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Portfolio Theory – Lecture 2 The discounted cash flow model 1. Introduction Note: Please read over these notes before the class. As we noted last week we are going to systematically apply the scientific method in our study of finance. Recalling that the very first step in the scientific approach is to start with a problem, a question, or a puzzle, that is what we do here. Step 1 of the scientific method. Problem. How do we find out the fair value of a share? More generally, how can we find the fair value of any asset, for example the fair value of other paper assets such as a corporate bond, or physical assets such as real estate, buildings and machinery, or intangible assets such as a company’s reputation, or its brands and trademarks? Step 2 of the scientific method. Build a model (i.e. a theory or a hypothesis). We have done that step already, i.e. the textbook world or the rational expectations model is the main model we will be applying in this course, although later on we will be improving this model somewhat to include certain behavioural factors when we give our introduction to the new field of behavioural finance. Step 3 of the scientific method. Explain the puzzle in the model. When we solve the problem in the model we are giving the theoretically correct solution to the problem. This does not necessarily mean that this theoretically correct solution will actually work in the real world, since the theoretically correct solution is nothing but a description of what happens inside the model world. However, a good model will be a good approximation to reality, so we would expect that in this case the theoretically correct solution will also be reliable in practice. As we will see in this chapter, the theoretically correct technique to find the fair value of an asset is to use what is called the discounted cash flow technique, or DFC technique. In other words, the rational self-interested investors of our textbook world model use the DCF technique to work out the fair value of assets. It turns out that this theoretically correct solution also works very well in practice, and in fact is so useful in practice that DCF is the most important technique in the whole of finance. However, DCF is not a perfect technique and it is more useful in some situations than others. This reminds us that although the textbook world is a good approximation to the real world it is, like all theories and models still only an approximation to reality. Step 4 of the scientific method. Empirical research. Empirical research means looking at the world to see how well the theoretically correct solution actually works in practice. In this chapter we will be focussing on explaining what the DCF model is. However, we will be seeing many successful applications of the DCF model throughout the course, and whenever we do this we are really applying step 4 of the scientific method to the asset valuation problem. Step 5 of the scientific method. Applications. When we find a theoretically correct solution that works well in practice we can apply it with confidence. In the course of your business career and in managing your personal finances you will find many situations where DCF techniques are called for, for example finding the cheapest mortgage, negotiating the sale of purchase of commercial property, deciding how much to pay for an investment, etc. etc. Also of course, since DCF is the language of finance you will need to understand the DCF technique well enough not to get ripped off in financial negotiations. 2. Some examples Before discussing the DCF model in its most general form, we first of all look at a few examples of the model in action. Example 1: (Bond valuation) You are employed as an analyst on the bond trading desk of a small Austrian bank. You are considering whether to buy US$250,000 worth of corporate bonds of Glassblower Chemicals. Some background and jargon Companies can borrow money in two ways. The first is simply to borrow from a bank. For example, under an interest only loan Glassblower might borrow $10m for ten years at ten percent interest from ABC Bank, paying $1m (= 10% of $10m = 0.1 $10m) interest at the end of years 1 to 10 and also repaying the principle of $10m at the end of year 10. The second way of Glassblower to borrow is to issue corporate bonds. A corporate bond is a piece of paper entitling the owner of the bond to certain payments to be paid by the issuer of the bond. To raise the money this way Glassblower might issue 10,000 bonds each with a face value or nominal value of $1,000, with a 10% annual coupon or nominal interest rate and a maturity of ten years. How does Glassblower sell these bonds? Glassblower is a chemical manufacturer, not a financial institution with expertise and contacts in the financial markets. To issue the bonds Glassblower approaches an investment bank, since helping corporations to raise finance is one the main things investment banks do. Glassblower engages Sharkey, Sleight and Hand Inc. (SSH) to undertake the financing. SSH advises that in the present market the bonds can be sold for $1050 each, and agrees to act as underwriter for the issue. Underwriting the bonds means giving a guarantee that the bonds will be sold for at least $1050, and that if SSH is unable to sell the whole issue of bonds for this price, SSH will buy any remaining bonds itself at the $1,050 price. For these services SSH charges a fee of $400,000. Let’s suppose the bond issue is successful. Glassblower receives $10,100,000 (= 1,000 bonds 2 $1,050 $400,000 fee). SSH successfully sells the bonds to its network of institutional and private clients for $1,050 each and pockets the fee of $400,000). What do the buyers of the bonds receive for their $1,050? Suppose a pension fund buys 1,000 bonds for $1,050,000 (= 1,000 bonds $1,050). For each bond, at the end of years 1 to 10 the fund receives the coupon payments of $100 (= the 10% coupon the $1,000 face value), and also receives the face value of $1,000 per bond at the end of the maturity of 10 years. (Note that the terms coupon, face value and maturity are just part of the legal jargon which tells us what cash flows the holder of the bond is entitled to. Note also that the face value of the bond is usually different from the market price of the bond, i.e. the price at which buyers and sellers are prepared to trade the bonds. In our example the face value is $1,000 while the market price $1,050). Corporate bonds also trade in the secondary market. For example, after three years the pension fund might decide to sell its Glassblower bonds in the bond market. How much will it get for each bond? Well that depends on how much investors believe the bonds are worth. Like in all markets the seller wants to receive the highest price while the buyer wants to pay the lowest price. In trying to find the fair price both the buyer and the seller will apply the discounted cash flow model (DCF). An important fact about the reliability of the DCF model in practice is that it is applied universally in the bond market, i.e. the bond market is one of the those situations where our textbook theory works so well that it is the standard model used by all bond traders, buyers and sellers all the time. Now back to our example. Should you buy $250,000 of these bonds for your bank? It is now four years since the bonds were issued, and the remaining maturity of the bond is six more years. The fourth coupon payment has just been made, so there are six more coupon payments due, at the end of years 1 to 6. At the end of year 6 the bondholder will receive the $1,000 face value. But will we actually receive these payments? These payments are really just promises to pay, i.e. Glassblower promises to make these payments, and is legally obliged to make the payments if it has the money. But what if Glassblower is half way to going bust? To work out the fair value of the bonds we need to assess Glassblower’s financial strength, i.e. we need to assess the risk of the bonds. You have just taken a call from Marco Ravioli, a bond salesman from Skalliwag Brothers Plc an investment bank based in London active in the bond market. Ravioli tells you the bonds are currently trading at $930, that this is cheap, and that he can get you $250,000 of these bonds at this price. He is going to ring back in one hour. What do you tell him when he calls? The first thing you do is check on your Bloomberg screen for the current price at which Glassblower bonds are trading. The idea here is that people trading in the bond market are mainly institutional buyers and sellers, and that basically they know what they are doing. If the bonds are currently trading at the $930 price Ravioli claims, then that is what these smart investors believe they are worth. In that case $930 is probably pretty close to fair value. 3 Unfortunately Glassblower is a small company and its bonds do not trade much, and in fact there is no recent quoted price for these bonds. The problem you have is that there is not a large market for Glassblower bonds, as there is for, say, bonds issued by the UK or US governments. Also you do not trust Ravioli. Ravioli is trying to make money for Skalliwag, not for you. Also you realise that the company you represent is small. You are not one of Ravioli’s big clients, so if he rips you off on this deal he may not be too concerned that you will not be doing business with him again. The institutional bond market is not like the retail market. The government will not step in to bail you out if things go wrong. The rule of the institutional bond market is caveat emptor, buyer beware. Solution: Since there is no market price for Glassblower bonds we work out the fair value of the bonds ourselves by applying the DCF model. If you own one of Glassblower’s bonds this will entitle you to the following cash flows: End of Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Cash flow $100 $100 $100 $100 $100 $1,100 To find the value of these cash flows, i.e. how much the bond is worth you need to apply the discounted cash flow model in order to find the present value. The present value is the value of the cash flows right now, i.e. the value of the bond. Note that in finance the only thing we care about is cash flow, not profits. In finance we only care about profit figures if they help us to get a handle on the underlying cash flows. This is the rational approach. Profit is only a number that appears at the end of a profit and loss account, worked out by applying the rules of accounting. Over the long term total profits roughly equals total cash flows, but in any particular year the profit and cash flow generated by the firm can be very different. One of the main differences between finance and accounting is that in finance the focus is always on cash. The present value of a cash flow is less than the value of a cash flow in the future, i.e. a promise of $100 in one years time is not worth $100 hard cash today. There are three reasons for this. Firstly, a cash flow today can be spent today or invested today. Secondly, inflation will eat away the value of a future cash flow. If inflation is running at 5% you need $105 dollars next year in order to buy what could be bought today for only $100. Thirdly, a future cash flow is risky, in other words you might not get the $100 you are entitled to since Glassblower may go bust over the coming year. In the DCF model we reduce the value of the future cash flows by applying the appropriate discount rate. The discount rate takes into account the three reasons given above for why a promised $100 in the future is worth less than $100 today. This is how it is done. First of all we need to estimate the discount rate that is appropriate for Glassblower bonds. This involves making a judgement about the financial strength of Glassblower, since the risk of the bonds depends directly on the risk of the company that is to pay the cash flows on the bond, and the discount rate itself depends on the risk of these cash flows. Fortunately we do not have to do analyse the financial strength of Glassblower ourselves, because bond rating agencies like 4 Moody’s and Standard and Poors have already done it for us. Looking up the rating of Glassblower bonds you see that Moody’s currently give the Glassblower bonds a Ca rating. Ca sounds pretty good. However, according to Moody’s definition of Ca: ‘Bonds which are rated Ca represent obligations which are speculative in a high degree. Such issues are often in default or have other marked short-comings’. Clearly the financial strength of Glassblower has deteriorated since the bonds were first issued four years ago. Ca rated bonds are also known in the trade as junk bonds. You now look at the prices at which other Ca bonds are trading in the market. You find that Ca bonds maturing in around 6 years currently have discount rates of 20%, i.e. 20% is the discount rate appropriate to the level of risk of a Ca rated bond. So 20% is the discount rate you should use to find the present value of Glassblowers bonds. This is how we do DCF calculations: The value of one Glassblower bond = the present value of the promised cash flows discounted at 20% = $100/1.2 + $100/1.22 + $100/1.23 + $100/1.24 + $100/1.25 + $1,100/1.26 = $667.449. $667.449 is miles away from the $930 fair value Ravioli told you, the lowdown, lying, *%@*&!. You probably have a pretty good idea now what to say to Ravioli when he calls back. If you pay $667.449 for each bond rather than the $930 Ravioli asked you for, on a deal of $250,000 worth of bonds you save your bank from a loss of $70,364. (If you buy $250,000 of bonds at Ravioli’s price of $930 then you will get 268 bonds since $250,000/$930 = 268.8172 and you cannot buy fractions of a bond. For these 268 bonds you pay $249,240 = 268 $930. These 268 bonds are really only worth 268 $667.449 = $178,876. So your loss is $249,240 - $178,876 = $70,364). Discount rates and required rates of return There are quite are few different names used for discount rate in finance. The main ones are discount rate = rate of return = expected rate of return = required rate of return. In applications of DCF to the bond market we also use the terminology discount rate = interest rate = market interest rate = yield = market yield. In our course we will use discount rate most of the time. Another way to understand the discount rate in this example is to ask ourselves ‘what rate of return are we getting on our investment if we pay $667.449 for a Glassblower bond, and if Glassblower actually pays us the promised coupon payments and principle?’ The answer is we get a 20% annual return each year for six years. The cash flows we receive are lumpy, since we do not get exactly the same amount every year. 5 There is a big payoff at the end of year 6. In the first year we get only a 14.98% income return, since the $100 coupon payment after year 1 is just 14.98% of the $667.449 (0.1498 $667.449 = $100) we paid for the bond. However, at the end of year 1 the present value of the remaining cash flows has gone up, so in addition to the income gain we also receive a capital gain, i.e. the value of the bond goes up as we get closer and closer to the end of the bond’s life. Let’s check this. At the end of year 1 there are 5 more payments to come. End of Year 2 Year 3 Year 4 Year 5 Year 6 Cash flow $100 $100 $100 $100 $1,100 The value of the bond at the end of year 1 is now the present value of the remaining future cash flows, i.e. the value of the bond at the end of year 1 = $100/1.2 + $100/1.22 + $100/1.23 + $100/1.24 + $1,100/1.25 = $700.94. This is an increase of $33.49 (= $700.94 $667.449). The total return after 1 year therefore = income gain + capital gain = $100 coupon + $33.49 increase in bond value = $133.49. Now $133.49 is indeed a 20% return on our investment of $667.449 (0.20 $667.449 = $133.49). So, the discount rate of 20% is the exactly the same thing as the rate of return we require for taking on the risk of owning Glassblower bonds. Discounted cash flow calculations can be done either using a calculator or by using tables. Note that in calculations we always use decimal numbers rather than percentage figures, so that we use 0.20 not 20% in the calculation (0.20 = 20/100). However, when we have a series of cash flows that are all the same we can do the calculation more quickly using tables. We do this in the lecture. Using tables to do the present value calculation Example 2 (share valuation): Sears, Roebuck and Company is a major retailer. Use the DCF model and the information given below to work out the fair value of one share of Sears. 6 If you buy and hold a share the only cash flows you receive will be the series of dividend payments D1, D2, D3, ... paid out by the firm. The value of the share is just the present value of this series of dividend payments. So to find the value of the share you need to discount the dividend payments to find their present value using the appropriate discount rate. There are two problems, estimating the dividend payments and estimating the discount rate. Both of these problems are difficult in practice. ESTIMATING THE DIVIDENDS The following simple approach is used by many professional analysts. To use this approach you make the following assumption: The dividend payments will grow at a constant rate g. This assumption is more reasonable for some companies than others. It is a reasonable assumption for stable companies such as Sears, Roebuck and Company, but not so good for a high-growth company such as a high-tech computer software manufacturer. Suppose you work out that Sears dividends will grow at a rate of around 6% per year into the foreseeable future. Suppose also that you estimate that Sears next dividend will be $2.00 per share to be paid in one years time. Then the dividend payments you will receive will be $2.00 in one years time, $2.12 (= $1.00 1.06) in two years time, $2.2472 (= $2.00 1.062) in three years time, and so on forever. To find the present value you now need to discount these payments at the right discount rate. Assume that the discount rate is 12%. (Later in the course we will see how to estimate discount rates for shares using the Capital Asset Pricing Model. Until then discount rates will just be given to you as part of the information in the examples we discuss). Then the present value of the dividend stream is: D1/1.12 + D2/1.122 + D3/1.123 + ... = $2.00/1.12 + $2.12/1.122 + $2.2472/1.123 + ... Now how do we discount an infinite stream of cash flows? The answer is to use the Gordon Growth Model, also called the Gordon Growth Formula, which gives us a formula for calculating present values of a constantly growing series of cash payments. The Gordon Growth Formula is: P0 = Present Value = D1/(k - g), where D1 is the next dividend which we receive in one periods time, k is the discount rate and g is the growth rate. Applying the formula to Sears stock gives us: Value of one share = present value of dividends = $2.00/(0.12 - 0.06) = $2.00/0.06 = $33.33. To apply the Gordon Growth Formula we need estimates for D1, k and g. ESTIMATING D1 and g. Estimating D1 is not too difficult. Sears is a stable company which usually tries to pay dividends in a predictable manner. Many stable companies have a fairly predictable payout ratio. The payout ratio is the proportion of accounting earnings paid out as dividends. Suppose in the past Sears payout ratio is 0.6. So nearly every year Sears pays out 60% of its accounting profits as 7 dividends. All we need to do now is to estimate next years profits. This is not too difficult for large, well-known and stable companies. Financial analysts spend a lot of time guessing next years profits of blue chip stocks. Blue chip companies are also helpful in providing quarterly earnings reports and other information through their public relations departments. They also issue profits warnings if they believe they will not be able to meet the markets expectations for the reported end of year earnings. So if we estimate next years dividend to be $2.00 per share this estimate will probably be quite reasonable. Now we need to estimate g. Again this is not impossible for stable companies. Here is one common way of doing it. Looking back over the last ten years we see that Sears return of equity has been 15%. That is most years Accounting Profits/Book Value of Equity has been 15%. Since Sears is a stable company we think that the future will be the same as the past. Since Sears pays out 60% of profits that means 40% of profits will be ploughed back into the company for investment. So the ploughback ratio is 40%. Since Sears will get a 15% return on profits ploughed back, this means that profits will grow at 15% x 40% every year. Therefore dividends will also grow at about 15% x 40% every year. The formula is: g = growth rate of dividends = ploughback ratio x return on equity. Return on equity is given by the formula: Return on equity = Accounting profits/Book value of equity. We now have D1 and g. To use the Gordon Growth Formula, P0 = D1/(k - g), we need also to have an estimate for the discount rate k. We know that k depends on the level of risk of the dividends. But what we want is a technique that can actually allow us to put a precise numerical value on k. At the moment we do not have such a technique. Later in the course we will discuss the CAPM, the Capital Asset Pricing Model. CAPM will give us a simple formula for estimating discount rates. To understand how difficult the problem of estimating k really is, you need to know that CAPM is one of the most important discoveries in the whole of finance. The discoverer of CAPM, William Sharpe was awarded the Nobel Prize for his work in finance. In this example we have assumed that k is 12%. The Gordon Growth Model therefore gives us an estimate for the value of the share as P0 = $2.00/(0.12 - 0.06) = $33.33. Should we trust this answer of $33.33? For example, suppose Sear's shares are actually trading in the market right now at $25.00 per share. Does this mean we should buy the shares at $25 because we have worked out they are really worth $33.33? No. If the calculated price of $33.33 is different from the price at which Sears shares are trading in the market it is more likely that our estimates of k and g are wrong than that all the buyers and sellers in the market are wrong. Probably the best way to estimate the fair value of Sear's shares is to use the current market price. For some shares however, there is no market price. For example, when a new company is floated on the stock market, or when a government owned enterprise is privatised. In that case we need a model to tell us at what price the stock should be offered for sale in the market. The discounted cash flow model is the best model available. Unfortunately it very often gives the wrong answer. There is nothing wrong with the model; the problem is that it is very difficult to estimate future dividends and to estimate the discount rate that appropriately reflects the level of risk. 8 Note: This is why the DCF model is not used so much in valuing stocks in practice. Example 3 (capital budgeting) We can use the same present value approach to decide which capital investments a company should undertake. Note: Corporate finance is the other main area where the DCF model is used all the time. DCF is the standard project evaluation technique used by nearly all large UK listed companies. You will see more applications of the DCF model to the corporate investment decision later in the course when we focus on corporate finance in the last part of the unit. You are the financial executive of Mears, Mutjack and Co. Mears is a shirt manufacturer. The company wishes to replace its cutting machine with a new machine. The new machine should be more efficient than the old machine and is expected to result in after tax cash savings of $10,000 per year for the next five years, five years being the expected life of the new machine. The new machine will cost $40,000. Should you buy the new machine? Buying the new machine is really just the same as buying a bond or a share. To get the cash flows of $10,000 per year for the next five years you have to pay $40,000 now. If the present value of the series of $10,000 cash flows is more than the cost of $40,000 you should buy the machine. If it is less you should not. So all you need to do is to discount these cash flows back at the appropriate discount rate. What is the right discount rate? Later in the course we will show you a standard technique called the Capital Asset Pricing Model for estimating the appropriate discount rate. Until then discount rates will just be given to you as part of the example. Suppose the discount rate on Mears shares is 11%. Then the value of the cash flows is: $10,000/1.11 + $10,000/1.112 + ... + $10,000/1.115 = $36,959. This is less than the cost of the machine, i.e. the present value of the cash savings you get from the machine is less than what you have to pay out to acquire the machine. So you should not go ahead with the purchase. Summary of the discounted cash flow model The general rule for applying the DCF model is this. To find the value of any asset go through the following steps: STEP 1: Estimate the CASH FLOWS the asset will generate. STEP 2: Estimate the DISCOUNT RATE. The discount rate will depend on the risk level of the cash flows. The greater the risk, the greater the discount rate. Later on we will develop the CAPM which gives us the theoretically correct technique for calculating the discount rate. Until then just remember that the discount rate is used as a measure of the level of risk of the asset, and that the higher the level of risk the 9 greater is the discount rate. This makes sense, the higher the discount rate the lower is the present value of the cash flows, i.e. a risky asset is worth less than a safe one. Also, the discount rate is the same thing as the required rate of return on an asset, i.e. if we buy a risky asset we want a greater rate of return to compensate for taking a greater risk. STEP 3: Find the PRESENT VALUE of the cash flows. The present value is the value of the asset. The model can be used to find the value of any asset. It is used to value bonds, shares and capital projects. Bonds Value of a bond = present value of cash flows generated by the bond = PV = C/(1 + k) + C/(1 + k)2 + ... + (C + F)/(1+ k)n C = the coupon payments F = the face value n = the number of periods. So if coupon payments are made annually the maturity of the bond is n years. k = the discount rate. The discount rate reflects the riskiness of the cash flows. The higher the risk level of the bond the higher k will be. In bond terminology k is also called the interest rate and the yield on the bond. PV = the present value of the cash flows. So PV is the fair market value of the bond. In a rational market like in the textbook world the bond should sell for its present value. This method of valuing bonds is used literally all the time in the real world. It is the standard technique that everyone in the bond market uses all the time. Shares Value of a share = D1/(1 + k) + D2/(1 + k)2 + D3/(1 + k)3 + D4/(1 + k)4 + ... D1 = the expected dividend at the end of period 1. Di = the expected dividend at the end of period i (we assume here that the next dividend to be paid will be paid at the end of year 1). k = the discount rate. The discount rate reflects the level of risk of the expected dividend payments. So k will be high for a risky stock such as a construction company. k will be low for a safe company such as a utility. If we estimate that the dividends will grow at a constant rate of g (as a decimal number) then the Gordon Growth Model gives us the present value of the share as D1/(k g). 10 The DCF method is not used so much for valuing stocks in the real world, and when it is used it tends to be used in conjunction with other methods. This is because we have less confidence in our estimates for the future dividends and for the discount rate for stocks than for the cash flows and discount rates for bonds. The present value is very sensitive to small changes in the discount rate and the estimated growth rate of dividends so DCF applied to stocks tends to give us only a ballpark figure. Capital projects Net Present value = NPV = CF0 + CF1/(1 + k) + CF2/(1 + k)2+ ... + CFn/(1 + k)n CF0 = the expected cash flow at the start of the project. Usually this will be negative. CF1, …, CFn are the expected cash flows over the life of the project. k = the discount rate which reflects the risk level of the project. NPV = the net present value of the project. NPV is the total value of the project, net of all costs. If NPV is positive we are getting something for nothing. So a positive NPV means that we should proceed with the project, (assuming we believe the estimates for the expected cash flows and the discount rate). If NPV is negative we should not proceed with the project. The DCF model is used all the time in corporate finance in evaluating the firm’s capital investment projects. The discounted cash flow asset valuation model is the most important technique in modern finance. To repeat the general rule: General rule To find the value of ANY asset (1) Estimate the future cash flows generated by the asset (2) Find the appropriate discount rate (3) Find the present value of the expect cash flows THE PRESENT VALUE OF THE EXPECTED CASH FLOWS IS THE VALUE OF THE ASSET THE DISCOUNT RATE IS DETERMINED BY THE RISK Required: Use present and future value tables or a calculator to answer the following questions. Note: Some of these questions are quite tricky, even for those who are good at this sort of thing. Generally the questions get harder as you go on. 11 Don’t panic! To understand the applications of discounted cash flow to the rest of the course, it is only the very basic DCF calculations you need to be able to do. However, try to do some of the tougher questions as well, and come along to the seminars with any questions or problems you had. See Arief if you get really stuck on this material. (1) You are considering depositing £300 in an account for eight years with an interest rate of 10%. The present value in this case is £__________. A. 300 B. 567.31 C. 578.99 D. 612.13 E. 643.08 (2) The value of £300 in eight years if it earns 10% compounded annually will be £___________. A. 300 B. 567.31 C. 578.99 D. 612.13 E. 643.08 (3) You borrow £2,000 from a loan shark for six years at an interest rate of 75% per year with annual compounding. The future value of the loan after six years will be £__________. A. 46,786.87 B. 57,445.80 C. 59,786.97 D. 61,777.34 E. 63,796.46 (4) Suppose you convince the shark to charge simple interest instead of compound interest. In this case the future value of the loan will be £__________. A. 11,000 B. 9,000 C. 24,500 D. 6,000 E. 2,600 (5) A customer deposits £1,000 in a savings account earning 14%. The value of the account in one year if the bank uses semi-annual compounding will be £__________. A. 1,144.90 B. 1,155.87 C. 1,167.13 12 D. 1,186.26 E. 1,998.10 (6) Suppose the bank uses quarterly compounding instead. In this case the value of the account in one year will be £__________. A. 1,098.23 B. 1,112.34 C. 1,127.91 D. 1,147.52 E. 1,458.30 (7) The future value of £2,000 invested for eight years at 12% with monthly compounding is £__________. A. 2,500 B. 3,300 C. 4,456.78 D. 5,665.13 E. 5,198.55 (8) The annual yield of an investment, or the __________ rate of interest is the simple interest equivalent to a stated rate that is compounded. A. direct B. effective C. respective D. correspondent E. basic (9) The present value of £200,000 to be paid in year 10 if the discount rate is 8% will be £__________. A. 92,639 B. 92,889 C. 93,882 D. 94,234 E. 95,124 (10) The present value of £10,000 to be paid in four years if interest rates are 10% compounded quarterly is £__________. A 6,736.25 B. 6,123.98 C. 6,856.29 D. 7,012.99 E. 7,234.86 (11) Find the present value factor for an interest rate of 15% and 10 periods. 13 A. 0.2472 B. 0.2657 C. 0.3561 D. 0.4479 E. 0.5960 (12) If one speaks of an annuity without any qualification, a (an) __________ is being discussed. A. simple annuity B. first annuity C. time annuity D. annuity due E. real annuity (13) You plan to save £1,700 per year for 24 years as a retirement fund and expect to earn 14% per year on all invested funds. The first payment is made at the end of this year, year 1, and the last payment is made at the end of year 24. You will have accumulated £__________ at the end of 24 years. A. 34,567 B. 38,990 C. 112,453 D. 214,509 E. 269,720 (14) You want to create an account from which you can withdraw £550 per month for 20 months. Assuming that the withdrawals are to start one month from now and that interest rates are 1% per month, you have to put £__________ into the account today. A. 2,318 B. 9,925 C. 23,180 D. 25,201 E. 25,234 (15) A five year annuity of £1,000 costs £4,000. The implied rate of interest is between __________ and __________ %. A. 1, 2 B, 4, 5 C. 7, 8 D. 10, 11 E. 13, 14 (16) Consider a lump sum of £900,000 paid into an account that is to be used to generate an eight year annuity. With interest rates of 10%, the annuity payment can be as large as £__________. 14 A. 132,897.12 B. 145,664.58 C. 168,700.44 D. 198,024.33 E. 201,502.17 (17) Assume a retiree has £90,000 and needs £12,000 per year to live. If interest rates are 8% per year, how long will the funds last? A. 11 years (with a little left over) B. 14 years C. 16 years (with a little left over) D. 23 years E. 25 years (18) The future value of a four year annuity of £1,200 with interest rates of 10% per year is £__________. A. 3,310 B. 3,240 C. 3,398 D. 5,569 E. 5,688 (19) The future value of an annuity of £85 for 17 periods, with an interest rate of 12% is £__________. A. 3,209.45 B. 4,155.14 C. 5,938.30 D. 5,839.09 E. 6,938.22 (20) The present value of an annuity of £40 for 25 periods, with an interest rate of 14% is £__________. A. 274.92 B. 394.02 C. 422.39 D. 551.27 E. 612.34 Question 2 Use the discounted cash flow model to find the value of a bond with the following characteristics. 15 The bond has an annual coupon of 12%, a face value of £100 and a maturity of 10 years. The market interest rate for bonds of this level of risk is 8%. Question 3 A share in ABC Company is expected to pay a dividend of 20 pence in one years time. The company is expected to maintain this level of dividend payment for the foreseeable future. The discount rate for the firm's equity is 18%. Estimate the value of one share of ABC. Question 4 As for question 2 except that now the dividends are expected to grow at a rate of 5% per year forever. Question 5 As for question 3, except that the dividends are expected to grow at a rate of 20% over years 2 and 3 and at 5% from year 4 onwards. Question 6 Bull plc has a dividend payout ratio of 70%. The next dividend to be paid in one year’s time is expected to be $1.50. Over the past ten years Bull's return on book value of equity has averaged 18%. Assume that Bull has a required return on equity of 10%; i.e. assume that investors require a 10% rate of return as compensation for taking on the risk of investing in Bull's shares. Use this information to estimate Bull's share price. Present value of £1 receivable in n years time Years Discount rate as a percentage n 1 2 3 4 5 6 7 8 9 10 1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091 2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 0.8573 0.8417 0.8264 3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 0.7938 0.7722 0.7513 4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 0.7350 0.7084 0.6830 5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 0.6806 0.6499 0.6209 6 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 0.6302 0.5963 0.5645 7 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 0.5835 0.5470 0.5132 8 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 0.5403 0.5019 0.4665 9 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 0.5002 0.4604 0.4241 10 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 0.4632 0.4224 0.3855 11 12 13 14 15 16 17 18 19 20 1 0.9009 0.8929 0.8850 0.8772 0.8696 0.8621 0.8547 0.8475 0.8403 0.8333 2 0.8116 0.7972 0.7831 0.7695 0.7561 0.7432 0.7305 0.7182 0.7062 0.6944 3 0.7312 0.7118 0.6931 0.6750 0.6575 0.6407 0.6244 0.6086 0.5934 0.5787 4 0.6587 0.6355 0.6133 0.5921 0.5718 0.5523 0.5337 0.5158 0.4987 0.4823 5 0.5935 0.5674 0.5428 0.5194 0.4972 0.4761 0.4561 0.4371 0.4190 0.4019 16 6 0.5346 0.5066 0.4803 0.4556 0.4323 0.4104 0.3898 0.3704 0.3521 0.3349 7 0.4817 0.4523 0.4251 0.3996 0.3759 0.3538 0.3332 0.3139 0.2959 0.2791 8 0.4339 0.4039 0.3762 0.3506 0.3269 0.3050 0.2848 0.2660 0.2487 0.2326 9 0.3909 0.3606 0.3329 0.3075 0.2843 0.2630 0.2434 0.2255 0.2090 0.1938 10 0.3522 0.3220 0.2946 0.2697 0.2472 0.2267 0.2080 0.1911 0.1756 0.1615 Present value of an annuity of £1 payable at the end of each of n years Years Discount rate as a percentage n 1 2 3 4 5 6 7 8 9 10 1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091 2 1.9704 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355 3 2.9410 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869 4 3.9020 3.8077 3.7171 3.6299 3.5460 3.4651 3.3872 3.3121 3.2397 3.1699 5 4.8534 4.7135 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908 6 5.7955 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553 7 6.7282 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684 8 7.6517 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5348 5.3349 9 8.5660 8.1622 7.7861 7.4353 7.1078 6.8017 6.5152 6.2469 5.9952 5.7590 10 9.4713 8.9826 8.5302 8.1109 7.7217 7.3601 7.0236 6.7101 6.4177 6.1446 11 12 13 14 15 16 17 18 19 20 1 0.9009 0.8929 0.8850 0.8772 0.8696 0.8621 0.8547 0.8475 0.8403 0.8333 2 1.7125 1.6901 1.6681 1.6467 1.6257 1.6052 1.5852 1.5656 1.5465 1.5278 3 2.4437 2.4018 2.3612 2.3216 2.2832 2.2459 2.2096 2.1743 2.1399 2.1065 4 3.1024 3.0373 2.9745 2.9137 2.8550 2.7982 2.7432 2.6901 2.6386 2.5887 5 3.6959 3.6048 3.5172 3.4331 3.3522 3.2743 3.1993 3.1272 3.0576 2.9906 6 4.2305 4.1114 3.9975 3.8887 3.7845 3.6847 3.5892 3.4976 3.4098 3.3255 7 4.7122 4.5638 4.4226 4.2883 4.1604 4.0386 3.9224 3.8115 3.7057 3.6046 8 5.1461 4.9676 4.7988 4.6389 4.4873 4.3436 4.2072 4.0776 3.9544 3.8372 9 5.5370 5.3282 5.1317 4.9464 4.7716 4.6065 4.4506 4.3030 4.1633 4.0310 10 5.8892 5.6502 5.4262 5.2161 5.0188 4.8332 4.6586 4.4941 4.3389 4.1925 17