Basic Financial Management: Investment PDF Analysis

Practice to basic ﬁnancial management In this pdf I will get into detail on the di8erent chapters. I will get into great detail and I will try to understand it. The chapters in question are 1-9 Subject Test Book Introduction Chapter 1 Part 1: Discounting and valuing bonds and stocks Discounting 1-2 Chapter 2 Valuing bonds and stocks Chapters 3,4 Valuing bonds and stocks 3 Chapters 3,4 Learning assistant session 1 Part 2: Portfolio theory and Cost of Capital Risk and return over time, Portfolio theory Chapter 7 Portfolio theory 4-5 Chapter 7 Cost of capital and Project risk Chapter 8 Cost of capital and Project risk Chapter 8 Cost of capital and Project risk 6 Chapter 8 Learning assistant session 2 Part 3: Project investment analysis Cash ﬂows and WACC 7 Chapter 6, 9 Cash ﬂows and WACC 8 Chapter 6, 9 Proﬁtability of investment projects 9 Chapters 5, 6.4 Investment analysis (part 1) Investment analysis (part 2) Summary and exam preparation 1 2 Chapter 2- how to calculate present values 2-1 how to calculate future and present values Calcula6ng future values - Money can be invested to earn interest. - A dollar today is worth mor than a dollar tomorrow - So if you invest 100 in a bank that gives you a return of 7%. You will earn 7*100= 107 and your investment grows with such - By the second year you will earn also last years 0.07. this is what we call compound interest. Your wealth will grow at a compound rate and the interest you earn is called compound interest - The higher the interest rate is , the faster your savings will grow Calcula6ng present value - We suppose that you will recive a cash ﬂow of ct dollars at the end of year t. the present value of this future payment is o Pv= ct/ 1+r^t - The r in this formula is called the discount rate and the present value is the discounted value of the cash ﬂow cr - We also can write this as 1/1+r^t. this is called the discount factor and measurs the present value of one dollar recived at the end of year t. - The longer you wait for your money , the lower the present value is ( so wait long, you will have a lower present value) Valuing an investment opportunity - It is a given fact that we don’t always know which investment will be worth it in the long run. - You can invest in the project or pay out to th shareholders who can invest on their own o For example you earn a 7% proﬁt by inves6ng for one year in safe assets. the opportunity cost of capital for the oﬃce investment is 7. This is the rate of 3 return that the companys shareholders could get any inves6ng on their own in risk free government debt. - How much the investment is worth and how much it will to the shareholders welth is to look at how much the project produces as cash ﬂow at the end of one year. - In order to ﬁnd the present value of this we discount that cash ﬂow by the opprun;y cost of capital which is o Pv= c1/1+r - The present value of something can also be seen as the market price What Is net present value - Well the net present value is equal to the present value minus the required investment - An example of this is 747665 which is the present value of an bulding- the ini6al investment of 700 000 which will give us an npv of 46 664. - By looking at the net present value we can determine things such as with the example shown above. It makes a net contribu;ons to shareholder value and increases their wealth - The formula of this is o C0+c1/1+r - C0 is the cash ﬂow at the 6me which is today and is nega6ve. So an investment and therefore a cash ou^low. Risk and present value - Although we make diﬀerent hopes for what we want our business to achieve it is something that is quite hard to mange. We only assume, but nothing is concrete. You can never be be certain about things such as the resale value. - If the cash ﬂows are uncertain , your calcula6on of the npv is wrong. - We therefore a saying which is “a safe dollar is worth more than a risky dollar”. Most investors dislik risky ventures and wont invest in them unless they see a future with a high return. Because it is important to rember that a riskeier venture will lead to the decrease in the fact that they will be able to pay and the higher the return that they will demand. 4 - The concept of present value and the opportunity cost of capta6al s6ll hold for risky investments. The reason fro this is because you s6ll need to discount the payoﬀ by the opportunity cost of capita, but because the cash ﬂow that is being discounted is the expected cash ﬂow and the opportunity cost of capital is now the rate of retrun available in ﬁnancial markets on inestments of similar risk to the projects - However, it is quite important to remember that not all investment are equally risky. Some are more riskier than others. - Cash ﬂows can be treated as if they are known with certainity or by talking about expected cash ﬂows and expected rates fo return without worrying how riks is deﬁned or measured. What is the rate of return - The rate of return (RoR) is used to measure the profit or loss of an investment over time. Present values and rates of return - In order to ﬁnd out how much an investment costs , we need to ask the shareholders how much they would need to invest in a risky secturies to achieve the same payoﬀ - Rate of retrun formula o Proﬁt/investment - We have two rules: o Net present value rule: this tells us that we only accept a investment with a posi6ve net present value o Rate of return rule:this rule tells us that we accept invesmtnet that oﬀer us a higher rate of return in excess of their opportunity cost of capital - If we apply both rules in the same way, we will end up with the same answer most 6mes. Calcula6ng present values when there are mul6ple cash ﬂows - Present value is expressed in tcurrent dollars. So we can add them together - So present value of cash ﬂow a+b is equal to the present value of cash ﬂow a plus the present value of cash ﬂow b - The formula for this is o 5 - This is what we call discountd cash ﬂow and we can also wrife it as this - - The 3 rfers to the sum of sries of dicountd cash ﬂows. And in order to ﬁnd the net present value we add the nega6ve ini6al cash ﬂows - Financial managrs turn to ﬁnancial calcuatos when doing this The opportunity cost of capital - If investment is uncertain, your shareholders woul be giving up the opportunity to earn an expected return of for example 12 % from an equally risky invesmten 6nt he stock market. The opportunity cost of capital is therefore in this case 12% - When we discount the expected cash ﬂows by opportunity cost of capital, you are essen6aly asking how much investors in ﬁncancial markets are prepared to pay for a secturitru that produces a similar stream of future cash ﬂows. - The intrest rate on a loan has nothing to do with the risk of a project. It tells us the good health of your business. It also important to remember that whether we take a loan or not , you s6ll will face the choice between two diﬀerent h6ngs that are equally risky to invest in the stock market. 2-2 how to value perptu6tes and annuites How to value perpetui6es - It is an investment that pays the same cash ﬂow every year In perpetuity with the ﬁrst cash ﬂow happen one year from now - Let c be the cash ﬂow paid out each year. We see that the annual rate of return on a perpetuity is the equal to the promised annual payment divided by the present value o Return= cash ﬂow/ present value , r= c/pv - The disocount rate of a perpetuity is o c/r-> cash ﬂow / rate of return 6 - two warnings about the formula o this formula looks like the presnt value formula. Howver, the perpetuity has a value of 1/ r o furthermore, this formula tells us the value of a regular stream of payments star6ng one priod from now. So if the cash ﬂow starts in year t+1, th the formula gves you the present value as of year t. § so : the formula gives you the value of the perpetuity one yar before the ﬁrst cash ﬂow arises o a perpetuity that starts right away ,is known as a perpetuity that is due. In order to ﬁnd a perpetuity that is due, we add to th ﬁrst years cash ﬂow to a regular perpeity o a perpeity due is worth 1+r 6mes as much as a regular perpeityu. Some6mes w need to calculate the value of perpetuity that does not start to make payments for several year how to value annui6es - an annuity is an asset that pays a ﬁxed sum each year for a speciﬁed number of years. - Examples: equal payment house mortage - You can value an annuity by calcula6ng the value of each cahs ﬂow and ﬁnding the total. But we can use a simple formula which says the intrest rate is r - The expression is the brackets is the t-year annuity factor. It shows the present value of 1$ a year for each of t years. - The annuity is worth less because the cash ﬂows are ﬁnite, and as the value of t increases, the subtrac6on term declines - When t inﬁnite , the annuity and perpetuity formula are the same Calcula6ng annual payments – example 7 - Pv= annual loan* 4- year annuity factor= 1000 - Annual loan payment= 1000/year annuity factor - Intrest rate is 10% a year - Annual loan payment.= 1000/3.170= 315.47$ Future value of an annuity - We need to ﬁnd the future value of a level stream of payments. - To calculate the future we need to mul6ply the present value by (1+r)^t. the general formula for the future value of a level stream of cash ﬂows by 1$ for t year is : future value of annuity= present value of annuity of 1$ a year *(1+r)^r - If you can ﬁnd the present value fo any series of cahs ﬂow, you can calculate its futue value by mul6plying it by 1+r^r - Future value at the end of year t= present value*(1+r)^t 2-3 how to value growing perpetui6es and annuites Growing perpetuites - You need to value a stream of cash ﬂows that grow at a constant rate. - If we call the growth rate in cost g, we can write down the present value of this cream of cahs ﬂow as following - We have a much simplier way to do this o Pv of growing perpetuituty ) c1/ r-g - If you want to provide a perpetual stream of income that eeps pace with the growth rate in costs, the amount that you must set aside today is… Growing annuites - Pv of growing annuity =c*1/r-g 2-4 how intrest is paid and quouted 8 - Apr : it tells us how much intrest is paid over the course of the year ignoring how ooen it is pai. It dosent tel us hw much intrest an investor accumulates over the year - Ear: the eﬀec6ve annual rate taeks into account how oon the apr Is paid and the aﬀect of compounding. this takes both componets into account , it converts the apr into an eﬀec6ve rate assuming compounding once a year. - When intrest is paid more frequently, the eﬀec6ve annual rate is higher than th apr Con6nuous compounding - Instead of compounding intrest monthly or semiannually, the rate could be compounded weekly ( m=52) or daily( m=365). There is no limit to how frequently intrest could be paid. - Payments are spread evenly an conitosuy through out th year so that the intrest rate is con6nisoly compounded. - M will become inﬁte - If we reverse the calcula6on, we obtain the present alue of 1$ recived at the end of year t whn con6noulsy compounded rate is r o Pv= 1/e^rt 9 Chapter 3- valuing bonds 3-1 using the present value formula to value bonds If you own a bond you will get a ﬁxed set of cash payoﬀs every year un6l the bond maturiess, you will collect intrest payments. When the bond maturires, you will get one ﬁna interest payment called the face value of the bond The present value of a bond is found by= pv(bond): pv( annuity of coupon payment)+ pv8( ﬁnal payment of principal). in esscence, it depends on the opportunity cost of capital. Bon dprices are expressed as a percentage of face value We have something called the bonds yield to matuotryr this is th intrest rat of the bond. The yield to maturity blends the retrun from the coupon payments with the increasing value of the bond over its remaning life. So lets say that a bond that is priceed belows its face value is said to sell at a discount is called a discount bond. Investors who buy these recive a capital gain over the life of the bond, so the yield to maturiry on these bonds is always more than the coupon rate. a bond that is pried above face values sells at a premium is known as a premium bond. Investors hre face a capital loss over the life of the bond so the yield to maturity is less than the coupon rate. and a bond that is priced at face value is known as a par bond and the yield. The only way to ﬁnd yield to maturity is trial and error. You guess at a gﬁguere and calculate the present value of the bonds paymnts. If the present value is greater than the actual prices, your discount rate must be low to and you need to try at a higher ate. Yield to maturity is the retrun to an investors who buys th bond at the asked price and hold it un6l it matu6rs. Intrest is semi anuall, yield on us bond are quouted as twice the semi annual yiel. 10 Semiannual coupons and bond prices Long term issues of debt are known as bonds and issues that mature in 10 years or less at the 6me of issu are known as notes. The treasury also issues short term debt maturing in a year or less. Short term secturies are known as treasury bills. You cant buy treasury bonds or notes on the stock exchange. They ar traded by bond dalrs who quote prices at which they are prepared to buy and sel. Ask price is the lowest price at which a seller will sell the stock If we call the ith cash payment ci, the annual yield to maturity y an the number of intrest payments each year n, then the general formula for a bonds present value is 3-2 how bond prices vary with yields Fixes coupoun rate is the ﬁxed annual rate at which a guaranteed-income security, typically a bond, pays its holder or owner. It is based on the face value of the bond at the 6me of issue, otherwise known as the bond's “par value” or principal. As the yield changes the bond price. The higher a yield is , the lower the price will become. Bond prices and yield must move in opposite direc=ons. The yield to maturity which again measures the intrest rats on a bond is deifned as the discount rate that explains the bond price. When the bond falls, intrest rats must writ. And when the intrest rates goes up and the bond prices will fall. A change in intrest rats has only a modst impact on th value of near term cash ﬂows but a much greater impact on the value of distant cash ﬂows. Therefore the price of a long term bond is aﬀected more by changing intrest ates than the price of short term bonds. Dura6on and intrest rate sensi6vity Changes in intrest rates have a greater impact on the prices of long term bons than on those of short term bonds. 11 A coupon bond that matures in year 30 makes payments thorugh each of the years. It will therefore be bad to denote it to being a 30 Toyear bond which will be th average 6me to each cash payment is less tha 30 years Duea6on is the weighted average of the 6mes to each of the cash payments. The weight for each year is present value of the cash ﬂow received at thee 6me divided by the total present value of bonds here 6 is the number of years to the ith payment. Pv(ci) is the present value of the ith payment and pv is the present value of the bond In order to measure how bond prices chang when intrest rates change we need to use modifed dura6on which is the dura6on divided by one plus the yield maturity - Modiﬁed dura6on= dura6on/ 1+yield It measurs the percentage change in bond price for a 1 prcentage point change in yield. The deriva6ve of the bond price with respect to a change in yield to matuirtuy is dpv/dy = - modiﬁed dura6on 3-3 the term structure of intrest rates A single discount trate is okay but there are also 6mes when you need to recognixe that the short term intrest rates may be diﬀerent from the long ter rates. It would therefore be worth it to discount each cash ﬂow at a diﬀerent rate The rela6onship between short and long term intrest rates is called th term strcuutre of intrest rates. In the earlier year, the term structure sloped downward. long term. Intrest 12 rates were lowr than short term rates. In later year the pafern was reversed and long term bonds oﬀered a much higher intrest rate than short term bonds To gind the present value of a loan of 1$ needs to discount the cash ﬂow by the one year rate of intrest r1 - Pv = 1/(1+r1) The r1 is called the one year spot rate. and in order to ﬁnd the present value of a loan that pays 1 dollar at the end of two years, we need to iscoutn by the two year spot rate , r2 - Pv= 1/(1+r2)^2 The ﬁrst year’s cash ﬂow is discounted at today’s one-year spot rate, and the second year’s ﬂow is discounted at today’s two-year spot rate. So each spot rate is used to discount a single cash ﬂow occurring at a single point in 6me Once we have yield we use it to value other two year annui6es, but we cant get the yield to maturity un6l we know the price. And the price is based by the spot interest rates for dates 1 and 2. Spot rates comes ﬁrst. Yields to maturity come later aoer the the bond prices are set. Spot rates, bond prices and the law of one price The law of one price states that in a compe66ve market, two assets that are the same must sell for the same price. If two assets each make a safe dollar payment on th same future date, then these payments must be worth the same today and should therefore be discounted at the same spot interest rat Managers who want a quick summary measure of intrest rates bypass spot rates and look in the ﬁnancial press at ﬁnancial prss at yields to maturity. They are also called the yield curve which plots spot rates. They muse the yield to maturity on one bond to value another bond with roughly the same cupon and maturity Measuring the term strucutr We deﬁne the spot rate rt as the rate of intrest on a bnd that makes a single payment at 6me t. such bonds exists and are called stripped bonds. 13 Why the discount fator declines as futurity increases the two year discount factor df2 = 1/(1+r2)^r is less than the one year discount factor -> df1= 1. There is no such thing as a sureﬁre money machine. The technical term for money machine is arbitrage. This is always on the lookout for cases where two iden6cal sets of future cash lows are selling at diﬀerent prices. In a well func6on markets wher the cost of buying and selling are low, arbitrage opportunifes are not there by inestors who try to take advantages of them. The value of a dollar tomorrow must be less than that of a dollar the day aoer tomorrow as long as you can hold cash to earn a non nega6ve return. So a dollar recived at the end of two year is worth less than a dollar recived at the end of one year. 1+r2^2 will be greater than 1+r1 Expec=ons theory of the term structure A year from now the treasuries measure and you can reinvest the proceeds for another one year period and enjoy whatever intrest rate the bond market oﬀers then. The intrest rate for the second year may be high enough to oﬀset the low return in the ﬁrst year. You ooen see an upward slopingg term structure when future intrest rates are expected to rise Expecta6on theory states that in well func6on bond markets investment in a series of short maturity bonds must oﬀer the same expected rturn as an investment in a single long maturity bond. if that is th case, investors would be prepard to hold both short and long maturity bonds. This theory implies that the only reason for an upward slopig term structure is that investors expect short term intrest rates to rise and the only reason for a declining term structure is that they expect short trm rates to fall. In short The expecta6ons theory cannot be a complete explana6on of term structure if investors are worried about risk. Long bonds may be a safe haven for investors with long- term fixed liabili- ties, but other investors may not like the extra volatility of long-term bonds or may be concerned that a sudden burst of inflation may largely wipe out the real 14 value of these bonds. These investors will be prepared to hold long-term bonds only if they offer the compensation of a higher rate of interest. Intrest rate risk The theory leaves out risk. If you are confident about the future level of intrest rates you will choose the strategy that offers the higest return, but if you re not sure you will opt for a less risky strategy even if it means giving up some return. Prices of a long duration bond are more volatile than prices of short duration bonds. For som investors , extra volatility of long duration bonds may not be a oncern. The volatiltiy of long term bonds does create extra risk for investors who do not have such long term obligations and these investors will prepare to hold long bonds only if they offer the compensation of a high prospective return. The term strcuture here will be upward sloping even if intrest rates are expected to remain the same Inflation risk You can reduce exposure to inﬂa6on by inves6ng short term and rolling oer the investment. You never know the future short term intresr rrates but you know that the future intrest rates will adapt to inﬂa6on. If inﬂa6on takes oﬀ you will be able to roll over your investment at higher intrest rates If inﬂa6on is important source of risk long term investors, borrowers must oﬀer some extra incen6ve to induce invstors to lend long. 3-5 real and nominal intrest The change in cpi from one year to the next measurs the rate of inﬂa6on. Inﬂa6on touced a peak at the end of world war 1. Prices can fall as well as rise an the us exprecies severe delfa6on in the great depresion where prices fell by 24% in three years Economics contradt nominal which is current dollars with real or constant dolars. The formula for conver6ng nominal cash ﬂows in a future period t to real cash ﬂows today is 15 - Real cash ﬂow at date t= nominal cash ﬂow at date t/1+inﬂa6on rate^t The real formula for calcual6g the real rate of return is - 1+r (real) = (1+r(nominal)()1+r) o I is the inﬂa6on rate Indexed bonds and the real rate of intrest Most bonds promise you a ﬁxed nominal rate of intrest. The real intrest rate that you recive is uncertain and depdns on inﬂ6on because if the inﬂa6on rate is. High the real return on the bonds will be lower. You can nail it buy buying something calle a indexed bond that makes cash payments linked to inﬂa6on. The real cash ﬂows on 6ps are ﬁxed but the nominal cash ﬂows increase as the cpi increases. The real yield to maturity measurs the extra goods and servies your investment will allow you to buy. What determines the real rate of intrest? The real rate of intrst depends on peoples willingess to save the supply of capital and the opportunites for porducive investment by goverments and business aka the demand for capital Example - suppose that investment opportuni6es improve. Firms have more good projects, so they are willing to invest more than previously at the current real interest rate. Therefore, the rate has to rise to induce individuals to save the addi6onal amount that ﬁrms want to invest.Conversely, if investment opportuni6es deteriorate, there will be a fall in the real interest rate. inﬂation and nominal intrest rates iriving ﬁsher tells us that a change in the expected inﬂa6on rate causes the same propor6onate change in the nominla intrest rate. it has no eﬀect on the required real intrest rate. if changes in prices are associated with changes in the level of industrial aci6vity then in inﬂa6naory condia6ons I might want want mor or less than 103 apples in a years 6m to compenesate for the loss of 100 today. 16 Investors hae for the most part demanded a higher rate of intrest when inﬂa6on was high. Fishers thory provides a useful rule of thumb for a ﬁnancial manager. If the expected inﬂa6on rate changes, it is a good bet that there will be a corresponidng change in the nominal intrest rate. so a strategy of rolling over short term investmnts aﬀords some protec6on against uncertain inﬂa6on Corporate bonds Governments can default on debt when they can’t pay it back, and it happens more ooen than you’d think. The riskiest type is foreign currency debt—like borrowing in U.S. dollars— because the country can’t just print dollars to repay. If the economy crashes, they might run out of money and default, like Venezuela did in 2017 on $65 billion. Debt in a country’s own currency is safer because they can print money to avoid default, but that ooen causes inﬂa6on. For example, Russia in 1998 couldn’t manage its crisis, defaulted anyway, and its currency lost two-thirds of its value. Then there’s the Eurozone, where countries like Greece use the euro but can’t print it. When they borrow too much, they can’t ﬁx the problem themselves and need bailouts. Greece defaulted in 2012, causing $100 billion in losses and a deep economic crisis. So, even governments can run out of op6ons—and money. 17 Chapter 4- valuing stocks 4-1 how stocks are traded Trading results for cummins Market cap ( short for market capiztalia6on. Most of the traing on nyse an nasdaq is in ordinary common stocks but other sectreu6es are traded also inlcuding preferred stock and warants. Investors can choose from hundrs of exhcnage traded funds which are por^olios of stocks that can be bought or sold in a single in trade. Most e^ are not manged thy simply aim to track a well known market index Market price vs book value Book values are values reported on the companys books. These include al puvlic companies that publish quarlty and annual ﬁnancial statements. book value of equity→ the net value of the total assets that common shareholders would be entitled to get under a liquidation scenario does that mean that inestors were three times as optimistic as the accountans who prepared the financial statments?→ no, becasue accountans do not try to forecast performance. instead they calculate the book value of equity by recording the companys cumulative past investments in its business. less an allowance for depreciation and then subtracting debt and other libabiites the resulting histroical cost of inestments is not a good easure of current value. investors care much more about future earnings than historical costs. book values do not incorporate inflation. countires with high or volatile inflation often require inflation adjusted book value. book values usally exclude intangible assets such as assets and therfore do not capture going concern alue 18 going concern value → created when a collection of assets is organized into a healthy operarting business book values can be useful benchark. they might also be useful clues about liquidation value. liqudation value is what investors get when a failed company is shut down and its assets are sold off. book values of hard assets like land, buldings, veichals and machinery can be indicate possible liquadtion values. Intangible “soft” assets can be important even in liquidation, 4-2 valu6on by comparables There are two appoaches when it comes to valuing stocks. The ﬁrst invols iden6fying a sample of similar ﬁrms as poten6al comparables and then examning how much investors in thse comparable companies are prepared to pay per dollar of earnings or book value. They see what the business would be worth if it traded at the comparables price earings( p/e= p/eps) or price to book value (p/b) ra6os. This valua6on approach is called valua6on by comaparables or comaprable companies analysis The second approach is to forecast and then discount the business dividends or future cash ﬂows. Valua6on by comparables works best if the relevant ra6os cluster togther by industry and for similar companies in the same industry Valua6on by com- parables is most useful when you don’t have a stock price enterprise value→ defined as equity market cap + debt. in other words, as the total value of the firm to debt and equity investors combied. earnings per share in the p/e ratio are calucalted after deducting interest expense and therfore, depend on the aount of debt and the company has issued. 19 Earnings per share aka the e in the p/e ratio are calculated after deductuin interst xpesene and therfore depend on the amount of debt the company has issued. Ebit and ebitda are calculated before the deduction and depdns of the firms operating profitability not its amount of debt financing. EBIT and EBITDA measure a ﬁrm's opera6ng proﬁtability independent of debt ﬁnancing. They are unaﬀected by accoun6ng diﬀerences in tax expense calcula6ons, with EBITDA also excluding varia6ons in deprecia6on and amor6za6on methods. 4-3 dividends and stocks prices Pv share of stock= pv expexted future dividends per share. this present value can be wrifen as : here p0 is todays price, divt is the exptexted dividend per share at future date t and r th disocount rate. the pvs of all future dividends are added up from date t= 1 to inﬁity , this is because a stock does not have a matu6ryr date and coud last forever. The discount rate r is the expected rate of rturn demanded by investors in the ﬁrms stocks. This is called the cost of euity because hsareholders provide equity ﬁnanicng, they do so when buying newly issued shared ansd when the ﬁrm retains and reinvests earinings that otherwsie could be paid out. The cost of equity is the the same as shareholders opporutnity cot of capital also deﬁned as the expectd retrun on other secturies with the same risks as the companys stock This equa6on is th same as the discount cash ﬂow or dividdend discounted model of stock prices. This formula does not requir the ﬁrm to pay dividends immenditlaty and it does not require investors to hold a common stock forever. When investors sell they hope to get more for the stock than they paid for it and therfor will gain a capital. The discount model says nothng about capital gaisn 20 dividends and capital gains the rate of return is equal to the expected dividend per share div 1 + the expected price appreca6on per share p1-p0 all dividend by the price at the start of the year po - Todays price: you can predict todays prie if you are given investors forecasts of year end dividend and price and the expected rtrun oﬀered by other equally risky stocks Among the thousands of traded stocks there will be a group with essen6ally the same risks. This is caleld the risk class. All stock in this class have to be priced to oﬀer the same expected rate of return. At each point in 6me, all stocks of equivalent risk are priced to oﬀer the same expected return. This is a condi6on for equilibirum in well func6on capital market. If all stocks of equivlent risk oﬀer the same rat of return that rate of return becomes the cost of equity for each one of the stocks in the equielant risk lass. The horizion period h could be inﬁntelity distant. Common stocks do not expire of old age. as h apporaches inﬁ6ty the present value of the terminal price ought to apporach 0. We can forget about the terminal price eternily and express todays price as the present vlalue of a perptual stream of cash dividends Two veriosn of the dividend discount model We can write it as 21 or The ﬁrst one projects dividedns to inﬁty and the second one projects dividends year by year to a horizion date h and then adds ph to capture the value of dividedns aoer the horizion date. The end value ph is called the terminal value or horiion value. 4-4 dividend discount model applica6ons Expected diidends grow at a constant rate. such investment would be an example of the growing perpeptufy. In order to ﬁnd the present value we must divide the ﬁrst years cahs payment by the diﬀerences between the discount rate and the growth rate This is the constant growth discount cash ﬂow model. We can use the model only when the growth rate is less than the discount. And as g apporahches r the stock price becomes inﬁite. r must be greater than g if growth really is perpetual po is explained as next years expected dividedn div1, the projctd growth trend g and the expected rate of return on other secturies of equivalent risk r. The expected return equals the dividend yield which is div1/ p0 plus the expected rate of growth in dividnds (g). An alterna6ve apporach to es6ma6ng the long run growth starts with the payout ra6o, the ra6o of dividends to earnings per share. The plowback ra6o = 1- payout ra6o= 1-div/eps 22 Return on equity= epd/book equity per share. This formula: g= plowbakc rate* roe is called the sustainable growth rte. There are dangers in analyizing any singe ﬁrm stock wirth the constant growth dcf formula. The ﬁrst assump6ons is that reguar future growth is at best and apporaciama6on. The second assump6on is that even if it is an accptable approxima6on errros can creep into the es6mate of g. In a well func6on capital amrket investors capitalize the dividends of all sectutries in american risk class at exactly the same rate. however, any es6mate of r for a single sommon stock is noisy and subject to error. Good practice avoids relying on cost of equity estimates from just one company. Instead, it uses a sample of similar firms and averages the results for a more reliable benchmark. We have a mul6stage dcf model. - Constant growth: the constant growth df formula is an useful guid. Naïve trust in the formula has led to many ﬁnancial analysis to absolute crazy - It is quite hard to es6mate r by analysis of one stock only. Dcf models with two or more stages of growth We need a dcf model with more degres of fredoom. This model accomodates any pafern of year by year dividedns from t= 1 to the horizion period t= h. we also have the dividedn growth rate which is given by: plowback ra6o* roe. In the end proﬁtability will fall and the ﬁrm will respond by inves6ng less. The return of on equity will devline over 6me. We can caluite the cost of cequity by using trial and error to ﬁnd the value of r that makes p0 equal to 50. Cylical growth in proﬁtabiity 23 The growth rates can vary for many diﬀerent reaosn. Some6mes growth rate is high in the short run because the ﬁrm is quite proﬁtable but because it is recovering form an episode of low proﬁtbility. Prac6vle 6ps: when you use the dividend discount models you always end up forexas6ng growth. Earnings and dividends can grow for two reasons - Investments and expac6on; if the growth Is moderate you may be able to use th perptual growth formua. But if the rate of expanion and the near term dividend growth rate are high- greater than the growth rate of economy over all then the grwoth rat will decline. You need dcf model with two or more growth rates. - Increasing proﬁtability: earings increase if roe increases, even if the rate of investment is steady. but roe will not increase forever. You need a two stage dcf 4-5 income stocks and growth stocks Investors dis6nguisk growth stocks from income stocks. They buy growth stocks for the xpecta6on of capital gains and they are more intrestd in the future growth of earings rather than in next years dividends. They buy income stocks or their near term eraning and dividneds. Return on a perptuityi is the same as the yalry cash ﬂow dividnd by the present value. So the expected return on our share would be equal to the dividend by the share priece aka the dividend yield. And since all earings are paid out as ﬁvidends, the expected return is the same as the earings per share dividedn by the share price. Exepctect retrun with no growth= diiend yield= earings -price ra6o The expected return ofr growing ﬁrms can be the same as the earings – price ra6o. The key is wheter earings are reinvestd to proide a retrun equal to the cost of equity. A compnay that has the same risk as the exisi6ng business will make no contribu6on to the companys value. Its reutrn is equal to the opportunity cost of capita. This will not have an aﬀect on the share price. The reduc6on in value causes by the nil dividend in year 1 is the same oﬀset by the increase in value caused by the extra ividends in later years. 24 The earings price ra6o measured in eps1 next years expected earnings is the same as th cost of equity only when the new projects npv= 0. This is important because we cannot confuce the cost of equaity with earings price ra6os. We can think of stock price as the value of average earings under no growth polity, plus pvo which the net present value of growth opportunites the earings price ra6o is therfore the same as It will uderes6mate r if pvgo is por6ve and overes6mate it I pvgo is nega6ve. Pvgo is deﬁned as th enet value today aka the npv of all investments the company is expcted to make in the future. Pvgo is pos6ve when the company can be expected to invest at retruns higher than the cost of equity. the faster the growth rate the befer. Growth stocks are stocks of companies where the pvgo is pos6ve and accounts for good frac6on of stock prices. Investors in growth stocks are afracted by the capital gains they will get if expansion succedes. Income stocks are stocks of companies where the pvgo accounts for small frac6on of stock price. Companies like these may grow but are not able to invest at returns that are above the cost of equity. The price of incoem stocks depends on future earnings and dividends from exsis6ng assets Share price= present value of level stream of earnings+ present value of growth opportuni6es: eps 1/ r+ pvgo 4-6 valua6on based on fre cash ﬂow 25 Present value of equity= market capitlaxa6on = present value of expected fututre free cash ﬂow Free cash ﬂow is the same as the aoer tax cash ﬂow that is generated by the companyies opera6ons aoer subtrac6ng investment that is requird for growth. Free cash ﬂow is the same amoutn of cash that is avaliable for payout to all shareholders. Rapid growth is a good things not bad. Valua=on formaat This equa6on is the he same as dcm , expet that free cash ﬂow takes place of expected dividends, pvh stands in for free cash ﬂow in priods h+1,h+1. Valua6on horizion are chosen arbitralt. Es6ma6on horizion value Two ways - Horizion value based on p/e ra6os. - Horiizion values based on price book ra6os - Horizion value based on dcf 26 Chapter 5- net present value and othr investment criteria 5-1 a review of net present value rule C0=the current cash ﬂow. Market value is the same as price per share 6mes the number of shares outstandind. Net present values compe=oteis Most corpora6ons calcualte net present value when deciding on investment projects. It is also not the only investment criterion that companies use and frims ooen also look at other things. Things such as irr is also looked at Irr rule will give the same number as the npv when choosen correctly. Five points to remember about the npv Accept projects with positive NPV, reject those with negative NPV. Include all forecasted cash flows—ignoring any leads to poor decisions. Time value of money matters—a dollar today is worth more than a dollar tomorrow. 27 NPV depends only on the project's cash flows and opportunity cost of capital—not company profits, other projects, or accounting methods. NPVs are additive: If you do two projects, the total NPV = NPV of A + NPV of B. This means you can’t “hide” a bad project by combining it with a good one—if B has negative NPV, it will lower the total NPV, and you'll correctly reject it. Other methods might mislead, but NPV ensures sound decisions even when projects are bundled. 5-2 the payback and accoun6ng rat of return rules The payback rule A projects payback period is found by counting th number of years it takes befroe the cumulative cash flow equalt the initial investment, it is showing us how long it takes for a company to make up for the initial investment. The payback rule stats that a projct should be accepted if the payback priod is less than some specifed cutoff period. The payback rule is kind of misleadng - It ignonrs all cash flows after the uctoff date. - It gives equal weight to all cash flows before the cutoff date. the rule says that project b and c are the same but c’s cash inflows have occurred earlier and has a higher nepv - The choicse of a cutoff period is arbitiarty. the choice depends on the cutoff period 28 Accouting rate of retrun Npv depns only on the cash flows of the project and the opportunity cost of capital, but when compaies report to shareholders ,they do not show the cash flows. The report accountign profits and assets aka the profits and th assets that are carreid on the firms books. Shareholders calcualte profiability measures such as the profit divied by the assts. This measure is called th accounting rate of retun or book rate of retun and is calcualtad as : average profits/ average assets 5-3 the internal rate of return rule the irr is a better measure and is recomaned in many cases. There is no ambigutiy in defining the true rate of return of an investment that generates a single payoff after one period: rate of retun= payoff/ investment- 1 discount rate = c1/-c0-1 the c1 is the payoff and the -c0 is the required inesmtent. So… the discoutn rate that makes the npv= is the rate of return. We say that the project rate of retrun is the discount rate that gives a 0 npv. The disocunt rat is also known as the internal rate of return. Calcualting the irr 29 The irr is defined as the discount rate that makes npv = 0. In order to find the irr for an investment prokect lasting t years we must solve for irr The irr rule The internal rate of reutn rule says that the firm should accept an investment proect if the opportunity cost of capital is less than the internal rate of reutn. if the irr is less than ex 2808 irr then the project has a postive npv when discounted at the opportunity cost of capital. If it is the same as the irr the project has a 0 npv. And if it is greater than th eirr the projcts has a negative npv because shareholders could earn a higher retrun by investing their funds in th capital market. We should therfor compar the opportunity cist of capital with the irr on your projects. In essence we look for a postiv npv. The rule will give us te same answer as the npv when the npv of a prokect is a continuolsy declign function of the discount rate What is mutually exluisve projects : firms often have to choce between severa alternative ways of doing the same job or using the same facility. So they need to chooce between mutually exclusive projects. The verdict on irr Although IRR (Internal Rate of Return) has its ﬂaws, it is s6ll more reliable than payback period and accoun6ng rate of return, which are considered crude and ooen misleading. While IRR is less straigh^orward than NPV (Net Present Value), it generally leads to the same decision when used correctly. Most large companies now use discounted cash ﬂow methods, and many prefer IRR. However, relying too much on IRR—especially for comparing projects—can be problema6c. It may lead managers to favor high-IRR, short-term projects 30 that don't truly maximize value. Despite its issues, IRR remains popular likely because it easily compares to the cost of capital and is seen as useful even when that cost is hard to es6mate precisely. 5-4 choosing capital investments when resources are limited The capital budgeting has rested on the propostion that the wealth of a frims shareholers is highest if th firm accepts every project that has a postive npv. Lets say that there are limitation on the amount of capital avaibalbe for th firm. Hre you need to pick projects that is in within th comapanys resourcs yet gives the higest possible npv. In order to do this you need to pick the project that offers the highest npv per tollar. This is called the profiabuility index and is given by : Npv/ investment If a project has a postive profitability index it also has a postive nov. therfore firms use the the pfi to select proejcts when capital is not limited. the pfi can be mislading when choosing between projects How important is capital ration in practive? The NPV rule assumes that the company can maximize shareholder wealth by accepting every project that is worth more than it costs. But that may not be possible if capital is limited. A solution is to calculate each project’s profitability index, which is the project’s net present value per dollar of investment. You then pick the projects with the highest profitability indexes until you have run out of capital. Sadly, this procedure fails when there are other constraints on project choice. Where there are well-functioning capital markets, hard capital rationing is rare. Many firms do use soft capital rationing, however. That is, they set up self-imposed limits when they are con- cerned about the volume of proposed investments. 31 Chapter 6:making investment decision with the npv rule 32 6-1 forcas6ng a projects cash ﬂows The largets investment inolves the acqustion of intangivle assets. The us bank invest annualy in new it projects. An investment of any assets creates wealth if the discouned value of the future cash flows exeeds th up front cost. When you are faced with a problem regarding what to discount, we stick to five rule - Discount cash flows, not profits - Discount incremental cash flows and ignore non incremental flows - Treat inflation consistenlty - Separate investment and financing decions - Forecast cash flows after taxes Rule 1: discount cash flows not profits The npv depdns on thee expected future cash flow. Cash flow is the difference between cash recied and paid out. profit shows how well the company is performing. Capital expenitutes The accountant takes the cash expenditutre and divies then into two groups, current and captial expenditutres. When calcuating the pfoti the current expenditutes are deducted from the current revnues NOT CAPITAL EXPENDITUTRES. You don’t lose the capital expenduitre altough you use it to buy capital. Depreciating capital expenditutres makes so much sense when judging firm performance but will affect the npv. Example: An investment costs $2,000 and returns $1,500 in year 1 and $500 in year 2. If you mistakenly calculate NPV using accoun6ng income (which includes deprecia6on), you get a misleading posi6ve NPV of $41.32. In reality, you're just ge|ng your $2,000 back, so the NPV should be nega6ve at any posi6ve discount rate. The error comes from using accoun6ng 33 proﬁt instead of cash ﬂow—deprecia6on isn’t a real cash ou^low, but the ini6al investment is. Capital expenditutes aﬀects cash ﬂow but not proﬁts whie depcreca6on aﬀects proﬁts but not cash ﬂow. Working capital Net working capital is the diﬀerence between a companys short trm assets and its short term liabili=es. Short term assets are accounts recibale aka unpaid bills and invensto6res. short trm liabili6es are account payables aka bills that are not paid and accruas which is liabili6es for wages or taxes that have been recied but not paid. Most projects entail and addi6onal investment in working capital Increase in current assets such as inventory or receibales reprsenst cash expenditutre. An increase in current liability shows a cash inﬂow. Most projects entail an investment in net working capital.each period change in working capital should be recognied in yhe cash ﬂow forcast. Working capital is a common source of confucion in capital investment calcula6on. the most common mistaks are - Forge|ng about working capital en6rely - Forge|ng that working capital may change during the life of the projct - Forge|ng that working capital is recovered at the end of the projects. Because when the project comes to an end , invento6res are run down and any unpaid bills are paid oﬀ and you recover your investment in working capital. This makes up for cash inﬂow Rule : discount incremental cash ﬂows and ignore non incremental cash ﬂows The value of projects will depend on all the addi6onal cash ﬂows that follow from project acceptance. There are 4 things we need to look for 34 - Include the projects eﬀects on the ﬁrms other business: many new projects have an eﬀect on the ﬁrms exis6ng business. Some6mes a new project will help the ﬁrms exisi6ng business. - Recongnize aoer sales cash ﬂows: managers should forecast all incremental cash ﬂows generated by an investment. Because some6mes these incremental cash ﬂows last for so many years. Many depend on the revnues that come aoer their producst are sold - Remember salvage value: when a project ends you may be able to sell the plant and equipment or redploy the assets elsewhere in the business. If the things are sold you pay tax on the diﬀerence between the sale price and the book value of th asset. This shows the a posi6ve cashf low to the ﬁrm. If the asset can be usd elsewhere the cash saed from not having to buy a new asset is also a posi6ve cash ﬂow - Include opportunity cost : the cost of a resource may be relevant to the investment decsions even when no cash changes hands. In order to ﬁnd this you take the diﬀerence between the cash ﬂows with the projects and those without th projects Furthermore, we need to be aware of two things - Bewar of allocated overhead costs: overheads include itmes such as rent and superisotry salaries. principl fo cash ﬂows says that in invesmentapprasil should be inldued only the extra expenses that would result from the project. A project may generate extra overhead expenses, but it may not. - Forget sunk coss: they are past and irrevsible ou^lows, because they are bygones, they cannot be aﬀected by the decision to accept or reject the projects and should be ifnore. Rule 3: treat inﬂa=on consist Interest rates are seen as nominal rather than real terms. If the discount rate is stated in nominal term, then consicity requires that cash ﬂows should be es6mated in nominal terms 35 taking account of trends in seling price , the cost of labor and so on. This makes it so that we apply a single assumed inﬂa6on rate to all componts of cash ﬂows. Tax savings from depreca6on do not increase with inﬂa6on , they are constant. Real discount rate=. 1+ nominal discount rate / 1+ inﬂa6on rate – 1 Discount nominal cash ﬂows at a nominal discount rate , discount real cash ﬂows at a real rate. never mix real cash ﬂows with nominal discount rates or nominal ﬂows with real rates Rule 4: sepreate investment and ﬁnancing decision You should not subtract the debt proceeds from the required investment nor recognize the intrest and principal payments on the debt as cash ou^lows. You should view the project as if it were if al equity ﬁnanced trea6ng all cash ou^lows required for the projct as coming from stcokholders an all inﬂows as going to them. This procedure focues on the project cash ﬂows not the cash clows that is accocisated with alterna6ve ﬁnanching schmes Rule 5: forecast cash ﬂows aoer taxes Cash ﬂows should always be es6mated on an aoer-tax basis, as taxes are a real expense like wages or materials. Some ﬁrms mistakenly use pretax cash ﬂows and try to compensate by increasing the discount rate, but there’s no reliable way to adjust for this. It’s also important to use actual cash taxes paid, not the tax ﬁgures shown in ﬁnancial statements, since accoun6ng methods (like straight-line deprecia6on) ooen diﬀer from tax rules (like accelerated deprecia6on). Summary of this: class notes rule 1- cash flows and not profits 36 cash flows o difference between cash recived and cash paid o relatd to project profits o current reenues- current expenditutes o accounting data o how well is firm performing capital expentiturs in accounting o Purchase of assets is spread over time and deducted over forecasted life of asset o Annual charge for depreciation o Depreciation is used to calculate profit o Treated as annual cost for number of years o While money was spent when asset was purchased we have an initial investment today, and we see it spread over time due to depreciation capital expentuties using npv State capital expenditures when they occur Not later when they show up as depreciation Capex affects cash flows but not profits Depreciation affects profits but not cash flows To go from accounting profit to cash flow Add back depreciation and subtract capex net workign capital = short term( current) assets- short terms(current ) liabilites Short-term assets o Accounts receivable (customer’s unpaid bills) o Inventories of raw materials and finished goods Short-term liabilities o Accounts payable (firm’s bills that have not yet been paid) o Accruals (liabilities that have been occured but not yet paid, e.g. liabilities for wages and taxes) => Accounts receivable + inventories – accounts payable Information also available at company level 37 o A measure of company’s liquidity and its ability to meet short-term obligations, as well as fund operations of business At start of project’s life cycle Most projects entail investment in net working capital = cash outflow Each year’s change (increase and decrease) in working capital should be recognized in cash-flow forecasts At end of project’s life cycle o Recovery of some of investment = cash inflow important to remeber Do not forget to include working capital Working capital may change during lifetime of project o Any increase or decrease must be taken into account Do not forget that working capital may be recovered at end of project rule 2- relevant cash flows= incremental cash flows 1- stand alone principle Consider as if project/investment would be mini-firm, with own revenues and costs, and cash flows Include impact and effects (spillover) of project on firm's other business(es) o Positive or negative Overall future cash flow that comes about as DIRECT consequence of decision to take that project o Any cash flow that exists regardless whether project is undertaken, is NOT relevant At moment it actually occurs => (Firm’s cash flows with the project) – (Firm’s cash flows without the project) what are some postive and negative effects of projects on firms othr buiness Example Positive effect o Launch of Lego movie affected positively sale of Lego sets o Movie costed $60 mn o Direct income of $250mn in the US and £31 mn in the UK o Sale of Lego sets increased with 11%, more than 3 times the figure of 6 previous years 38 Example Negative effect o New product will cut into sales of older versions of product o Launch of new Polestar will affect sale of previous Polestar-versions 2- after sales cash flows First, sale of product => inflow of cash Then, revenues from service and spare parts Example o Selling cars generate inflow at date of sale o Customers need to come back for repairs and maintenance every year, which generates inflows as well 1. salvage value Positive o Selling plant or equipment or redeploy assets o If asset can be used elsewhere in firm, cash saved from not having to buy new asset is also positive cash flow o But ! Take taxes on sale price into account – only if higher than book value Negative o Shutdown costs o Example Future reclamation costs of nuclear plants, mines 4- opportunity cost What it requires to give up benefit o An asset that we already have, but that will be used for new project Take difference between cash flows with project and those without project Opportunity cost equals market price Example o Piece of land available o Can be used to build new plant and start new business o But it can be sold as well o Opportunity cost = cash flows generated by new project – sales price what is not incremental ? 39 Allocated overhead costs Example o Supervisory salaries, rent, insurance costs, property taxes, heat and light But Not all overhead costs to be ignored – include extra expenses needed to start new project Sunk costs o Past and irreversible o Example Marketing, research expenses, training costs, new software already acquired and installed rule 3- treat inflation consistently Nominal interest rate o Use it consistently and use nominal cash flows as well o Take into account trends in selling prices, cost of labor and materials,... Likely not same/one single assumed inflation rate to all components of cash flow Final result of NPV calculation is same whether you use ‘nominal’ or ‘real’ data! o Real cash flow 〖CF〗_R => use real interest rates r_R o Nominal cash flow 〖CF〗_N => use nominal interest rates r_N o Inflation i rule 4- seperate investment from investment from financing decision Regardless actual financing o Consider project being all-equity financed o All cash outflows coming from stockholders and all cash inflows going to them o If project is financed partly with debt Do not substract debt proceeds from required investment Do not take installments on loans into account § Do not recognize interest and principal payments on debt as cash outflows 40 Focus is on project cash flows, not cash flows associated with alternative financing schemes rule 5- after tax cash flows taxes are expenses→ cash flow estimation after tax! 6-2 corporate income taxes Corpora6ons may also need to pay tax to a reginoal govermenet. Tax rates change over 6me some6mes ram6clay Deprecia=on deduc=ons When calculating taxable income, firms can't deduct capital expenditures but can deduct depreciation. Most countries require straight-line depreciation, spreading the cost evenly over the asset's life. This creates a tax benefit called the depreciation tax shield. Some countries, like the U.S., allow accelerated depreciation, letting firms deduct more in earlier years. From 2018, U.S. companies could use 100% bonus depreciation, allowing immediate write-off of qualifying investments, though this is being phased out by 2027. Not all investments qualify—real estate must use straight-line, and R&D must be amortized over five years since 2021. U.S. firms maintain two sets of books: one for the IRS (using accelerated depreciation) and one for shareholders (often using straight-line). This boosts early tax savings while maintaining higher reported profits. Analysts must distinguish between the two, especially since only tax books matter in capital budgeting. 6-3 a wokred example of a project analysis The three components of project cash ﬂows Total cash ﬂow = cash ﬂwo from capital investment+ opera6ng cash ﬂow+ cash ﬂow from investment in working capital Opera6nc ash ﬂows: rev-expenses-taxes 41 6-4 how to choose between compe6g projects 4 problems - invstment 6ming problem : should I invst now or wait - the choice between long and short livd equipment. Should you save money today by choosing something that will not last long - the replacment problem, when should exisi6ng machinry be replaced - the cost of exces capa6cty. What is the cost of using equipment that is not being used problem1: the investmnt =ming decision a pos6ve npv does not mean that it is the best understanden now. It might be more valuble if undertaken in the future. Timing is not diﬃcult when cash ﬂows are certain. You must examine alterna6ve start dates for the investment and calcualte the net future value at each of these dates. Npv of investmnt if undertaken at date t= nt future value at date t/ 1+r^t Pv of t year anniuity 42 Chapter 7: introduc6on to risk, diversiﬁca6on and por^olio selec6on 43 7-1 the relationship between risk and return how are risk and return linked? treasury bils are about as safe an investment as you can make. there is no risk or default, and their short maturity means that prices are relativley stable. an investor who wishes to lend money for three months can achiee a perfectly certain payoff by purchashing a treasury bill maturing in three months bond prices fall when intrest rates rise and rise when intrest rates fall. o stocks are riskier still becasue their alue depends mont he evvolution of the stock market and on the performance of the issuing companies over a centruy of capital market history investment performance concides with inutitive risk ranking an investment in long term goverment bonds would have produced an average nominal return. why do we look back over such a long period to measure avaergae rates of return o becasue the annual rates of retrun for stocks fluctuate so much that aerages over short period are unreliable. the retrun in any particular year was affected by one off events using historical evidence to evaluate todays cost of capital if there is an investment project that has the same risk as the stock market, represented by the s&p 5. what rdate should you use to discount this projects forecasted cash flows o you should use the expected future rate of return on the s&p. its the future return thats relevant becasue the alternative to investing in the projects is to pay out the cash as dividends and allow shareholderes to invest in the s&p how do we predict this future retrun o one apporach is to assume that the future will be like the past and so toays investors can expect to recive the average historic rate of retrun 44 § however, rm wont be stable over time. remeber that its the sum of the risk free intrest rate,w hich we label as rf, and a premium for risk. o we predict that there is a stable risk premium on equitites , so that the expecteed futrure risk premium can be measured by the average past risk premium, however this might not hold § for example, investors might be less averse to holding stoxk today than in the past. the growth in mutual funds and etfs haas made it easier for indivuals to diviersy away risks, or perhaps pension funds can diiersy by investing overseas. o there is not a corect way to caluclate the average past retrun. art omes in two forms § the first is knowing how far back to go when gathering past data. perhasps stcok retruns 100 years ago were from a differenr era, when it was much harder to diversify so inestors required a higher risk premium to hold stocks § the second way is known how much to adjust historic data when forecasting the future. there is now formula to tell us what will happen next, nor how much we should in crease pr decrease the historical average by analysis future risks. that is where the skill of investing lies. it is also why professional investors can disagree widley on the future direction of stocks, even though they all see the same data o this explains why there is so muh trade in finanical markets. one inestors is willing to buy and the other is willing to sell, yet both sides belive they´ve got a good deal becasue htey have diferent predictions for the future. the diffiuclty of estimating expected retrun is particulary important for growth stocks, which pay no divididens today. since the bulk of their present value is partuclary senseitive to the discount rate to sum up o estimating the risk premium is just as much of an art as a science. history will contain valubale clus, bu the future may be different from the pat. 7-2 how to measure risk 45 varience and standard deviation so inestors require a higher retrun for takin on more risks, but whats relevant way to measure risk? o on a histogram we have superimpose a bell shaeped normal distrubution. when measured over a short interval, the padt returns on any stock conform failry closley to a normal distriubution normal distributin can be completley defined by two numbers. this is caputured by the average or expected value. the second is spread of the distribution , which is measured by the variance or standard deviation o the vareince of teh market retrun is the expected squared deviation from the expected retunr § Variance" ( ~r i ) =the expected value of ( ~ r i - r i ) ^2 o ~r i is the actual stcok i and ri is the expected return. the standard deviation is the suare root of the varience o standard deviation of ~r i = calculating risk consider the following game. o you start by investing 100. then two coins are flipped. for each head you get back your starting balence plus 20%, and for each tail, you get back your starting balence less 10%. so you have tour equaly likely outcomes § head+ head: you gain 40% § head + tail: you gain 10% § tail+ head: you gain 10% § tail + tail: you lose 20% o there is a cahance of 1 in 4 that you make 40% , a change of 2 in 4 that you will make 10% and a change of 1 in 4 that you will loose 20% o the expected retrun= 0.2540+0.510+0.25*-20= +10% 46 estamiting future risk just as investors are concered with the future expected return, its the future risk of retruns that they also care about. however, looking at the volitlaity of retrun in the past may be a helpful guide to the volailtility of retruns that may occur in the future. data on past volatility are only a starting point for estimating the volaitliy of retruns that might occur in the future. we may ned to adjust past volatility if we think the future will be different. market turbalence over shorter daility, weekly or monthly periods can be amazingly high. on balck friday 1987 the market fell by 23% on a single day. 7-3 how diversification reduces risk the market portfolio is made up of indiual stock, so why dosent its risk reflect the average risk of these stocks. the answer is that diversifiaction reduces risk example o The standard deviation of your portfolio won't be a simple weighted average because diversification reduces risk. With 60% in Southwest and 40% in Amazon, your portfolio risk will be lower than the weighted average of their individual risks. figuere 7.10 the entires in these diagnoal boxes depend on the variences of stocks 1 and 2. the entires in the other two bozes depend on their covareince. o the covareince measuires the degree to which the two stocks covary in other words move together. o its the product of teh correlation coefficent and the two standard deviations the two standrd deviation define the risk of each stock in isolation, not how they covary together. 47 o the key variable that determines covarience is the correlation coefficent. § a postive correlation coefficent indicats that when stock 1 goes uo, stock 2 tends to go up. if p12 is at the highest value of +1 , we have a perfect postive correlation § when stock 1 goes up, stock 2 always goes up proportoionally. a negative corelation coefficent indivates that when stock 1 goes up, stock 2 goes down. if p12 is at its lowest value of-1, we have a perfect negative correlation. § when stock 1 goes up a,stock 2 always goes down. § when p12 is 0 , the returns on the two stcoks are wholly unrealted. just as we weight the varience nby the square of teh proportion invested so we weight the covarience by the product of the two propotionate holdings x1 and x2 o once you’ve complete these four boxes, you simply add the entires to obtain the portfolio varience specific and systematic risk specifc risk is known as iddiosyncratic riks , residual risk , unique risk or unsystematic risk. these are risk that affect only one company or only companies in a single industry or of a particular type. systematic risk are risks that are shared by most other business. you cant diversify these risk away, no matter how many other companies you move into. systematic risk is known as undiversiable risk total risk is the sum of specific and systematic risk holding a portfolio of two businneses does not get rid of all risk. your still exposed to the possibiliyy of a recession which is a systemtaic risk. diversification with many stocks 48 the method for calculating portfolio risk can easily be extened to portfolios of three or more sectures. 7-4 systematic risk is market risk if investors have the same information, the logic of systematic risk and diversification leads them to invest in th emarket portolio which contains all stokcs. investors know what the market portfolio is they can onsrve its retruns and volatility. they dont have to compute market risk by adding up millions of boes the total risk of an indivual company is measured by its standard deviation. if you held that stock in isolation , thats the amounts of risk youd bear. investors dont hold indivual stocks. they hold portfolios. the risk that a stcok ads to a diversifed portfolio isnt its total risk but only the systematic risk thats shared with the rest of the portfolio. specific risk is diviersifed aways we only worry about a stock s systematic risk how to measue is o systematic risk is the risk thts shared with the rest of the portfolio so we first nee to know what the rest of the portfolio is before we can estimate it. investment opportunity set o reflects the opportunities available to the firm due to the nature of its investments. o The red dotted line in the figure represents an extreme case where the returns of two stocks are perfectly negatively correlated (correlation = -1). In such a scenario, a combination of the two stocks could eliminate all risk. In practice, however, you’re not limited to just two stocks. For example, you could select a portfolio from 10 stocks, as listed in Table 7.5, with their expected returns and standard deviations shown. efficent portfolios and efficent fortiner o efficent portilio § if there is no other portfolio that offers higher returns for a lower or equal amount of risk o efficent frontier 49 § the set of optimal portfolios that offer the highest expected return for a defined level of risk or the lowest risk for a given level of expected return. portfolio choice with borrowing an lending When you combine a risky asset with a risk-free asset, the result is a straight line of possible risk-return combinations. For example, if portfolio A has a 10% return and 30.5% risk, and Treasury bills offer a 2% return with no risk, investing half in each gives a 6% return and 15.2% risk. If you borrow money at the Treasury bill rate and invest everything in A, your expected return increases to 18%, with a higher risk of 30.2%. The Sharpe ratio measures the extra return earned per unit of risk. By combining the risk-free asset with a stock portfolio, investors can maximize their Sharpe ratio by choosing the tangency portfolio (T)—the optimal mix of stocks. Instead of picking different stocks, investors adjust risk by borrowing or lending at the risk-free rate while holding T. This leads to all investors holding the same stock portfolio but at different risk levels. the slope of the sharpe ratio as long as investors have the same information and make the same assessments of expected returns, standard deviation and correlations , then everyone sees th same invstment opoportunity set and efficient frontier and everyone sees the same tagency portfolio t. o if all investors have access to the same information alls hould hold the same portfolio stocks § why § Investors should diversify rather than take on specific risk. Instead of picking single stocks like Southwest Airlines (for risk- seekers) or Coca-Cola (for risk-averse investors), both should hold the tangency portfolio (T)—the most efficient mix of 50 stocks. Risk-tolerant investors (lions) can increase risk by borrowing to invest more in T, while risk-averse investors (chickens) can reduce risk by holding T alongside Treasury bills. The key takeaway: everyone should hold the same stock portfolio (T) but adjust risk through borrowing or lending. o all investors should hold the same stock portfolio. investors should differ in how much of that portfolio they hold. market risk if investors share the same information all will hold the portfolio of stocks with the higest sharpe ratio o that is the portfolio that is tangent to a lening borrowing line the market portolio is the portfolio of all stocks in the economy, where the weights corrwspond to the fraction of the overall market that each stco represents two stages o the first is to select the best portfolio of stocks, which is the portfolio t with the higest sharpe ratio o the second step is to blend this portfolio with borrowing or lending to match the investors willingness to bear riks. § each investor holds just two investments. the market portfolio t and the amount of lending or borrowing. § this result is known as the two fund sepreation therom becasue investors hold the enitre stock market, all unsystematic risk is diversified away. o “all of the remaing systematic risk is market risk § the line through t is therfore known as th capital market line. § this shows the trade off between risk and retrun that an inestor faces when he decids how to allocate his saving between lending or borrowing and the market portfolio § the sharpe ratio of the capital market line is 51 § the equation for the line § § m now replaces t and stands for market § p stands for portfolios along the capital market line 7-5 should companies diversify diversification reduces risk and therfore makes sense for investors. diversification is good for investors, but it does not mean that companies should pursue it. this is becasue investors can diversify themselves. becasue investors can diversify on thir own account, they wont pay any extra fro a firms that is diviersisfied. if it costs more , no investor would buy the conglomerate becasue hed simply buy energy and media companies seperatly this is a cruciual result becasue it implis value addivity. if the market values assets. a( an energy business) at pv(a) and asset b( a media business) at pv(b)) o value addivity means that the market value of a firm that holds only these two assets is § pv(ab)= pv(a(+ pv(b) Value additivity allows managers to evaluate new investments independently. A company considering a new project, B, only needs to calculate its present value (PV) since the company's total value is the sum of its parts. Whether B diversifies risk or amplifies existing exposure doesn’t affect this calculation. This principle applies to any number of assets, ensuring a straightforward valuation approach. chapter 8 - the capital asset pricing moel 8-1 market risk is measured by beta 52 market risk is the risk that a stock shares with the market. we measure it by estimating how senstive a stock is to market movments→ how much its price changes, when the market moves up or down. o this is calles beta which is deifned as here the om is the coverice between the stocks return and the markets return and o2m is the varience of markets return what is the itunutio behind this definition o the market risk of a stocks depend on how it covaries with the market § so the covarience oim in th numerator make sense. but why do we scale it by o2m? o there are two ways to see this 1. simple decompostion. to find the market risk of sectury , we decompose its total risk as measured by its standard deviation into its market risk and specific risk o the market risk is pimoi, becasue the correlation coefficent pm measures the part of total risk oi that comes from the market portfolio 3. the second way to explain this is by what it means the best estimate of beta, the regression coefficent that captures the avrage effect of the market rtrun on amazons return is the covarience divided by th varience, bi=oimio1m. o beta measurs the sense6vity of a stocks return to the markets retrun- how much a companys stock retun goes ip, on aberage when the market gos up by 1% while beta surs the sslope of the best fit line, an alternative variable, r squared measures the goodness of fit of the best fit line- how close the returns are to the line or how little specific risk thre is o r squared is calculates as r2= oim 2/oiom or alternal6vley the square of the corrlea6on coeﬃcient stocks with betas more than 10 move more than one for one with the overall market. Ex: when the economy does well consumers will buy luxury goods but the scale back when 6mes are 6ght. Stocks with a beta between 0-1.0 s6ll go up with the market but less than 53 one for one. In a boom, companies will produce more goods and use more elxectricity so that the beta is s6ll posi6ve. However. This sensi6vity will be low. Betas can be nega6ve. Such a stock does well when the economy does badly. The market por^olio of all stocks so the average stock has a beta of 1.0. The market poreolio All investors who share the same informa6on will hold the same portofolio of stocks since all th stocks must be held all investors must hold the market portolio. In reality, investors can invest in many types of risky assets such as bonds around the world. They can invest in nonreadable assets such as private equity. The true market por^olio contains al the worlds risky assets not just th us stocks. There is no index that measures the value of all risky assets. Investors use an approxima6on for the market por^olio. Then the beta of an assets such as stock an es6mate of how much that asset moves with the proxy for the market por^olio. Most investors por^olio are concentrated in their home countries stocks so most of investors use a domes6c index as the markt proxy. Why betas detmine poreolio Two crucial points about secturiy risk an por^olio risk - the risk of a well diversiﬁed por^olio is given by its market risk - the market risk of a stock is measured by its beta - the beta of a por^olio is the weighted average of the betas of the individual stocks within that prtofotlio the intui6on for why the total risk of a por^olio is lower than the average of the total risk of the indivual stocks. That because some of the total risk of an indivudal stock is diversifable 54 and so goes away when you add it to a porrfolio. However, beta measurs undiversifable risk to begin with theres no diversiﬁca6on eﬀect when adding a stock to a por^olio. The beta of a por^olio is simply the weighed average beta of the stocks in the por^olio. The weight for each stock holding is its market value as a percentageof por^olio value the beta of a well diversiﬁed por^olio in turn detrmins it risk. 8-2 the rela6onship between risk and return this is the equa6on of the market line. This gives ust eh expected retun for an eﬃcient por^olio that combines lending or borrowing the market por^olio. This isn’t apply for induvial stocks because they are not eﬃcient as standalone assets. Part of their risk can be diversiﬁed away so the investor should not be rewarded for bearing it. The capital market line doesn’t apply for them An investors will gt rewarded for bearing market risk because this risk is something that you get rid oﬀ. Therefore we need something new, to an induvial stock aka I, we have to replace the total risk by market risk only What is the market risk of stock i: We can also write t as 55 Expected risk premium on stock= beta*expected market risk premium. The equa6on above shows us the capital asset rpcing moel. It shows a rela6onship between risk and retrun that’s important to ﬁnance. The models meassge is both starlign and simple. It tells us that in a compe6ve market, the expected risk premium on any secu6y or por^olio not only the eﬃcient portolio is that its beta mul6plied by market risk. This shows us the security market line. We see that the treasury bulls hae a beta of 0 and a risk premium of 0, the market por^olio has a beta of 1 and a risk preomium of …. The risk premium of any security or por^olio is the propo6nal to its beta. The capm formula shows that the risk premium of a stock depends on its market risk and only on its markt risk. Diviersifable risk should not aﬀect expected retrun nor should any other frim charachterics6cs. What if a stock did not lie on the sml An investor can always obtain an expected risk premium o bets*rm-rf by holding a mixture of the market por^olio an the risk free assets. In a well func6on market, nobody will hold a stock that gives you an expected risk premium that is less than the beta*rm-rf. 56 If there are stocks that oﬀer a higher expected risk premium and lie above th sml line, then the stocks would oﬀer a higher return than the other stocks of the same systema6c risk. We would see investors plie into those stocks, pushing the price up and the expected return down un6l they lay on the security market line and oﬀered the appropriate return for their beta. The stock in essence must lie on the security market line and command an expected risk premium of: The capital markt line and the security market line Both lines have expected return on the ver6cal axis, ut the capital market line has a total risk on the horizon axis while the security market line has market line( bi). The cml applies to eﬃcient por^olios only. This is because ineﬃcient por^olios that bear speciﬁc risk lie below the line. They carry more risk than the eﬃcient por^olios that give the same expected return. Investors want these stocks because the risk is diversiﬁable.. risk goes away when held as part of a well diversiﬁed por^olio The sml applies to all stocks an por^olios. This is because the horizontal axis is the market risk. an since risk is undiversiﬁable every ounce must be rewarded else an investor would not be willing to bear it. Investors needs to be given the risk premium (rm-rf) on th en6re amount of market risk. Eﬃciency only mafers at a por^olio lvel. An investors may be willing to hold ineﬃcient assets which is why induvial assets lie below the cml , because these speciﬁc risks washes away when you hold them as part of a por^olio. An eﬃcient por^olio is comprised of induvial ineﬃcient assets 57 What is the logic behind the capital asset pricing model? - Efficient portfolios provide the highest expected return for a given level of risk and are preferred by investors. - If investors can borrow or lend at the risk-free rate, the optimal portfolio is the one with the highest risk premium-to-standard deviation ratio—this becomes the market portfolio. - Risk-averse investors combine the market portfolio with the risk-free asset, while risk-tolerant investors may invest more heavily, even borrowing to do so. - Investors hold portfolios, not individual stocks, so only systematic risk matters; individual (unsystematic) risk is diversified away. - Systematic risk is the stock's sensitivity to overall market movements, measured by beta. Why does a high beta and high returns go together High beta stocks payof more than low beta stocks in economic booms when investors marginal utility of money is low and lss in reccsion, when marginal utilityu is high investors compentsate by requring a higher expected rate of retrun from high beta stocks. The opportunity cost of capital from high beta investent is also higher A high beta stock pays off most when the market is booming. Investors hold the market their portfolio oes well in a boom when there probably dpomg well persoanly and their marginal utility of more money is quite low. They are able to pay th erent etc. When the market craches investors protfolios goes down at the same time when there probaly not doing so well personally and the marginal utility of more money is high, if a stock has a high beta, then its also likely to suffer more in a market crach. The stock loses more money at the time when you need it. 58 A low beta stock is less sensetative to the market. It dosent do well when the market I booming an the marginal utlity of more money is low. The strength of the low beta stock is in bad times. When the portfolio is underperforming you need the money and the marginall utilot is high. When the low beta stock is likely to deliver more than high beta stocks. When choosing stocks you don’t car about how much you get aka expected retrun but when you get it aka wheter returns airse in good or bad times. That’s what the beta measures 8-3 does the capm hold in the real world? - Investors choose their porfolios based on expected return and risk measurd by varience of return - All investors hav the same es6mates of the mean returns, ariences and covariences of all assets - Investors trade in perfect captail markets there are no ﬁrc6ons such as taxes transca6on costs or restric6ons on short sales. - Investors are pric takers and cannot inﬂuence prices - The supply of all assets is ﬁxed How larg is the retrun for risk? 59 Are returns unrealted to all other charachtersi=cs 60 Although returns are unrealted to idiosyncra6c risk, they are related to other frim charachteris6cs. The red line shows the cumu- la6ve diﬀerence between the returns on small-ﬁrm and large- ﬁrm stocks from "*$+ to $%$%. The green line shows the cumula6ve diﬀer- ence between the returns on high book-to-market-value stocks (value stocks) and low book-to-market- value stocks If the return to the strategy are due to much rather than a risk that investors care about they should not be persit going forward. The capm makes sense. It seems to generate sensible es6mates of costs fo capital. However it dosent hold perfectly in the real world. 8-4 some alterna6ve theorires Arbitrage pricing theory Apt comes froma diﬀerent thing. It dosnt ask which por^olios are eﬃcient.it assums that each stocks risk premium depends on pervaise macroeconomic factors such as the factors that drive small stocks 61 Stock returns are assumed to obey the following simple rela6onship Noise stats for a diversifable risk. This theory doesn’t say what the factors are. Some stocks are more sensi6ve to a par6cular factor than other stocks. The apt agrees with the capm that there are two sources of risk for an indivual stock. The ﬁrst is th risk that is speciﬁes to the company. This is because the risk can be elimated by diversiﬁca6on , and dosent aﬀect a stocks risk premium. The second is th risk that stems from the macroeconomic factors. The apt stats that the expected risk premium on a stock depends on the expected risk premium associated with each factor and the stocks sens6ty to each factor the formula becomes:r-r A diversiﬁed por^olio with zero sensi6vity to all macroeconomic factors has no risk premium and must earn the risk-free rate. If it didn’t, arbitrage opportuni6es would exist—investors could earn risk-free proﬁts by exploi6ng the price diﬀerences. This is the core idea behind Arbitrage Pricing Theory (APT). A por^olio’s risk premium increases in pro

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