Lecture 4: Financial Math & Cash Flow Valuation II C. L. Mattoli (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 1 Intro Last week we looked at the concept of time value of money. The basic idea that you should have carried away is that any money has different values at different times. Last week, we began slowly and considered the case of only one cash flow. There are many situations, investments and securities that promise to pay a bunch of cash flows in different years. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 2 Intro In this lecture, we will continue our discussion and analysis of time value, and we will apply those concepts to these more complicated situations. We already have developed the basic equation for time valuation. In this lecture we will use that equation to form more complicated aggregate equations to value aggregated cash flows. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 3 Intro We will also study some more theory and discuss the additional assumptions that we must make for this next step in time valuation. This will form a major basis for everything else that we shall study from here on, so it is crucial that you understand and can use these concepts and equations. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 4 Rate of return: Interlude A rate of return is a specific example of a growth rate. A growth rate is simply the percentage change (%Δ) of anything, A: %ΔA = (A1 – A0)/A0. In finance time value, the growth rate is over time, and we usually annualize it and call it the annual rate of return: r = [(FV – PV)/PV]/n where FV is PV’s value in the future, and we divide by n because the growth term is a HPR. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 5 Rate of return: Interlude From that definition of return, the PV/FV equation follows by solving for PV or FV: FV = PV(1 + nr) It answers the question how will an amount grow through simple interest. In compound interest, we implicitly assume that intermediate cash flows are reinvested. Thus, under compound interest we must reinvest any cash flow that we get from the investment, during our holding period (HP). (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 6 Rate of return: Interlude The simple example is a savings account at a bank where you are paid periodic interest on the money, and you leave the whole thing in the account. In that case, your money is automatically reinvested. In other cases, it is an implicit assumption, as we shall discuss toward the end of the lecture: the reinvestment assumption. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 7 Rate of return: Interlude The further questions are why will it grow, and why do we need it to grow? It will grow because we rent it out to other people as debt or equity for their business. We want it to grow because we give up consumption; inflation erodes buying power; and we want to earn money and stay ahead of inflation when we do our deferred spending. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 8 PV/FV for Multiple CF’s (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 9 Intro PV/FV is one of the most important concepts in finance, and, you will have to admit, it does not seem too complicated. The next step is to be able to find FV/PV for a stream of cash flows, instead of just one. The step is simple, but it involves some technical issues. We begin by looking at FV, as we did in the case of single cash flows, but first … (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 10 Time lines: a Pictorial device A time line is a diagrammatic representation of cash flows, either received or paid, or both. It gives you a way to picture the process. PV FV1 FV2 0 1 2 This diagram relates a present value to two future values. The FV’s could also represent payments in future years. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 11 FV for multiple CF’s Suppose that we make a $100 deposit, right now, to a bank account earning 10%/yr., a second $100 deposit, a year from now, and we keep the money in bank til 5 years from now. Then, we can break it down into FV1 = $100(1+10%)5 = $161.05, for the first deposit, and FV2 = $100(1+10%)4 = $146.41, for the second, since it will be in bank for only 4 years. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 12 FV for multiple CF’s The total FV is just FV1+FV2 = $307.46, five years from now. An example of a situation in which payments are made into an investment account year after year, is contributions to a retirement fund. Then, the future value would be the money that you have at the end of the extended period of years for your retirement. We can make a general formula for cash flows invested in year n, and held in the account til year m. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 13 FV for multiple CF’s For each cash flow that you put into an investment account, n years from now, and keep in the account til m years from now will earn return for m – n years. So, the future value of each of those cash flows will be FVn(1 + r)(m-n). Then, your total future value will be the sum of those valued in year m, and we can write a complicated-looking equation, but not really complicated, as follows … (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 14 FV for multiple CF’s The general formula for cash flows, CFm, invested in year m, and held in the account til year n as: The symbol, , is used to denote the sum of the objects to its right, indexed by m, over the specified range of m, in this case, m = 0, 1,2, …,n. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 15 FV for multiple CF’s Thus, for example, with n = 2 and m = 0 to 2, we have, written out in full = CF0(1+r)(2-0) + CF1(1+r)(2-1) +CF2(1+r)(2-2) = CF0(1+r)2 + CF1(1+r)1 +CF2 Get used to this notation because it will be used throughout the course. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 16 FV for multiple CF’s The formula looks complicated, but you should, instead, focus on the meaning of FV of a cash flow. FVn = PV(1 + r)n means that PV is invested for n years at r rate of return. When we look at multiple cash flows, we have to figure out how long each one is invested, which can be tricky. Again, we can make it clearer, if we use time lines. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 17 2-dimensional time lines and FV The use of time lines can be really helpful in looking at situations of PV and FV, and you should use them to organize your thoughts and equations. For FV, we use a 2-dimensional time line, as shown below, to help you better understand what is happening with FV of multiple cash flows. Basically, each deposit grows from the time it is put in the investment account til the end of the investment horizon. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 18 Two-dimensional time line Cash Flows, C, are received at the times shown, and the grey cells represent the years that each cash flow receives interest. FV(5 years of deposits, Cm , start year0) = C0(1+k)4 + C1(1+k)3+ C2(1+k)2 + C3(1+k)1 + C4(1+k)0 0 1 2 3 4 Totals C0 C0(1 + k) C0(1 + k)2 C0(1 + k)3 C0(1 + k)4 C0(1 + k)4 C1(1 + k)2 C1(1 + k)3 C1(1 + k)3 C2(1 + k)2 C2(1 + k)2 C1 C1(1 + k) C2 C2(1 + k) C3 C3(1 + k) C4 C3(1 + k) C4 Sum (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 19 MCF FV example Let us look at a detailed example of FV of MCF’s. Assume that you deposit into an investment account $100, now; $200, in 1 year; and $300, 2 years from now. How much will you have in the account, 3 years from now, if you earn 10%/year, on your investment? Note: the first deposit will earn interest for 3 years, the second will earn interest for 2 years, and the third will earn interest for 1 year. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 20 MCF FV example Thus, the total that you will have in the account at the end of 3 years will be the sum of: $100(1 + 10%)3 = $133.10, $200(1 + 10%)2 = $242, and $300(1 + 10%) = $330, for a total of $705.10. That also means that, over the period, you have earned $705.1 – ($100+$200+$300) = $105.10 in total interest income. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 21 Annuities: a special MCF case 0 A special case of multiple cash flows is annuity payments. An annuity is a stream of cash flows, A, that are all equal and evenly spaced in time, e.g., yearly, quarterly, monthly, etc. We show an n-year annuity in a time line, here. $A $A $A $A 1 2 3 4 (C) 2008-2009 Red Hill Capital Corp., Delaware, USA $A … n 22 Annuities: a special MCF case There are annuities sold by insurance companies. More importantly, this type of even cash flow is actually useful, also, for valuing certain other types of securities, which we shall see in the next lecture. By tradition, an ordinary annuity CF stream begins one period in the future, as is shown in the previous slide. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 23 Annuities: a special MCF case Happily, there is a compact equation for the future value of an annuity, FVA. (1 k ) n 1 FVA A k This type of equation, with compounding is of the class called geometric equations, and the method that leads to the compact equation is actually quite simple. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 24 PV of Multiple CF’s (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 25 PV of MCF’s Since we can find PV for any future CF, and PV is right now, time = n = 0, then, if we have PV’s for a bunch of future CF’s, the PV of the sum of the CF’s is the sum of all of the PV’s of those CF’s. The general formula for a stream of CF’s, CFi, discounted at rate, k, is given by n PV i0 n CF i (1 k) i CF i (1 k) i i0 (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 26 PV of MCF’s Notice that we have begun our summation at i = 0. which is more general than the usual equation that you might see. However, some investment instruments, for example, a so-called annuity due, begin right now. Another example is the value of a stock that you buy today and it just happens to be paying a dividend, today. In the equation, CF0/(1+k)0 = CF0 since anything to the “power” 0 is equal to 1. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 27 MCF PV example Assume that we will get payments: $100, one year from now, $300, 3 years from now, and $500, 6 years from now, and our required rate of return (RRR) is 10%. Then, the PV is PV = $100(1+10%)-1 + $300(1+10%)-3 + $500(1+10%)-6 = $90.91 + $225.39 + $282.24 = $598.54. What that means is: if someone offers you a right to receive those payments, and you want to earn 10% on your money over the years, then, you should pay, at most, $598.54 for that right. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 28 MCF PV example That is the true use of PV: to find out what you should pay for an investment, like stocks, bonds, real estate, or any business investment (intrinsic value). You get future cash flows for giving out money, in the present, and you want to know how much to pay, if you want to earn a certain rate of return on investment. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 29 The Reinvestment assumption in PV There is one technical issue in PV MCF, which is sometimes conceptually difficult for students to grasp, but it is not too difficult to understand, if you try. If you will receive just one future CF, FV, and you want to earn a certain APR, k, compounded annually on your initial investment for n years, then, you should pay PV = FV/(1+k)n. In that way FV = PV(1+k)n , so FV is PV getting interest, k, compounded over n years. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 30 The Reinvestment assumption in PV Indeed, compounding is all about the situation in which you continue to reinvest the interest, return, or cash flows that you get during each interest period (yearly, monthly, etc.). Thus, there is already a reinvestment assumption, in the case that we have only one initial cash flow, PV, that grows into FV by compounding, or reinvesting the intermediate period returns. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 31 The Reinvestment assumption in PV Now, consider two CF’s, $10, paid 1 year from now, and $110, paid 2 years from now. Assume that your RRR is 10%, then, PV = $10/(1+10%) + $110/(1+10%) = $100. So, you invest $100 and get $110+$10 = $120 over the 2 years. However, if you take PV(1+k)n = $100(1+10%)2 = $121, not $120. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 32 The Reinvestment assumption in PV We said that we wanted to earn 10% on our initial investment, and we discounted the future CF’s to get PV, the investment we should make to get our RRR, but it appears that we didn’t get that. The problem is, actually, that everything needs to be valued at the same point in time, in finance, when adding values. However, we added $10 in year 2 to $110 in year 3 to get $120 total, but values in year 2 and 3 are not on an equal footing. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 33 The Reinvestment assumption in PV We have to make an adjustment: value year 2 CF, in year 3, by pushing it forward as FV($10 in year 2) = $10(1+10%) = $11, in year 3. Then, we will have $11+$110 = $121, and we will have truly earned 10% per year compounded for 2 years on our initial investment of $100. What it means is that you have to reinvest intermediate (in the middle of your holding period) CF’s at the same interest rate and let them also earn that rate over the life of the investment. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 34 The Reinvestment assumption in PV That is commonly known as the reinvestment assumption in PV MCF’s. We said that PV and FV are just flip sides of the same concept, and the reinvestment assumption in the case of compounding and MCF’s is what makes FV and PV of MCF’s correspond, in the same way as PV and FV, in the case of only one CF. It is a simple matter of taking PV(1+k)n = CF1(1+k)n-1 + CF2(1+k)n-2 + … + CFn-1(1+k) +CFn = FV. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 35 The Reinvestment assumption in PV That equation is, then, just the FV of all of the CF’s valued at the end of the holding period. Moreover, it is not unreasonable to require that cash flows coming over the years should be reinvested. It would be stupid not to reinvest, if we were investing, in the first place. Thus, the reinvestment assumption is not only necessary, but it also makes good business investment sense. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 36 The real point: one time We usually talk about PV, the value right now, or some FV. The real point is that, in finance, we realize that money has a time value. Because of that, if we are to value things and we want to compare their values, then, they all have to be valued at the same time. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 37 The real point: one time We could value them all, 3 years from now, for example, if we wanted to. Moreover, if we had all CF’s valued 3 years from now, we could just add them up, and discount the sum back to now, PV, by dividing that summed number by the PV factor for 3 years from now. Or we could push them forward into another future year. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 38 The real point: one time We just need to have them all valued in the same year. In that manner, for example, if we are offered different cash flows in different future years, we can compare them, in the present, by looking at their present values. A simple example of that is consider two choices: $100, now, or $110, a year from now. Which is a better offer? (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 39 The real point: one time Answer: if you can make more than 10% on your money by investing it for a year, you will have more than $110. Thus, which is better, in this case, depends on your opportunity rate of return, which you can also use to find out the PV of the $110, a year from now. The other thing that we can do with the scheme is to compare values of the same cash flow at different times. Then, we are looking at the question: what is or was the growth rate of that cash flow over time, and we call that the rate of return. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 40 Annuity PV Again, we can discuss annuities, which are just the same amount of CF paid for a number of periods, evenly spaced in time. Thus, if the first payment will come in 1 year from now and the last comes n years from now, we have: PV = A/(1+r)1 + A/(1+r)2 + … + A/(1+r)n n i0 n A (1 k) i A (1 k) i i0 (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 41 PV annuities Fortunately, again, geometric math allows us to get a compact reduced equation for the PV of even, evenly spaced in time, CF’s, A, for n years, present value of annuity: PVA. 1 (1 k) n PVA A k PVA $A $A $A $A 0 1 2 3 4 (C) 2008-2009 Red Hill Capital Corp., Delaware, USA $A … n 42 PV annuities The other fortunate thing is that annuities can be found either pure or as part of the CF streams from actual investment instruments. In addition to annuities sold by financial companies, coupon bond debt securities pay a coupon interest payment that is the same dollar amount, period after period, in addition to the final payment of principal. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 43 PV annuities So, part of the valuation of a coupon bond will involve the PVA of coupons. The other part will be the discounted value of the final payment. By having a simplified formula we will have a simple way to figure out the price that we should pay for those future even cash flows, PV, given our own RRR =k. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 44 PV annuities Consider an example. A lottery contest promises to pay a $5,000,000 prize. That sum will actually be paid out over 20 years with equally-divided annual payments. Thus, the prize will pay you $250,000 a year annuity payments for 20 years. Not a bad annual income, but are you really getting $5,000,000? (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 45 PV annuities As students of finance who understand that money has time value, we are not that easily fooled. If our opportunity rate for investment is 6%, then we know that the prize is actually worth PVA, right now. Putting in the numbers, we find 1 (1 6%) PVA $250,000 6% 20 $2,867,480 .31 Just over half the advertized value. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 46 Annuity language. As we mentioned earlier, annuity payments that begin one year from now are traditionally called ordinary annuities. If the CF’s begin right now, time = 0, we call it an annuity due. If payments begin, m years from now, and payments are made for n years, it is called an nyear deferred annuity with deferment m – 1 years after ordinary. We call an infinite annuity payment stream that begins 1 year from now a (ordinary) perpetuity. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 47 Annuity relationships Then, for example, we can use our PVA ordinary to get the right values for the others. An n-year annuity due is like a payment of A, right now, and n-1 payments in future years, so we can write 1 (1 k) ( n -1) PVAD A A k PVA+A 0 1 A A 2 A 3 A 4 A 5 A 6 A A 7 …… n – 1 .... n (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 48 Annuity Relations The ordinary annuity equation just gives a value of a bunch of cash flows, one year before the first cash flow. Although we can use the equation to find PV, if the first CF is 1 year from now, the equation doesn’t really know what time it is. Equations just have rules and inputs. Put n, A, and k, into the PVA, and it values the n CF’s one year before they began. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 49 Annuity Relations Then, for example, we could even use ordinary PVA to get annuity due, in another way. Take PVA for n payments, beginning right now, and it will give a value one year before now (t= -1). Next push that forward from a year ago to now as PVA(1+k). Thus, another way to write PV annuity due is PVAD = PVA(1+k). (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 50 Annuity Relations If you take the time, you can see that the two equations for annuity due are the same in the end (they have to be). We can also use this n-period annuity equation to value an infinite number of equal payments. Interestingly, the value is finite, not infinite. To value a perpetuity payment stream that begins one period from now, we can use PVA, again. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 51 Annuity Relations PVP, present value (ordinary) perpetuity, is just the limiting case of PVA, as n goes to infinity. Looking back at the PVA equation, as n , the term (1+k)-n = 1/(1+k)n, goes to zero, so from the PVA equation, we can write: PVP = A/k. Note also that we refer to this as an ordinary perpetuity because even though infinity is infinity, PVP gives a value for infinite payments beginning 1 period from now. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 52 Annuity Relations If payments were, for example, to begin right now, we would have to add a payment and get the equivalent of annuity due for perpetuity, or PVPD = A + PVP. It is an important point to note. For deferred annuities, we think in terms of how many years after an ordinary annuity do the payments begin. So, if the first payment begins 4 years from now, that is actually 4 – 1 = 3 years after an ordinary annuity would begin. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 53 Annuity Relations In general, we use the PVA for the right number of payments. That gives us a value one period before the first payment. You look at a time line to find the year PVA gives a value, and you discount that value back to time = 0 to get the present value of a deferred annuity. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 54 Annuity Relations Since it begins in year m, that will be m – 1 years after an ordinary annuity. PVA will give a value in year m – 1, one year before the first payment, in year m. Thus PVA deferred to year m = PVA/(1+k)m–1 to bring the PVA value in year m – 1 back to time = 0, true PVA deferred to year m. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 55 Annuity Relations We show a simple pictorial example description for annuity beginning 2 years from now, in the next slide. The value will be PVA deferred payment year 2 = PVA(n-1 years)/(1+k) to bring the value back one year to the present, t = 0. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 56 Deferred annuity pictures 1year deferred annuity using PVA PVA $A $A 0 1 2 PVA defer 1 PVA(n-1) $A $A $A 0 1 2 3 4 (C) 2008-2009 Red Hill Capital Corp., Delaware, USA $A $A n-1 $A … n 57 Installment loans Another application of the annuity concept is installment loans. An installment loan is a loan that is paid off in equal periodic installments: annuity payments. Thus, the value of the loan payments at the time of borrowing is the PV of the installment payments, discounted at the interest rate that the banks wants to get on the borrowings. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 58 Installment loans PVA = amount borrowed, now. Typically, since we know how much we want to borrow, and we know how much interest the bank wants to charge us, we can find the amount of the installment payments, given the length of time that we want to borrow for. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 59 Installment example Assume you want to borrow $10,000 for 5 years, and bank wants 8% interest on its money. We use the PVA equation, and solve for A: 1 (1 k) n A PVA/ k Thus, 1 (1 8%) A $10,000/ 8% 5 $2,504.56 (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 60 Inside Installment Loans: Amortization What is really happening in an installment loan situation is this. Each payment actually represents interest owed on the outstanding loan balance, plus some payment of principal on the loan. In that manner, each payment also reduces the principal amount outstanding. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 61 Inside Installment Loans: Amortization Successive payments represent less and less interest and more and more principal payment, until the last payment completely liquidates the loan. We show a so-called loan amortization schedule for the example loan discussed, above, in the slide, below. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 62 Inside Installment Loans: Amortization The first year’s interest due to be paid on $10,000 outstanding principal is 8%/yr x $10,000 = $800. The actual even annuity installment payment that we figured is $2,504.56, so that means that $1,704.56 = $2,504.56 - $800 is applied to reduce the principal to $10,000 – $1,704.56 = $8,295.44. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 63 Loan Amortization Example Over the life of the loan, total payments of $12,522.82 are made. Of that, we know the principal was $10,000, so interest payments totaled $2,522.82. Loan Amortization Table Year Payment Interest 0 1 2 3 4 5 Totals $2,504.56 $2,504.56 $2,504.56 $2,504.56 $2,504.56 $12,522.82 $800.00 $663.63 $516.36 $357.30 $185.52 $2,522.82 Principal $1,704.56 $1,840.93 $1,988.20 $2,147.26 $2,319.04 $10,000.00 Principal outstanding $10,000.00 $8,295.44 $6,454.51 $4,466.30 $2,319.04 $0.00 (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 64 Further notes on PVA To find the number of payments, given, PVA, A, and r, we can use the rule-ofthumb for doubling (from last lecture, chapter 4), or we can use trial and error, i.e., put in a guess, and see how close the answer is, and try again. The same applies to finding r, given PVA, n, and A. Trial and error. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 65 PVA: the True Inverse of FVA We want PV and FV to be inverse operations of one another. We saw, earlier, that there is an implicit reinvestment assumption in the PV of multiple CF’s to really get the compound RRR that you want. So assume that you pay PVA for an annuity payment stream. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 66 PVA: the True Inverse of FVA Then, take a factor of (1+k)n times PVA to see what we get. (1+k)n PVA = The future value of the present value of n A payments = (1+k)nA[1 – (1+k)-n]/k = A[(1+k)n – 1]/k = FVA. Thus, PVA and FVA are truly inverse operations. If you pay PVA for an annuity stream and you reinvest all of the payments at the same discount rate, you will have FVA at the end of the investment horizon, in n years. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 67 Compounding more than once a year (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 68 Intro There is no reason that we cannot compound more than once a year. Interest can be compounded semi-annually, quarterly, monthly, weekly, daily, or continuously, every nanosecond. What we are looking at, mechanically, is, for example, with semi-annual payments, half a year’s interest gets paid into your account at the end of 6 months, and you earn half a year’s interest on the principal plus the first 6 month’s paid interest. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 69 Per Period Concepts. To allow for compounding more than once a year, we don’t need a new PV equation. All we must do is to make some simple adjustments to the way we use the equations. The equations don’t know that they are using years for n or that we are using an annual interest rate for k. They are just equations with variables. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 70 Per Period Concepts. We give meaning to the variables. FV, for example, just takes a PV and compounds a return on it n times. Another way to use the equation is to take r as a per period interest rate, PPIR. There is no a priori reason that n has to represent years. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 71 Per Period Concepts. N is just a set period, which could be years, months, days, … , whatever. We have to just a bit careful, here. N and r are related by units, in the FV/PV equation. N is a period of time, and r is the interest rate that is earned over that period, a PPIR Then, for each period we compound interest on a PV for n of those time-periods. (or discount FV to PV) using the PPIR. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 72 Per Period Example For example, assume that you are given an annual interest rate, r, and interest is compounded interest every month for 2 years. First, find the PPIR = [r/yr.]/[12 months/year] = a monthly rate of interest. Then, compound that rate for 2 years x 12 months/year = 24 months (periods). Take, for example, APR = 12%/year. Then, PPIR = [12%/yr.]/[12 months/yr.] = 1%/month (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 73 Per Period Example Assume an initial principal (PV) of $20,000. After 2 years you will have FV = PV[1+(r/yr.)/(# periods/year)](n years x # periods/year) = $20,000[1+12%/12]2years x12months/year = $20,000(1+1%/month)24 months = $25,394.69. Compare that number to the number that you would get, if you just compounded interest once a year: FV = $20,000(1+12%)2 = $25,088, which is much less. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 74 Per Period Example Thus, the effect of compounding more and more times per year will be to get more and more future value than with less compounding. In that regard, if you invest money, you would like to have more compounding per year, but if you borrow, you would prefer less compounding per year. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 75 Annual Effective Rate We have seen that the earning rate is larger than it would be with annual compounding when interest is compounded more than once a year. We want to some how quantify that annual earning rate because we like to compare annual rates. So, assume that k = APR and m = # of periods in a year. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 76 Annual Effective Rate If we take PV and add compound interest for one year (m periods), we will have FV = PV(1+k/m)m at the end of a year. We can find the effective rate of return over the year from our basic equation for return that we have seen since the first lecture: r = Income/initial investment = (FV – PV)/PV. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 77 Annual Effective Rate Moreover, it will be an annual return, in this case, since we compound interest over m per year periods = 1 year exactly. Using the above equation for FV, we have reffective annual rate of return = [PV(1+k/m)m – PV]/PV = (1+k/m)m – 1 = reff. All that the annual effective interest rate (EAR) does is to give an equivalent annual earning rate of return that resulted from a nominal APR that is compounded more than once a year. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 78 Annual Effective Rate In other words: FV = (1+reff)PV for one year. For more than 1 year, say n years, FV = PV(1+reff)n = PV[(1+k/m)m]n = PV(1+k/m)n x m. The effective rate will always be higher than the nominal interest rate, except when the nominal rate is for the same period as the compounding period. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 79 Matching CF periods to compounding period As a final technical note, we tell you that you have to be careful to match compounding periods with the period of cash flows. For example, if CF’s come in quarters and compounding is quarterly, use the quarterly rate for making discounted CF’s. If compounding is quarterly and CF’s come only annually, you can compound 4 quarters, find the annual effective rate and use that as the discount factor for annual compounding. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 80 Bank Loans (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 81 Types of Bank loans Although you may not realize it, we have already seen some templates for various types of bank loans. A pure discount loan is a loan that has one balloon payment at maturity, so FV = PV(1+k)n. It is like a LT bank loan version of the ST CB’s we looked at earlier. We saw installment loans with even periodic payments. Principal borrowed = PVA of the payments. Just let us point out that payments might be monthly or quarterly, in practice, rather than yearly. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 82 Types of Bank loans A final type of loan is an interest-only loan, whereby only interest payments are made each intermediate year, with a final interest payment plus principal amount at the end. The principal remains unpaid til the end, so interest is charged on the same principal every period, so the interest payments will form an annuity stream. To value an interest-only loan, take PV = kP[1 – (1+k)-n]/k + P/(1+k)n = P – P(1+k)-n + P(1+k)-n = P, which is as it should be. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 83 Trailer (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 84 Present or Future Value? Present and future value are from the same basic equation, but we must understand what these concepts really mean. We calculate present value of future cash flows because we want to know the value of those future cash flows to us now. Then, we can decide how much to pay now for things like stocks, bonds, and other investments, which are always promises of money in the future. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 85 Present or Future Value? We calculate future value of money we have now or will get later to find out how much money that we will have at a future time. The basic difference is that present value tells us how much something is worth to us, now, while future value tells us how many dollars we will have at some point in the future but not how much it is worth to us now. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 86 The learning experience: the use of logic and pictures Our use of logic and pictures to find other annuity equations from the first one for ordinary annuities is just an example of how to attack PV and FV problems, in general. We have the basic equations for a one-payment cash flow. From it, we construct general equations for multiple cash flows. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 87 The learning experience: the use of logic and pictures Then, we derived specialized FV and PV equations for an ordinary annuity, using a simple trick. From those equations, we found equations for other kinds of annuities, using one- and two-dimensional time lines, the logic of the situation, and our understanding of the meanings of the original ordinary annuity equations. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 88 The learning experience: the use of logic and pictures In our future studies and in other problems, we will be given a set of cash flows, and we will be asked to figure out an answer. Often, we will not be asked to figure out PV or FV in the problem but part of the problem will be, first, to figure out which one we should calculate: FV or PV. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 89 The learning experience: the use of logic and pictures After we figure out which one, FV or PV, is called for in the problem, we set up a time line to see more clearly what the situation is and if we can use some of the simplified equations, like PV or FV annuity, in the problem, maybe only for part of it. Then, we set up a final equation based on everything that we can see about the situation. The point is: don’t just memorize equations, understand them, and then you will learn how to use all of them for any situation. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 90 Exam Caliber Questions 1. 2. An investment pays $50 per year for 100 years, except, in year 25, it pays $100, in year 50, it pays $200, and in the final year, it pays $500. If your RRR= 10%, how much would you pay for this investment, now? Explain the reinvestment assumption in the multiple cash flow PV/FV theory. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 91 Tutorial Problems Learning activity ● Attempt all of the chapter review and self-test problems and the critical thinking and concepts review questions on pages 142 to 145. Attempt questions and problems 1, 3, 4, 7, 12, 15, 19, 21, 31, 44. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 92 Next week Monday April 6 is a holiday Diane has informed me that Maurie has informed her that we should do lectures in the tutorials. I will let you know by email what the rooms will be next week. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 93 END (C) 2008-2009 Red Hill Capital Corp., Delaware, USA 94

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