Lecture 4: Financial Math & Cash Flow Valuation II

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

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

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

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

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

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

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

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

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. • 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

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

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

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

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

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

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

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

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 (C) 2008-2009 Red Hill Capital Corp., Delaware, USA

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

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

Annuities: a special MCF case • 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. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA

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

Annuities: a special MCF case • Happily, there is a compact equation for the future value of an annuity, FVA. • 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

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 (C) 2008-2009 Red Hill Capital Corp., Delaware, USA

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 (C) 2008-2009 Red Hill Capital Corp., Delaware, USA

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. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA

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

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

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

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 • Just over half the advertized value. (C) 2008-2009 Red Hill Capital Corp., Delaware, USA

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 n-year 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

PVA+A A A A A A A A A 0 1 2 3 4 5 6 7 …… n – 1 .... n 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 (C) 2008-2009 Red Hill Capital Corp., Delaware, USA

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

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

Lecture 4: Financial Math & Cash Flow Valuation II