SECTION HANDOUT #1 : Review of Topics

Transcription

1 SETION HANDOUT # : Review of Topics MBA 0 October, 008 This handout contains some of the topics we have covered so far. You are not required to read it, but you may find some parts of it helpful when you study for the midterm. For each chapter, I also list some of the homework problems that are most relevant. I. Valuing a Project and The NPV Rule From a financial perspective, a project is a series of cash flows. Suppose that Google would like to create and sell a new type of phone. To do so, it needs to invest in research labs and new plants in the hope of selling these phones in the future. For the financiers, this means that Google will incur some negative cash flows over the first few years and (hopefully! some positive cash flows afterwards. Should Google undertake the project? Well, this depends... first, on the project s cash flows: 0,, If they re risky, then our answer should depend on our best estimate of these cash flows. For statistical reasons, we often use the expected value as our best guess. second, on how valuable these cash flows are today. A dollar received tomorrow is worth less than a dollar received today. To see why, invest $ in a risk-free bank account. If the one-year interest rate is 5%, you ll have $.05 next year! Furthermore, since most of us do not like risk, if the future cash flows are risky, they should be worth even less today (i.e. they should be discounted at a higher market rate. Prepared by Sara Holland, 008. These notes reflect contributions that Vijay Balakrishnan and Sebastien Betermier made in previous years.

2 The market rate, such as our risk-free rate of 5%, is referred to as the discount rate, the hurdle rate, or the opportunity cost of capital. It s an opportunity cost because it s the rate of return of an alternative investment opportunity that has the same level of risk as our project. In the risk-free case, we can choose to invest in a risk-free savings account instead of our project. In the risky cases, we can choose to invest in risky stocks that are as risky as our project. If we assume that capital markets are working well and that these rates are fairly priced, we can then pick the rate that matches the risk level of our project and use it to discount all of the project s cash flows. The sum of the discounted cash flows is called the Net Present Value (NPV of our project. If we only include the positive cash flows, we obtain the Present Value (PV of our project. If the NPV is positive, then our project is a profitable opportunity! It is a better project that equally risky alternative projects. onversely, a negative-npv project is overpriced and not worth undertaking. Example (key++:.q,.q4,.p,.p (Questions are coded by chapter/type/number. For example, Quiz question from hapter is.q II. ompound Interest and PV alculations Finding the PV of a project requires recognizing that the value of a dollar today is not worth the same as a dollar tomorrow. In the same way, a dollar tomorrow is not worth the same as a dollar the day after tomorrow. To see this, think about investing $ this year at an annual interest rate of 0% (which is compounded annually. Next year you will get back your initial amount ($ plus an interest payment (0% * $ 0 cents, or $.0 in total. If you invest that $.0 again for one year, then you will have ($.0 (.0$. at year. Here we re assuming no taxes, no transaction costs, everyone has the same access to information, and perfectly competitive markets. These are very strong assumptions but we need to start somewhere! For the economists, this means that these are the rates that clear the financial markets in equilibrium.

3 The fact that the interest payment of 0 cents is reinvested at year is referred to as compound interest. With compounding, each interest payment is reinvested to earn more interest in subsequent periods. The interest rate may compound at different intervals such as annual, semiannual, and monthly periods. So, if we re given a monthly rate r monthly, how we do get the equivalent yearly rate r yearly? Simply by compounding the monthly return times! (+r monthly (+r yearly Let s apply this concept to the valuation of cash flows. Suppose that we ll receive a cash flow of $ in 5 years and the yearly interest rate is 5%. The equivalent 5-year return is the yearly return compounded five times: (+r yearly 5 (+r 5years What s the present value of $0 in 5 years? To answer this question, think of how much you would need to invest today in order to receive $0 in 5 years. You would need to invest $0 / ( + r 5years! Then, the present value is: 0 PV + r 0 ( + 5 years r yearly The present value of $ received in 5 years is also called the 5-year discount factor. Most projects will generate cash flows that occur over several periods. Recall Google s decision of whether or not to take on the phone project. Google probably expects to sell their new car for several years into the future. Hence, in order to decide whether or not to undertake a multi-period project, we need to value the entire stream of cash flows. How? By discounting each future cash flow at the correct rate, and then by adding all of the PVs! We just need to modify our present value formula: PV + r + ( + r + ( + r +...

4 (Note that for now we are assuming that the interest rate in each period is the same. When we have many future cash flows, this calculation can be tedious. Fortunately, when the future cash flows are constant or grow at a constant rate, we can use shortcut formulas for certain assets: perpetuities and annuities. PV of Perpetuity r PV of Annuity r ( + r PV of Growing Perpetuity r g ( + g PV of Growing Annuity r g ( + r These formulas have been derived assuming the first cash flow occurs at the end of the first period. For example, if you purchase an annuity with annual payments, your first cash flow will be paid in one year. T T Example:.Q,.Q,.Q4,.Q5,.Q6,.Q7,.P4,.P III. Interest Rates EAR vs APR In the last section we looked at how we can convert interest rates from long periods to short periods and vice-versa. For example, if we re using a monthly rate and we would like to compute the equivalent yearly rate, we need to compound the monthly return twelve times: (+r monthly (+r yearly This yearly interest rate is also called the Effective Annual Rate (EAR. Even though the EAR is the true rate (per year, it s rarely the rate you ll see quoted on car 4

5 loans or mortgage advertisements. Instead, you ll see an equivalent rate, the Annual Percentage Rate (APR, along with the frequency of compounding. Why? Well, even though the APR is not the true rate of interest, it makes it easier for customers to compare different deals. In fact, businesses are required by truth-in-lending laws to include the APR. Once we re given an APR with a frequency of compounding, we need to divide the APR by the frequency to get the true rate. That s just how the APR has been defined. For instance, a car loan might be quoted at 6% APR, compounded monthly. Then, the true monthly rate r monthly 6% / 0.5%. If, instead, you want to true yearly rate (the EAR, compound the monthly return times: EAR r yearly ( %. To go directly from the APR to the EAR, just merge the two previous steps: APR EAR + k where k is the frequency of compounding. Use the EAR to discount annual cash flows. k Example:.P6,.P8 IV. Interest Rates Nominal vs. Real It s important to distinguish between nominal cash flows and real cash flows. A nominal cash flow is the actual number of dollars you ll pay or receive, and a real cash flow is adjusted for the rate of inflation. Remember that if there is inflation next year, a dollar next year isn t worth as much as a dollar today, independently of the time value of money. Note that if the compounding period is per year, APREAR. 5

6 If the inflation rate is constant over time, then the real cash flow in year t is the nominal cash flow in year t discounted by the inflation rate: ( Real ash Flow t ( Nominal ash Flow ( + t Inflation Rate t Note that when t 0, both cash flows are equal. This is because in Finance, date 0 (i.e., today is our reference point. It s where we re standing on the time line. When we are valuing projects, we only care about current and future cash flows. The formula also applies to the interest rates. Suppose I invest $ today in a one-year risk-free account. Next year, I ll have a nominal cash flow of $*(+ Nominal Rate.. This is equivalent to having a real cash flow of $*(+ Real Rate. According to the formula above, we get: ( + Real Rate ( + Nominal Rate ( + Inflation Rate When we value a project, should we use nominal cash flows or real cash flows? It does not matter, as long as you use the appropriate discount rate! The rule is: discount real cash flows using the real interest rate, and discount nominal cash flows using the nominal interest rate. Example:.P V. Horizon - varying interest rates So far we have assumed the same value of r to discount each cash flow. In reality, interest rates are different for different horizons. For example, a -year itibank ertificate of Deposit (D account currently yields.75% per year (that s an APR, whereas a -year D, in which your money is locked up for two years, yields.00% per year (APR. Why is it the case that these rates are different? Well, there is a huge amount of academic research on this topic and no one has come up with an overwhelmingly 6

7 convincing answer. One theory is that it depends on our expectations about changes in future interest rates. If short-term interest rates are expected to increase, then the longterm rate will be higher the short-term rate. In the case where rates of different horizons aren t the same, each cash flow needs to be discounted at the appropriate discount rate: PV + r ( + r ( + r where the EARs r, r, r are called the spot rates. They are today s rates for investments of different horizons. The first period s cash flow is discounted at one-period spot rate and the second period s cash flow at the two-period spot rate. The set of spot rates of different horizons is called the yield curve. We ll see shortly that we can find these spot rate by looking at bond prices. VI. Bonds Now let s apply our knowledge of cash flow valuation to pricing bonds. The neat thing about bonds is that we know exactly the cash flows we ll receive. 4 Usually, bonds provide an interest payment to the holder for a specified number of periods and the face value at maturity. The face value is also called the principal or the par value. Bond prices are usually quoted as a percentage of the face value. This means that often times you can assume the face value of a bond is $00 or $000. The interest payment is called the coupon payment and it s usually quoted as a percentage of the face value. For example, the present value of a bond with a Face Value of F, a coupon payment of, and a maturity of three years, is given as: 4 That s for government bonds, like US Treasuries. Since US government has never defaulted on these bonds, they are considered as risk-free. With corporate bonds, there s a greater chance that the firm will default on its debt obligations and that you won t receive the cash flows you were promised. 7

8 PV + F r r r ( + ( + In a fair financial market, this Present Value should be the price of the bond. (i.e. the NPV of our investment is zero. The bond is said to be priced at par when its price equals the face value, at a discount when its price is lower than the face value, and at a premium when its price is higher than the face value. Bond prices and interest rates are inversely related. Oftentimes we are interested in the Yield to Maturity (YTM of the bond. The YTM is defined as the single rate y that gives the same PV as above: PV F + y ( + y ( + y We can think of the YTM as a kind of a weighted average of all the spot rates. It s informative in that it tells us the rate of return on our bond, assuming that ( we ll hold it all the way to maturity and ( we ll reinvest the coupons at the same yield rate. These assumptions are very strong and so we need to be extremely careful when we think of the yield as a measure of return. The YTM typically needs to be computed numerically, but in some cases, it s possible to derive it: Zero-oupon bonds. These bonds are also referred to as discount bonds because when there are no coupon payments, the price is the PV of the face value. If interest rates are positive (as they should be!, then the price must be lower than the face value: PV F ( + r t For zero-coupon bonds there is only one spot rate we need to use. This spot rate will also be the YTM. The Spot rates are all the same. Then the YTM is equal to the spot rates as well. t 8

9 The coupon bond is priced at par. Then the YTM equals the coupon rate. You can verify this fact by doing some algebra on the PV formula. Similarly, if the bond is priced at a discount (premium, the YTM should be higher (lower than the coupon rate. Example: 4.Q, 4.Q8, 4.P5, 4.P7, 4.P, 4.P VII. Forward Rates We defined spot rates as today s interest rates for investments of various horizons. For example, the two-year spot rate, denoted as r, is today s rate (per year for an investment in a two-year risk-free account. A forward rate is a different type of interest rate. It is the rate at which we would agree today for a loan that will take place in the future. Its value, however, should depend on the spot rates. To see why, let s go back to the example given in lecture. Suppose you take a one-year loan of $, at spot rate r, and you invest that $ in a two-year risk-free D account, which will yield spot rate r per year (those are true yearly rates, i.e. EARs. What are the cash flows associated with this investment? Today there are no cash flows, since you invested the $ that you just obtained from the loan. Next year, you ll owe ( + r dollars. Two years from now, you ll receive ( + r dollars. This looks exactly like a one-year loan we re agreeing on today and that begins next year! By definition, the rate of return on this loan should be the one-year forward rate at year, f, : or ( + r ( + f f,, ( + r ( + r ( + r Example: 4.Q7, 4.P0, 4.P 9

10 VIII. Stocks A stock is a security representing ownership of a corporation. ompanies raising funds issue shares of stock in the primary market. Once they have been issued, they are traded amongst investors in secondary markets like the New York Stock Exchange. If you wanted to buy shares in a firm, how much should you pay? When valuing bonds, we saw that we could use our now familiar Present Value formula to discount their cash flows. It s the same for stocks. Instead of receiving coupons and the face value, investors who own stock expect to get cash flows in the form of a stream of dividend payments. Then, in a fair financial market the price of the stock should be the PV of its expected dividend payments, 5 P 0 DIV DIV DIV r ( + r ( + r L Note that here our discount rate will generally be different from the riskless rate because generally we consider stocks risky assets. It will be the expected rate of return of an alternative investment that s equally risky (and fairly priced. This opportunity cost of capital is also called the market capitalization rate. Notice that there is no future price in our PV formula. Why? If we only expect to hold on to our stock for one year, why should the expected dividends in years,, 4 be in the PV formula? Well, if we sell next period, we ll receive DIV and P, P 0 DIV + P + r But the fair value of P can be found by discounting the stream of DIV and P, and the fair value of P can be found by discounting the stream of DIV and P, and so on! By rearranging terms, we can express our expected return as, r DIV + P P0 P 0 5 There is also some value for having some control over the firm as a shareholder, but we are not taking it into account in our PV formula. 0

11 IX. Dividend Growth Model To value a stock, we need to estimate all the future dividend payments. Sometimes it can be useful to make assumptions about the behavior of the future dividends. In doing so we ll lose precision, but we ll save some time. One of the simplest models is the dividend growth model: we assume that dividends will grow at a constant rate forever. This sounds like a growing perpetuity, so we can apply our formula for finding the present value of a growing perpetuity to price the stock, P 0 DIV r g where g is the expected growth rate of the dividends. If we have information on the dividend yield (DIV /P 0 and on the expected growth rate of dividends, we can estimate the market capitalization rate as, DIV r + g P 0 Example: 5.Q, 5.Q4, 5.Q5, 5.Q6, 5.Q6, 5.Q7 X. Estimating Long Run Growth Although we may use analysts estimates to determine the growth rate of dividends, we can also use the plowback ratio and the return on equity. The plowback ratio is the portion of the firm s earnings that are plowed back into the business instead of being paid out as a dividend. Hence, if the payout ratio is DIV/EPS, the fraction of reinvested earnings is given as, Plowback R atio Payout Ratio DIV EPS

12 We can then estimate the dividend growth rate as: g Plowback Ratio * ROE where ROE is the firm s Return on Equity: EPS ROE Book Value Per Share The ROE is a measure of the firm's profitability, which reveals how much profit a company generates (Earnings per share with the money shareholders have invested (Book Value per share. Whether the firm should retain some earnings in order to invest in new projects essentially depends on its ROE. If the firm s ROE is greater than r, the market capitalization rate or opportunity cost of capital, there exist positive NPV investment opportunities that can increase the value of the firm. Then the firm should plow back some of its earnings to finance these valuable projects. Shareholders will benefit from this decision because the PV of the higher future dividends will exceed the reduction of today s dividends. Example: 5.P8, 5.P9, 5.P6