 𝑃!"# − 𝑃!"

Issuing convertibles: Convertibles are issued due to (1) differences in risk perceived by managers and the market (e.g. markets will demand higher coupon rates for straight debt), (2) preventing the asset substitution problem (shareholders have lesser incentives to take on risky projects if debtholders are given the right to become shareholders too) (18) Mergers, corporate governance and control Valid sources of synergy/value

Source Description Restructuring and realigning managers’

Eliminating management inefficiencies (e.g.


incentives undertaking negative NPV projects) and obsolete products

Market power Consolidating market power by buying out the competition

Economies of scale (mostly through horizontal mergers)

Reduce costs through combined production, sharing central services (e.g. office management, accounting and financial control) and transferring technology or distribution platforms

Economies of vertical integration (vertical mergers)

Reduce costs through gaining control of vertical processes (e.g. owning the supply chain) or complementary resources

Reduction in taxes Firms may enjoy tax benefits (tax shield) through combined debt

Surplus funds Holding on to excess cash may cause the firm to become an acquisition target. Instead of paying out dividends, firms may choose to buy other firms instead

Dubious sources of value

Source Description Increasing financial slack Managers may want to

buy a company for its cash reserves so as to avoid raising capital to finance positive NPV projects – but it costs a dollar to buy a dollar

Diversification/combined stock has lower volatility than individual stocks

Diversification only lowers idiosyncratic risks, and there is little evidence to show that investors place a premium for diversified firms. Shareholders can diversify away idiosyncratic risks themselves and are hence no better off If diversification can reduce the probability of costly financial distress, firm value may increase and shareholders may be

better off Empirically, diversified firms have lower market-to-book asset ratios than its synthetic comparables (Lang and Stulz, 1994). Excess values are negatively related to future excess returns (Lamont and Polk, 2001)

Increasing EPS Overall EPS increases, but there is no real gain to the combined entity’s value (bootstrap effect). EPS cannot increase indefinitely, and EPS growth in the long run will be lower due to share dilution

Lower financing costs While interest rates on debt may be lower, the acquirer will now be responsible for the target’s debt as well – overall risk has increased

Equity-financed merger with premium (x%)

1 + 𝑥 𝑃𝑉! = 𝑃!"×∆NOSH… 1 𝑃𝑉! + 𝑃𝑉! + Synergies = NOSH! + ∆NOSH ×𝑃!" … 2

⟹ Solve for 𝑃!" and ∆NOSH for consideration

Increasing EPS through merger: EPS is not a good indicator of shareholder well being. A higher EPS may be the consequence of riskier cash flows, whereas a lower, “safer” EPS may precede a higher rate of growth in future earnings

𝐸𝑃𝑆!" =𝐸𝑃𝑆!×NOSH! + 𝐸𝑃𝑆!×NOSH!NOSH!" = NOSH! + ∆NOSH

Price-earning ratio: In the absence of synergies, P/E of the combined entity will be between the P/E ratios of the acquirer and target. Like EPS, P/E is not a good indicator of shareholder well being

𝑃𝐸 !"

=𝑃!"𝐸𝑃𝑆!"

Cash-financed merger gains: Due to asymmetric information, optimistic managers would prefer to finance the merger with cash. Pessimistic managers would prefer to finance the merger with equity, as they think that their shares are overvalued Value of combined firm: 𝑃𝑉!" = 𝑃𝑉! + 𝑃𝑉! + ∆𝑃𝑉!"


Gains to merger/synergies: ∆𝑃𝑉!" = 𝑃𝑉!" − 𝑃𝑉! + 𝑃𝑉!

Cost to merger: Consideration in cash − 𝑃𝑉! 𝑁𝑃𝑉 of merger:𝑃𝑉!" − 𝑃𝑉! − Consideration in cash

Equity-financed merger gains

Cost to merger:𝑃!"×NOSHEntitledConsideration in stock

− 𝑃𝑉!

Terminal values

Liquidation: 1 − 𝑡 LPrice − LCostSalvage value

+ 𝑡 𝑃𝑃𝐸

Perpetuity w/o growth:𝐹𝐶𝐹!!!

𝑟=

1 − 𝑡 𝐸𝐵𝐼𝑇!!!𝑟

Perpetuity w growth:1 − 𝑡 1 − 𝑔 𝐸𝐵𝐼𝑇!!!

𝑟 − 𝑔

Cash vs equity financing: In a MM world, there is no difference between financing a merger with cash or equity

Factor Explanation Equity valuation If shares are overvalued,

equity financing will be cheaper than cash financing, vice versa

Control If the bidding company has a large shareholder, it may want to finance the merger with cash instead to avoid diluting the key shareholder’s controlling and voting rights

Taxes A cash bid would result in target shareholders being subject to taxes

Takeover defense: Takeover defenses give managers greater bargaining power in extracting value from bidders. Protected from takeovers, managers can focus on long-term objectives instead of short-term positions. Long-term contracts can also be preserved. Stock prices generally fall following an amendment of a firm’s anti-takeover defenses

Method Description Greenmail Repurchasing the

company (indireclty bribing the acquirer to go away), with (empirically) a 16% premium

Poison pill Rights issue (ie. for existing shareholders) at steeply discounted prices

to reduce acquirer’s ownership. Share price falls 2% on average when poison pill is announced

Employee stock ownership plan (ESOP)

Employees are given the right to vote (in stock) for or against the merger

Supermajority provision Merger must be approved by a supermajority instead of the conventional 50%, making it more difficult for acquirer to pass its proposition

White knight Approaching friendly potential acquirers to compete in the bidding contest

Merger gains: (1) Acquirers’ stock price, on average, falls post merger. This may be due to excess management confidence which is downplayed by investors or a signaling mechanism indicating that the market is stagnant, (2) Target earn high percentage returns due to often large differences in market cap, (3) On average, combined entities are worth more than the sum of their individual entities (19) Initial public offerings (IPO) IPOs

Benefits Costs Provides market access and funds for investment Diversify investors: With market access, initial investors can diversify their holdings and reduce idiosyncratic risks Exit strategy for VCs and angel investors

Monetary costs: Investment banking fees, regulatory compliance, IPO underpricing, accounting for up to 20% of total costs Disclosure, loss of control and freedom: Managers are now accountable to public shareholders, with full disclosure and regulatory requirements

Role of investment banks: (1) Assist the firm in registering and meeting SEC requirements, (2) Provide credibility in backing the IPO, (3) Value and price the issue, (4) Absorb risk by underwritting the issue (buying the issue and selling them to the public) with appropriate price stabilizing mechanisms Uncertainty in issuance

Negative Positive Insufficient demand Excess demand: The


Price risk Criminal and civil liability (negligence and misrepresentation) Reputational risk

investment bank can (1) choose to exercise an over allotment option by issuing more shares at the offer price, (2) select investors through lottery, (3) fixing the percentage of investor types

Rights issue: Owning every A number of existing shares gives you the right to purchase B number of new shares at the issue price (often lower than the current price)

Value of right =𝑃0×𝐴 + 𝑃Issue×𝐵

𝐴 + 𝐵!New

− 𝑃0

Value of right =𝑃0 − 𝑃Issue 𝑁𝑁 + 1

, where 𝑁 =𝐴𝐵

Buying into a rights issue (as above, owning every A number of existing shares gives you the right to purchase B number of new shares at the issue price)

𝐴 + 𝐵 ×𝑃NewValue of new holdings

≥ 𝐴×𝑃!Value of old holdings

− 𝑃New − 𝑃!Value of right

⟹ Buy into rights issue if 𝑃New ≥𝐴 + 1 ×𝑃!𝐴 + 𝐵 + 1

⟹ 𝑃!×𝐴Old share capital

+ 𝑃Issue×𝐵Rights issue proceeds

= 𝑃New× 𝐴 + 𝐵New share capital

New shares issue: Issuing new shares to raise a target level of capital, assuming no issuance costs. As a result, a fraction of old equity owners’ holdings (assume previously 100%) are given up

𝑃New×∆NOSH = Target… 1 NOSH! + ∆NOSH ×𝑃New = 𝑃!×NOSH! + Target… 2

⟹ Solve for 𝑃New and ∆NOSH

NOSH!

NOSH! + ∆NOSHPost IPO old holdings

<NOSH!NOSH!

Pre IPO old holdings

= 100%

𝑃New×NOSH!

Old shareholders' wealth post IPO

< 𝑃!×NOSH!Pre IPO

if 𝑃New < 𝑃!

IPO returns: In the short run, IPO returns are risky and procyclical (during booms, IPO returns are high due to greater incidence of underpricing). IPO returns are lower with certainty (e.g. larger firms are underpriced less). Short term returns average 16%. In the long run, IPO returns are low – empirically lower than a portfolio of comparable stocks (34% vs 62%) Underpriced IPOs

Factor Description

Underwriter price supports

Underwritters take on risk by buying the issue before selling them in capital markets. As such, they would buy at a price lower than the offer price

Benefits underwritters’ clients

Underwritters’ clients can earn higher profits by buying the stock at issue and selling it soonafter for a quick profit

Increase firm’s ability to raise further capital

Low offering price raises the price when the stock is traded, enhancing the firm’s ability to raise further capital But old shareholders lose, because they have sold shares at a lower price than what they are worth

Winners’ curse Investors will only buy the stock if they believe that they will not be paying more than what it is worth

Asymmetric information Future profitability is not made known to investors. The IPO is underpriced to attract investors to buy into the issue

(20) Risk management and hedging Rationale for hedging: In a MM world, there is no place for hedging. This is because investors can hedge these risks themselves

Factor Explanation Cost of financial distress Hedging lowers the

expected cost of financial distress by lowering the probability of financial distress

Financial constraints Hedging lowers the risk of being financially constrained especially if investment opportunities and cash flows are countercyclical

Managerial incentives Hedging can improve managerial incentives in the presence of moral hazard


The case for insurance firms: Insurance companies mainly provide insurance against idiosyncratic risks. The remaining risk is passed on to shareholders through the securities market

For Against Expertise in estimating and pricing probabilities (assuming competitive insurance industry) Ability to pool and diversify risks by selling a spectrum of different policies

Administrative costs are incurred Moral hazard and adverse selection Risk pool may have correlated risks Insurance companies may not be able to adequately deal with large-scale, rare losses. Companies (e.g. BP) may resort to the stock market to insure against such losses (e.g. stock devaluation)

Documents

FM212 Highlights - LSE 2013