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Lecture 3: Financial Math & Cash Flow Valuation. C. L. Mattoli. Intro. This week we cover topics in chapters 4 & 5 of the textbook.

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Lecture 3: Financial Math & Cash Flow Valuation


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    1. Lecture 3: Financial Math& Cash Flow Valuation C. L. Mattoli (C) 2008 Red Hill Capital Corp., Delaware, USA

    2. Intro • This week we cover topics in chapters 4 & 5 of the textbook. • One of the most basic principles in finance is that a dollar, now, is worth more than a dollar, received later. Thus, money actually has a time value. • If you have money, now, you could put it into a bank account, earn interest, and have more money, later (future value). (C) 2008 Red Hill Capital Corp., Delaware, USA

    3. Intro • If you get money, later, you will have lost the opportunity to invest (opportunity cost). • In investment, we invest money, now, to get cash flows, in the future, whether we invest in equipment to make noodles or we buy the securities of companies. (C) 2008 Red Hill Capital Corp., Delaware, USA

    4. What is Investment???? • Although you may have heard your friends and the news talk about investment, what really is investment? • Common things that you think of are stocks and real estate, but still what is an investment about? • Take stocks, for example. You buy a share of stock of China Telecom. You pay money now (PV) for a piece of paper from China Telecom. (C) 2008 Red Hill Capital Corp., Delaware, USA

    5. What is Investment???? • For your present outlay, you are expecting future money. You might get cash dividend payments, once or twice a year. Plus you will get a cash inflow when you later, in the future, sell the stock. • The difference, by the way, between the price you buy the stock at and the price you sell it for, Plater – Pnow = Capital gain or loss, depending on if the price that you sell it at is higher or lower than when you bought the stock. (C) 2008 Red Hill Capital Corp., Delaware, USA

    6. What is Investment???? • The same applies to buying real estate, like a house, an apartment, or an office complex. You will hope to get a capital gain when you sell it, and, in the mean time, you can rent it out and get periodic future cash flows from people paying their rents. • Buy a noodle machine and make noodles to sell: you pay money out for the noodle machine and space to house the business. You make an investment, in the noodle maker and some flour, eggs, water, and salt, and you make noodles to sell at the market, every day. (C) 2008 Red Hill Capital Corp., Delaware, USA

    7. What is Investment???? • Again, you have a present outlay of money, you get future cash flows from the sale of your noodles, and you expect to come out, somehow, ahead. • The thing is: you pay out money, now, instead of putting it in bank and earning interest, and you want to be ahead of the future money that you would get for putting your money in bank instead of making some other kind of investment. (C) 2008 Red Hill Capital Corp., Delaware, USA

    8. What is Investment???? • At this point you should be beginning to perceive the problem. Money, at different times, will, somehow, not have the same value. • Therefore, finance asks the question: what is money, received later, worth to us, right now (present value of future cash flow). • These themes will be one of the foundations for further study, in the course. So, be sure that you understand them before you go on. (C) 2008 Red Hill Capital Corp., Delaware, USA

    9. Future Value (C) 2008 Red Hill Capital Corp., Delaware, USA

    10. Future Value: Simple Interest • In finance, we talk about (percentage) rates of return, and usually, annual percentage rates (APR) of return. • Interest (and interest rates) on savings in a bank is an example of a rate of return. • In that regard, if I put $100 in a bank account that earns a 10%/year interest rate of return, then, I will earn 10% of that $100, in a year, or 10%$100 = $10. (C) 2008 Red Hill Capital Corp., Delaware, USA

    11. Future Value: Simple Interest • Therefore, at the end of a year, I will have $110, in bank = $100, original principal, plus $10, interest earned on the principal at 10%/year. • We can put this into a simple equation form as FV1 = P + r P = P(1 + r), where r is the annual rate of return = interest rate, in this case. (C) 2008 Red Hill Capital Corp., Delaware, USA

    12. Future Value: Simple Interest • We have used the notation, FV1, to indicate that this is the value of your savings account, (1 year) in the future, and we call it the future value. • If you earn interest on that principal for n years, where n can be > 1 or n < 1, then, you will get rP for n years, or nrP. (C) 2008 Red Hill Capital Corp., Delaware, USA

    13. Future Value: Simple Interest • The simple interest equation becomes FVn = P + nrP • We collect terms inside a parentheses and get FVn= P(1 +nr). • In n years, you will have P(1+nr) dollars in future value. (C) 2008 Red Hill Capital Corp., Delaware, USA

    14. Future Value: Compounded • More common, actually, than simple interest is compound interest. • If I put $100 (P) in bank for a year, at the end of a year, I will have $110 [FV = P(1+1r)], in the bank. • If I leave that money, in the bank, I will earn interest on the whole thing, which, after the first year, is P + rP = P(1+r). (C) 2008 Red Hill Capital Corp., Delaware, USA

    15. Future Value: Compounded • At the end of 2 years, I will have FV2 = $110(1+10%) = $121 = $100(1+10%)*(1+10%) = $100(1+10%)2 = P(1+r)2. • What has happened is that we have earned interest, not only on the original P that I put in the bank, but also on the interest that we earned in the previous year. (C) 2008 Red Hill Capital Corp., Delaware, USA

    16. Future Value: Compounded • We can see also play around a little with the equation to understand what it is telling us. • So, take FV = [$100+$10](1+10%) = $100 + $10 + $10 + $1 = P+rP+rP+Pr2 = P+2rP+Pr2 = [P(1+2r)]+[(Pr)r]. • The first part of the equation, [P(1+2r)], is the amount that you would have gotten, if you were only getting simple interest. (C) 2008 Red Hill Capital Corp., Delaware, USA

    17. Future Value: Compounded • The second term, [(Pr)r], represents interest, r, on the interest $’s, rP, that you earned in the first year. • This is referred to as compounding, and you earn compound interest on your principal. • For any number of years, n, the future value equation with compounding of interest is given by FVn = P(1+r)n. (C) 2008 Red Hill Capital Corp., Delaware, USA

    18. Simple vs. Compound • You earn more money with compounding. Shown:the value of money versus time into the future for simple interest and compounding. Compound $ Simple N = years (C) 2008 Red Hill Capital Corp., Delaware, USA

    19. Present Value: the Inverse of Future Value (C) 2008 Red Hill Capital Corp., Delaware, USA

    20. Intro • If you have money, now, you will have a larger amount, in the future, because you can put it in bank or some other investment. • If you have $100, now, and your opportunity to invest, a rate that you can get from an investment, is 10%/year, a year from now it will be $110. (C) 2008 Red Hill Capital Corp., Delaware, USA

    21. Intro • Thus, $110, a year from now, has a value, now, it’s present value, of $100 because if You put $100 in bank now it would be $100, in a year from now. • We wrote, before: FV = (1+r)PV, and we just reverse that equation to get: PV = FV/(1+r), the present value of an amount of money that we would get in the future, a year from now. (C) 2008 Red Hill Capital Corp., Delaware, USA

    22. Intro • The PV of $110, received in a year, is PV = FV/(1+r) = $110/(1+10%) =$100. • We already knew that should be the answer because we knew that if we put $100 in bank, now, at 10%/year, we would have had $110, a year from now. • We refer to this as discounting a future amount back to the present. • PV and FV are just inverse relations of each other. (C) 2008 Red Hill Capital Corp., Delaware, USA

    23. PV of any future payment • Assuming that interest is compounded, the PV of an amount of money, FVn, that will be received n years from now, is PV = FVn/(1+r)n. • Again, all that we have done is to reverse the future value equation: PV = FVn/(1+r)n (C) 2008 Red Hill Capital Corp., Delaware, USA

    24. PV of any future payment • We usually refer to r as the discount rate or the required rate of return (RRR), and the PV is called discounted future cash flow. • It just tells us what a future cash flow is worth to us, today, given that we could invest (opportunity lost; opportunity cost) it, if we had it now,and earn r rate of return compounded to that future time. (C) 2008 Red Hill Capital Corp., Delaware, USA

    25. The language of interest rates • Interest rates are normally quoted up to two decimal places, e.g., 4.03% (= in numbers 0.0403). • The decimal places, which represent hundredths of percents, e.g., the above rate is 4 and 3 one-hundredths percent, are called basis points (bp). • Thus 0.03% is commonly called 3 basis points. (C) 2008 Red Hill Capital Corp., Delaware, USA

    26. The language of interest rates • Then, you might hear an interest rate quote, like 30 bp over the BAB rate. So, if the BAB rate is 4.50%, the quote would mean 4.50% + 30bp = 4.50% + 0.30% = 4.80% • Bank accepted bills (BAB’s) = a type of short-term corporate debt paper, called a bill, like a dollar bill, and payment is guaranteed by a bank. (C) 2008 Red Hill Capital Corp., Delaware, USA

    27. Time value • One thing that you should notice from this simple equation is that the value, now, of a future payment of money decreases, the further into the future that we receive it, i.e., as n gets larger. • The other important thing that you should notice is that, for given n, the PV decreases as r increases. Thus, we say that the PV and the discount rate are inversely related. As r increase, PV decreases, and vice versa. (C) 2008 Red Hill Capital Corp., Delaware, USA

    28. Time value • Often, in using the PV concept to value future money, we will know FV and n, and we will have to decide on an appropriate discount rate, r, to value the future payment, in the present. • As we shall learn, that will be based on opportunity to invest. • After all, the PV concept is based on the fact that, if we have money now instead of getting it in the future, we could invest it. (C) 2008 Red Hill Capital Corp., Delaware, USA

    29. Time value • Different people may, in fact, have different required rates of return for the same situation. • We shall also see that for other situations, we can read the market’s required rate of return for certain types of future cash flows. • In the next slide we show plots of PV vs. n for various r’s. • Various symbols are used for r, like k or j. (C) 2008 Red Hill Capital Corp., Delaware, USA

    30. Time value vs. n for varied r 0% 5% 10% 15% (C) 2008 Red Hill Capital Corp., Delaware, USA

    31. In Truth: Consumption vs. Saving • People give up buying things, now, to save or invest money, and get money in the future. • There is a reason that people should not just put the money under their mattress and hide it. • There is always inflation, in the world. • Inflation means that the $10 that you have now will buy you less in the future because the prices of everything are going up. (C) 2008 Red Hill Capital Corp., Delaware, USA

    32. In Truth: Consumption vs. Saving • If inflation is, for example, 10% per year, that means that $10 will only buy $10/(1 + 10%) of the same goods and services, next year. • Thus, if someone wants to not consume, now, they will want to consume as much or more, in the future. • That means that people will set a base rate for lending or investing their money equal to at least inflation. • As a result, we have a beginning gauge to see how much people will require for returns. (C) 2008 Red Hill Capital Corp., Delaware, USA

    33. Real life example: commercial bills • A commercial bill (CB) is a ST debt security that is issued by corporations. • A CB is called a discount security because it pays no formal interest payment, but, instead, it pays only its face value (FV) at maturity. • In order for someone to make money on their investment, therefore, they buy it at a discount (less than) face value, and their profit will be the difference between what they paid for it (PV) and what they get at the end: FV. (C) 2008 Red Hill Capital Corp., Delaware, USA

    34. Real life example: commercial bills • We can use this example security to study the PV/FV concept. • What follows is an example of how we can use thoughts and words to make up an equation. • The return from investment is equal to the money that you earn, E, from the investment. • The rate of return, r, from investment is the earnings, E, divided by the original investment, PV, at time = 0. (C) 2008 Red Hill Capital Corp., Delaware, USA

    35. Real life example: commercial bills • In general return can have two parts, actual intermediate time cash flow income, INCi, and the final profit or loss, the capital gain/loss, from selling, at a future date. • CGL, the capital gain or loss is just FV – PV, the difference between what you bought it at, PV, and what you sold it for, PV. • Thus the general rate of return equation will be ROI = [INC + CGL]/PV. (C) 2008 Red Hill Capital Corp., Delaware, USA

    36. Real life example: commercial bills • Then, for the CB, which has no INC term, but only a gain on sale at the end maturity, the rate of return on investment will be (FV – PV)/PV, the profit divided by the initial investment. • In finance, we usually like to look at rates of return for a one-year period. That is just, again, so that we can compare one return to another (remember, finance is a comparative science). (C) 2008 Red Hill Capital Corp., Delaware, USA

    37. Real life example: commercial bills • What we have found, above, in the CB equation, so far, is a returnover the period from the purchase date to expiration. • A return earned over the [period of time that you hold an investment is referred to as a holding period return (HPR). • So, assume that our CB matures d days from now. Then, we make our rate of return in d days. (C) 2008 Red Hill Capital Corp., Delaware, USA

    38. Real life example: commercial bills • To convert our HPR into an annual rate of return (APR = annual percentage return or rate), first we divided the HPR byd and get a daily rate of return. Then, we multiply the daily rate of return by 365 days to get an APR return, r. • Thus, converting the words, in the last point, into equation, we find r = annual rate = [(HPR for d days)/days = daily rate] x 365 days/year = annual amount of daily return = {[(FV – PV)/PV]/d}365 = [FV/PV – 1](365/d). (C) 2008 Red Hill Capital Corp., Delaware, USA

    39. Real life example: commercial bills • Thus, we have started from first principles, just knowing what we want, in words, and making that into a solvable equation. • That is how the problems in this course will be. You will be given, mostly, word problems, and you will have to find which equation will help you solve the problem. • We know FV, so we will usually be solving for either PV, given an r, or r, given a price, PV, that you paid for it. (C) 2008 Red Hill Capital Corp., Delaware, USA

    40. Real life example: commercial bills • Rearranging the equation, we get PV = FV/[1+r(d/365)]. • That is just the simple interest PV equation, PV = FV/[1 + nr), with n = d/365. • Commonly, the interest rate on a CB (and other debt securities) is called an annual yield or yield-to-maturity (YTM). • For example, assume that you purchase a $1,000 FV CB maturing (coming due for pay off) in 90 days, and your annual RRR is 8%. What is the price that you should pay for it? (C) 2008 Red Hill Capital Corp., Delaware, USA

    41. Real life example: commercial bills • PV = $1000/(1+(90/365)8%) = $980.66. • That is the amount that you should pay if you want to earn (RRR) 8% annual return. • In fact, in this simple case of CB, we can actually solve the equation for the YTM, by rearranging, again. • Solving, r = [FV/PV – 1]x365/d gives us the YTM, if we know PV and d. • Assume, for example, that a 180-day $1000 FV CB is priced at $900. What is its annual yield? (C) 2008 Red Hill Capital Corp., Delaware, USA

    42. Real life example: commercial bills • We have $900 = $1000/(1+(180/365)r). • Solving for r, we get r = [($1000/$900) – 1](365/180) = 0.2253 = 22.53%. • Thus, the annual yield is 22.5%. • Note: CB’s and other debt instruments are usually quoted on yield, not the actual dollar price. • Thus, if you call a broker to get the price of a CB, he will not say $950, he will say 8%. (C) 2008 Red Hill Capital Corp., Delaware, USA

    43. YTM more general concept. • In the CB example, we said that the rate earned is called the YTM. • For more general investments, we can talk of the YTM isa an “average” rate of return over the life of an investment. • Thus, if an investment earns r1, one year, and r2, another year, the average YTM rate would be: FV = PV(1 + r1)(1 + r2) = PV(1 + rYTM)2. • A similar definition will apply when we have more than one cash flow. (C) 2008 Red Hill Capital Corp., Delaware, USA

    44. BAB’s • A major form of short-term money market debt paper issued by companies is called commercial bills. • Commercial bills can be guaranteed by a bank. • Then, they are called bank accepted bills (BAB’s). • Since the bank is liable for payment, BAB’s trade on the credit rating of the bank, not the underlying company issuer. (C) 2008 Red Hill Capital Corp., Delaware, USA

    45. BAB’s • Because they are guaranteed by a bank, BAB’s are the only security that is considered to be intermediated (financing which involves an intermediary, like a bank) rather than direct financing, getting money directly from investors by selling paper, which is what a BAB really is. • However, even though the company issues securities to investors, since the bank is involved, it is strictly considered intermediated finance. • That is often a trick question in exams. (C) 2008 Red Hill Capital Corp., Delaware, USA

    46. How many days in a year? • Again, this might sound like a stupid question, but actually it is a trick question. • Australia uses 365 days in a year to calculate yields, which seems logical, and you might ask, what else would it be. (C) 2008 Red Hill Capital Corp., Delaware, USA

    47. How many days in a year? • The answer is that the U.S. and many other countries use 360 days (30 days per month x 12 months) in their quotes and calculations of yields. • Thus, if you get a quote on a CB of 8%, in Australia, it will not have the same meaning, or value, as an 8% quote in the U.S. (C) 2008 Red Hill Capital Corp., Delaware, USA

    48. Need 3 to get the 4th • There are 4 variables in FV/PV equation: PV, FV, r, and n. • So, we need to know 3 to get the 4th. • Thus, in problems, you will be given enough information to figure out 3 variables, and you will be required to calculate the one unknown. (C) 2008 Red Hill Capital Corp., Delaware, USA

    49. Need 3 to get the 4th • We might know PV, n, and r, then, we can get FV. If we know FV, n and r, we can find PV. Etc. • So, if we are told that we will receive $1,000, a year from now, and that, if we had money, we could earn 10%/year over the next year, then we know FV, r, and n, and we can solve for PV = $1,000/(1.1) = $909.09. (C) 2008 Red Hill Capital Corp., Delaware, USA

    50. Need 3 to get the 4th • The meaning of that is that $1,000, a year from now, is worth only 909.09 to us now, given our lost opportunity to invest. • In this simple situation of only 1 Cash Flow, in present and future, we can also solve for r, if we know, PV, FV, and n. • Then, if FV = PV(1+r)n , solve for r as: • (1+r)n = FV/PV • [(1 + r)n]1/n = [(1+ r)n/n] = 1 + r = [FV/PV]1/n • r = [FV/PV]1/n – 1 (C) 2008 Red Hill Capital Corp., Delaware, USA