Midterm Review. P resent value = P V =

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1 JEM034 Corporate Finance Winter Semester 2018/2019 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 = = 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: = C 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 N = C 0 + = 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 resent value of perpetuity = C r 1

2 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 (bond) = (annuity of coupon payments) + (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 (C 1) + 2 (C 2) + 3 (C 3) + + T (C T ) 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

3 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 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 nominal 1 + r real = 1 + inflation rate Primary Market market for the sale of new securities by corporations Secondary Market market in which previously issued securities are traded among investors 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, that the expected 3

4 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, all divided by the price at the start of the year : A general stock price formula: Expected return = r = DIV 1 + P 1 = H t=1 DIV t (1 + r) t + P H (1 + r) H Dividend yield = DIV 1 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: = EP S 1 r The earnings price ratio, therefore, equals EP S = r + GO ( 1 GO ) 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 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, = F CF r + F CF 2 (1 + r) F CF H (1 + r) H + H (1 + r) H 4

5 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 N = 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 of 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 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 5

6 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 Profits that more than cover the cost of capital are known as economic rents In a long run competitive equilibrium, no competitor can expand and earn more than the cost of capital on the investment Economic rents are earned when an industry has not settled down to equilibrium or when your firm has something valuable that your competitors don t have Under straight line depreciation, annual depreciation equals a constant proportion of the initial investment less salvage value If T denotes the depreciable life, then the straight line depreciation in year t is Depreciation in year t = 1 T depreciable amount 6

Midterm Review. P resent value = P V =

Midterm Review. P resent value = P V = 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