JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Midterm Review F uture value of $100 = $100 (1 + r) t Suppose that you will receive a cash flow of C t dollars at the end of year t The present value of this future payment is P resent value = P V = C t (1 + r) t The expression 1 /(1+r) t is called the discount factor It measures the present value of one dollar received in year t Net present value equals present value minus the required investment Return = profit investment Two equivalent decision rules for capital investment: Net present value rule Accept investments that have positive net present values Rate of return rule Accept investments that offer rates of return in excess of their opportunity costs of capital Suppose that you wish to value a stream of cash flows extending over a number of years The total present value is: P V = C 1 1 + r + C 2 (1 + r) + C 3 2 (1 + r) + + C T T 3 (1 + r) = C t T (1 + r) t This is called the discounted cash flow (or DCF) formula To find the net present value (NPV) we add the (usually negative) initial cash flow: t=1 NP V = C 0 + P V = C 0 + T t=1 C t (1 + r) t A perpetuity is an investment that delivers a steady stream of cash flows forever The annual rate of return on a perpetuity is equal to the promised annual payment divided by the present value: cash flow Return = present value, Consequently, r = C P V P resent value of perpetuity = C r 1
An annuity is an asset that pays a fixed sum each year for a specified number of years ( ) 1 P resent value of annuity = cash flow t year annuity factor = C r 1 r(1 + r) t F uture value of annuity = present value of annuity (1 + r) t P resent value of growing perpetuity = C 1 r g P resent value of growing annuity = C 1 r g C 1 (1 + g)t r g (1 + r) t Effective Annual Interest Rate interest rate that is annualized using compound interest Annual Percentage Rate interest rate that is annualized using simple interest If you own a bond, you are entitled to a fixed set of cash payoffs Every year until the bond matures, you collect regular interest payments At maturity, when you get the final interest payment, you also get back the face value of the bond, which is called the bond s principal P V (bond) = P V (annuity of coupon payments) + P V (final payment of principal) Bond prices and interest rates must move in opposite directions The yield to maturity, our measure of the interest rate on a bond, is defined as the discount rate that explains the bond price When bond prices fall, interest rates (that is, yields to maturity) must rise When interest rates rise, bond prices must fall A strip is a special type of Treasury bond that repays principal at maturity, but makes no coupon payments along the way Strips are also called zero coupon bonds Duration is the weighted average of the times when the bond s cash payments are received The times are the future years 1, 2, 3, etc, extending to the final maturity date, which we call T The weight for each year is the present value of the cash flow received at that time divided by the total present value of the bond Duration = 1 P V (C 1) P V + 2 P V (C 2) P V + 3 P V (C 3) P V + + T P V (C T ) P V Investors and financial managers track duration because it measures how bond prices change when interest rates change For this purpose it s best to use modified duration or volatility, which is just duration divided by one plus the yield to maturity: Modified duration = volatility (%) = duration 1 + yield Modified duration measures the percentage change in bond price for a 1 percentage point change in yield The relationship between short and long term interest rates is called the term structure of interest rates Spot Rate the actual interest rate today (t = 0) 2
Forward Rate the interest rate, fixed today, on a loan made in the future at a fixed time Future Rate the spot rate that is expected in the future Yield To Maturity (YTM) the internal rate of return on an interest bearing instrument The law of one price states that the same commodity must sell at the same price in a well functioning market Therefore, all safe cash payments delivered on the same date must be discounted at the same spot rate Suppose that you held a portfolio of one year US Treasuries Here are three possible reasons why you might decide to hold on to them, despite their low rate of return: 1 You believe that short term interest rates will be higher in the future 2 You worry about the greater exposure of long term bonds to changes in interest rates 3 You worry about the risk of higher future inflation The expectations theory of the term structure states that 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 if that is the case would investors be prepared to hold both short and long maturity bonds The expectations theory implies that the only reason for an upward sloping term structure is that investors expect short term interest rates to rise; the only reason for a declining term structure is that investors expect short term rates to fall If short term interest rates are significantly lower than long-term rates, it is tempting to borrow short term rather than long term The expectations theory implies that such naive strategies won t work If short term rates are lower than long term rates, then investors must be expecting interest rates to rise When the term structure is upward sloping, you are likely to make money by borrowing short only if investors are overestimating future increases in interest rates The Consumer Price Index (CPI) measures the number of dollars that it takes to pay for a typical family s purchases The change in the CPI from one year to the next measures the rate of inflation The formula for converting nominal cash flows in a future period t to real cash flows today is nominal cash flow at date t Real cash flow at date t = (1 + inflation rate) t The formula for calculating the real rate of return is: 1 + r real = 1 + r nominal 1 + inflation rate Fisher s theory: A change in the expected inflation rate causes the same proportionate change in the nominal interest rate; it has no effect on the required real interest rate Primary Market market for the sale of new securities by corporations Secondary Market market in which previously issued securities are traded among investors 3
Common Stock ownership shares in a publicly held corporation Exchange Traded Funds (ETFs) portfolios of stocks that can be bought or sold in a single trade The cash payoff to owners of common stocks comes in two forms: (1) cash dividends and (2) capital gains or losses Suppose that the current price of a share is P 0, that the expected price at the end of a year is P 1, and that the expected dividend per share is DIV 1 The rate of return that investors expect from this share over the next year is defined as the expected dividend per share DIV 1 plus the expected price appreciation per share P 1 P 0, all divided by the price at the start of the year P 0 : A general stock price formula: Expected return = r = DIV 1 + P 1 P 0 P 0 P 0 = H t=1 DIV t (1 + r) t + P H (1 + r) H Dividend yield = DIV 1 P 0 The fair rate of return on equity for a public utility ought to be the cost of equity, that is, the rate offered by securities that have the same risk as the utility s common stock The payout ratio is the ratio of dividends to earnings per share (EPS) P lowback ratio = 1 payout ratio = 1 DIV EP S Return on equity = ROE = EP S book equity per share Dividend growth rate = g = plowback ratio ROE Expected return = dividend yield earnings price ratio We can think of stock price as the capitalized value of average earnings under a no growth policy, plus PVGO, the net present value of growth opportunities: P 0 = EP S 1 r The earnings price ratio, therefore, equals EP S P 0 = r + P V GO ( 1 P V GO ) P 0 It will underestimate r if PVGO is positive and overestimate it if PVGO is negative The latter case is less likely, since firms are rarely forced to take projects with negative net present values Free cash flow is the amount of cash that a firm can pay out to investors after paying for all investments necessary for growth 4
The value of a business is usually computed as the discounted value of free cash flows out to a valuation horizon (H), plus the forecasted value of the business at the horizon, also discounted back to present value That is, P V = F CF 1 1 + r + F CF 2 (1 + r) 2 + + F CF H (1 + r) H + P V H (1 + r) H book income Book rate of return = book assets A project s payback period is found by counting the number of years it takes before the cumulative cash flow equals the initial investment The payback rule states that a project should be accepted if its payback period is less than some specified cutoff period The discount rate that makes NP V = 0 is called the internal rate of return (IRR) The internal rate of return is a profitability measure that depends solely on the amount and timing of the project cash flows The opportunity cost of capital is a standard of profitability that we use to calculate how much the project is worth The opportunity cost of capital is established in capital markets It is the expected rate of return offered by other assets with the same risk as the project being evaluated The internal rate of return rule is to accept an investment project if the opportunity cost of capital is less than the internal rate of return The rule will give the same answer as the net present value rule whenever the NPV of a project is a smoothly declining function of the discount rate There can be as many internal rates of return for a project as there are changes in the sign of the cash flows P rofitability index = net present value investment Capital Rationing limit set on the amount of funds available for investment Soft Rationing limits on available funds imposed by management Hard Rationing limits on available funds imposed by the unavailability of funds in the capital market Net present value of investment if undertaken at date t = net future value at date t (1 + r) t Equivalent Annual Cash Flow the cash flow per period with the same present value as the actual cash flow as the project Equivalent annual annuity = present value of cash flows annuity f actor Sensitivity Analysis analysis of the effects of changes in sales, costs, etc on a project Scenario Analysis project analysis given a particular combination of assumptions Simulation Analysis estimation of the probabilities of different possible outcomes Break Even Analysis analysis of the level of sales (or other variable) at which the company breaks even 5
A business with high fixed costs is said to have high operating leverage Operating leverage is usually defined in terms of accounting profits rather than cash flows and is measured by the percentage change in profits for each 1% change in sales Thus degree of operating leverage (DOL) is percentage change in profits DOL = percentage change in sales The following simple formula shows how DOL is related to the business s fixed costs (including depreciation) as a proportion of pretax profits: DOL = 1 + fixed costs prof its Expected P ortfolio Return = x 1 r 1 + x 2 r 2 P ortfolio V ariance = x 1 σ 2 1 + x 2 σ 2 2 + 2x 1 x 2 ρσ 1 σ 2 Diversification strategy designed to reduce risk by spreading the portfolio across many investments Unique Risk risk factors affecting only that firm Also called diversifiable risk Market Risk economy wide sources of risk that affect the overall stock market Also called systematic risk Market Portfolio portfolio of all assets in the economy Beta sensitivity of a stock s return to the return on the market portfolio Sharpe ratio = risk premium standard deviation = r r f σ The capital asset pricing model, the security market line: Expected risk premium on stock = beta expected risk premium on market, r r f = β(r m r f ) Investors like high expected return and low standard deviation Common stock portfolios that offer the highest expected return for a given standard deviation are known as efficient portfolios If the investor can lend or borrow at the risk free rate of interest, one efficient portfolio is better than all the others: the portfolio that offers the highest ratio of risk premium to standard deviation A risk averse investor will put part of his money in this efficient portfolio and part in the risk free asset A risk tolerant investor may put all her money in this portfolio or she may borrow and put in even more The composition of this best efficient portfolio depends on the investor s assessments of expected returns, standard deviations, and correlations But suppose everybody has the same information and the same assessments If there is no superior information, each investor should hold the same portfolio as everybody else; in other words, everyone should hold the market portfolio Arbitrage pricing theory: Return = a + b 1 r factor1 + b 2 r factor2 + b 3 r factor3 + + noise, 6
Expected Risk P remium = r r f = b 1 (r factor1 r f ) + b 2 (r factor2 r f ) + F irm value = P V (AB) = P V (A) + P V (B) = sum of separate asset values The values of debt and equity add up to overall firm value (D + E = V ) and firm value V equals asset value The cost of debt is less than the company cost of capital, because debt is safer than the assets The cost of equity is greater than the company cost of capital, because the equity of a firm that borrows is riskier than the assets Equity is not a direct claim on the firm s free cash flow It is a residual claim that stands behind debt Company cost of capital = r DD V + r EE V Interest is a tax deductible expense for corporations, so the after-tax cost of debt is (1 T c )r D, where T c is the marginal corporate tax rate After-tax W ACC = (1 T c ) r DD V + r EE V D Asset beta = β A = β D V + β E E V Cash flow = revenue fixed cost variable cost P V (revenue) = P V (fixed cost) + P V (variable cost) + P V (asset) P V (fixed cost) β revenue = β fixed cost P V (revenue) +β P V (variable cost) P V (asset) variable cost +β assets P V (revenue) P V (revenue) P V = C t (1 + r) t = CEQ t (1 + r f ) t, where CEQ t denotes certainty equivalent at period t There are two ways to calculate PV: Estimate the expected cash flows and discount at a rate that reflects the risk of those flows Estimate what sure fire cash flows would have the same values as the risky cash flows Then discount these certainty equivalent cash flows at the risk free interest rate A call option gives its owner the right to buy stock at a specified exercise or strike price on or before a specified maturity date If the option can be exercised only at maturity, it is conventionally known as a European call; in other cases, the option can be exercised on or at any time before maturity, and it is then known as an American call Whereas a call option gives you the right to buy a share for a specified exercise price, a put gives you the right to sell the share V alue of call option at expiration = max{market price of the share exercise price, 0} V alue of put option at expiration = max{exercise price market price of the share, 0} A fundamental relationship for European options: V alue of call + present value of exercise price = value of put + share price 7
This relationship holds because the payoff of Buy call, invest present value of exercise price in safe asset is identical to the payoff from Buy put, buy share This basic relationship among share price, call and put values, and the present value of the exercise price is called put call parity Any set of contingent payoffs that is, payoffs that depend on the value of some other asset can be constructed with a mixture of simple options on that asset When the stock is worthless, the option is worthless The value of an option increases as stock price increases, if the exercise price is held constant When the stock price becomes large, the option price approaches the stock price less the present value of the exercise price The value of an option increases with both the rate of interest and the time to maturity The option price always exceeds its minimum value (except when stock price is zero) The probability of large stock price changes during the remaining life of an option depends on two things: (1) the variance (ie, volatility) of the stock price per period and (2) the number of periods until the option expires If there are t remaining periods, and the variance per period is σ 2, the value of the option should depend on cumulative variability σ 2 t Other things equal, you would like to hold an option on a volatile stock (high σ 2 ) Given volatility, you would like to hold an option with a long life ahead of it (large t) Thus the value of an option increases with both the volatility of the share price and the time to maturity Delta of call option = Delta of put option = spread of possible option prices spread of possible share prices spread of possible option prices spread of possible stock prices The delta of a put option is always equal to the delta of a call option with the same exercise price minus one The delta of a put option is always negative; that is, you need to sell delta shares of stock to replicate the put The general formula for calculating the risk neutral probability of a rise in value is: p = Two ways to calculate the value of an option: interest rate downside change upside change downside change 1 Find the combination of stock and loan that replicates an investment in the option Since the two strategies give identical payoffs in the future, they must sell for the same price today 8
2 Pretend that investors do not care about risk, so that the expected return on the stock is equal to the interest rate Calculate the expected future value of the option in this hypothetical risk neutral world and discount it at the risk free interest rate Black and Scholes formula: V alue of call option = delta share price bank loan = N(d 1 ) P N(d 2 ) P V (EX), where d 1 = ( ) P log P V (EX) σ t d 2 = d 1 σ t, + σ t 2, N(d) = cumulative normal probability density function, EX = exercise price of option; P V (EX) is calculated by discounting at the risk free interest rate r f, t = number of periods to exercise date, P = price of stock now, σ = standard deviation per period of (continuously compounded) rate of return on stock 9