1 00:00:00,520 --> 00:00:02,640 Welcome back to corporate finance. 2 00:00:02,640 --> 00:00:04,750 Last time we talked about compounding or 3 00:00:04,750 --> 00:00:07,970 the process of moving cash flows forward in time. 4 00:00:07,970 --> 00:00:12,850 Today I want to present several useful shortcuts to compute the present value and 5 00:00:12,850 --> 00:00:17,320 future value of common streams of cash flows that we see often in practice. 6 00:00:17,320 --> 00:00:18,800 Let's get started. 7 00:00:18,800 --> 00:00:22,420 Hey everybody, welcome to our third lecture on the time value of money. 8 00:00:22,420 --> 00:00:24,330 So, last time we talked about compounding or 9 00:00:24,330 --> 00:00:27,303 the process of moving cash flows forward in time. 10 00:00:27,303 --> 00:00:29,370 [COUGH] Excuse me. 11 00:00:29,370 --> 00:00:33,540 To find their future value, whereas in our first lecture we 12 00:00:33,540 --> 00:00:37,660 moved cash flows back in time by discounting to find their present value. 13 00:00:37,660 --> 00:00:41,670 So, what I want to do today is I want to give you some useful shortcuts for 14 00:00:41,670 --> 00:00:42,820 computing the present value or 15 00:00:42,820 --> 00:00:47,590 the future value of some streams of cash flows that commonly arise in practice. 16 00:00:48,630 --> 00:00:49,870 So let's get started. 17 00:00:50,960 --> 00:00:53,710 The first thing that I want to talk about is an annuity. 18 00:00:53,710 --> 00:00:57,440 An annuity is a finite stream of cash flows of identical magnitude in equal 19 00:00:57,440 --> 00:01:01,420 spacing and time, and I've highlighted key elements of this definition. 20 00:01:02,420 --> 00:01:04,630 So here's a timeline representing an annuity. 21 00:01:06,570 --> 00:01:14,360 First key aspect or feature of an annuity are cash flows of identical magnitude. 22 00:01:14,360 --> 00:01:16,350 All of these cash flows are the same number. 23 00:01:18,290 --> 00:01:20,070 So identical magnitude. 24 00:01:20,070 --> 00:01:22,590 Secondly, this is a finite stream of cash flow. 25 00:01:22,590 --> 00:01:25,100 It ends at some point in time. 26 00:01:25,100 --> 00:01:26,140 Okay? 27 00:01:26,140 --> 00:01:29,540 And that might seem like an unnecessary or obvious assumption, but 28 00:01:29,540 --> 00:01:33,040 you'll see that we'll deal with infinite cash flow streams in a little bit. 29 00:01:34,670 --> 00:01:39,320 And then the last assumption is that the spacing between the cash flows 30 00:01:40,720 --> 00:01:42,380 has to be equal, so it's always a year. 31 00:01:42,380 --> 00:01:45,280 We always get the cash flow after one year, two months, 32 00:01:45,280 --> 00:01:47,700 whatever that spacing is, it has to be the same. 33 00:01:49,800 --> 00:01:52,520 And it turns out that this cash flow 34 00:01:52,520 --> 00:01:55,100 stream arises in a number of situations in practice. 35 00:01:55,100 --> 00:01:58,654 So insurance companies sell a product called an annuity, 36 00:01:58,654 --> 00:02:00,910 representing its cash flow stream. 37 00:02:01,980 --> 00:02:05,470 Home mortgages are an example of an annuity stream. 38 00:02:05,470 --> 00:02:10,315 Auto leases, certain bond payments and amortizing loans are annuities. 39 00:02:10,315 --> 00:02:13,590 So it's actually fairly common in practice. 40 00:02:13,590 --> 00:02:18,130 Now if we wanted to find the present value of these cash flows, 41 00:02:18,130 --> 00:02:20,190 we know how to do that. 42 00:02:20,190 --> 00:02:21,470 We could brute force it. 43 00:02:21,470 --> 00:02:27,660 We can take each cash flows and discount it back to today. 44 00:02:27,660 --> 00:02:31,200 So imagine I had a second cash flow here. 45 00:02:31,200 --> 00:02:35,190 I could go CF divided by 1 plus R squared. 46 00:02:35,190 --> 00:02:36,780 That would bring it back to today. 47 00:02:36,780 --> 00:02:42,060 I could take this cash flow, CF over 1 plus R to the T minus 1. 48 00:02:42,060 --> 00:02:43,660 That would bring it. 49 00:02:43,660 --> 00:02:47,080 And I do that for all the cash flows, and then I could add them up here. 50 00:02:47,080 --> 00:02:48,280 That would give me the present value. 51 00:02:49,280 --> 00:02:55,294 But that's a bit burdensome, especially when T is big. 52 00:02:55,294 --> 00:02:58,920 So, what I'd like to show you is a shortcut or a simple 53 00:02:58,920 --> 00:03:03,450 formula to compute the present value of this cash flow stream, and here it is. 54 00:03:03,450 --> 00:03:08,760 We take the cash flow, CF, divide it by the discount rate and 55 00:03:08,760 --> 00:03:13,390 multiply it by this term in parentheses here. 56 00:03:13,390 --> 00:03:15,168 Now, if I move the R over here, 57 00:03:15,168 --> 00:03:20,640 I can re-express the present value of the annuity formula as just 58 00:03:20,640 --> 00:03:25,990 the annuity cash flow times this term here which is called an annuity factor. 59 00:03:27,770 --> 00:03:32,130 That would give me the present value of this cash flow string. 60 00:03:34,230 --> 00:03:38,890 One thing to keep in mind though is in order for this formula to make sense, 61 00:03:38,890 --> 00:03:43,180 not only do all the features defining an annuity stream have to be true, but 62 00:03:43,180 --> 00:03:48,160 we are assuming that the first cash flow arrives one period from today. 63 00:03:50,140 --> 00:03:53,070 So, for example, if my cash flow stream looked like this. 64 00:03:53,070 --> 00:03:56,080 This is an annuity stream of cash flows, but 65 00:03:56,080 --> 00:04:00,050 this formula is not going to give me the present value of this cash flow stream. 66 00:04:00,050 --> 00:04:03,200 Actually, what I would have to do is I could just add CF, 67 00:04:04,610 --> 00:04:07,120 because this is the present value. 68 00:04:07,120 --> 00:04:07,620 It's coming, today. 69 00:04:11,307 --> 00:04:12,562 Let's do an example. 70 00:04:12,562 --> 00:04:15,351 How much do I have to save today to withdraw $100 at the end 71 00:04:15,351 --> 00:04:18,100 of each of the next 20 years if I can earn 5% per annum? 72 00:04:18,100 --> 00:04:21,000 Well, step one is draw a timeline. 73 00:04:21,000 --> 00:04:23,580 I'm trying to figure out how much I have to save today in order 74 00:04:23,580 --> 00:04:27,790 to pull out $100 every year over the next 20 years. 75 00:04:27,790 --> 00:04:30,440 Well, we know how to do that by brute force. 76 00:04:30,440 --> 00:04:33,720 We can simply discount all of the cash flows back into today's 77 00:04:33,720 --> 00:04:34,880 time units and add them up. 78 00:04:36,040 --> 00:04:38,530 More elegant solution that we just learned, of course, 79 00:04:38,530 --> 00:04:42,040 is to apply our present value of annuity formula. 80 00:04:42,040 --> 00:04:44,170 Right. The cash flow is 100, CF. 81 00:04:44,170 --> 00:04:47,870 The discount rate, R, is 5%. 82 00:04:47,870 --> 00:04:53,180 And the time of the cash flows is 20 years. 83 00:04:53,180 --> 00:04:55,770 Plugging all those numbers into the formula, and computing, 84 00:04:55,770 --> 00:05:00,480 we get the present value of these cash flows is $1,246.22. 85 00:05:00,480 --> 00:05:05,140 That's how much money I have to save today in order to withdraw the $100 every year. 86 00:05:07,753 --> 00:05:11,892 Now let's turn to something called the growing annuity, which is as the name 87 00:05:11,892 --> 00:05:16,730 suggests just like an annuity but for the fact that the cash flows are growing. 88 00:05:16,730 --> 00:05:20,440 So it's a finite stream of cash flows, okay. 89 00:05:20,440 --> 00:05:22,110 Evenly spaced through time, okay. 90 00:05:24,420 --> 00:05:29,060 But now the cash flows aren't constant, they're growing at a constant rate, g. 91 00:05:31,450 --> 00:05:37,060 And this type of cash flow stream pops up in a number of instances in practice. 92 00:05:37,060 --> 00:05:39,990 Certain income streams, for example, your work. 93 00:05:39,990 --> 00:05:44,520 You might imagine that your salary grows at some constant 94 00:05:44,520 --> 00:05:49,340 rate or approximately some constant rate g. 95 00:05:49,340 --> 00:05:53,620 Certain saving strategies, maybe you want to save a certain amount each year, but 96 00:05:53,620 --> 00:05:56,550 you want that amount to grow with your growing income stream. 97 00:05:57,750 --> 00:05:59,890 In corporate finance certain project revenue and 98 00:05:59,890 --> 00:06:05,310 expense streams will often grow at a near constant growth rate. 99 00:06:05,310 --> 00:06:09,480 So it's a really useful approximation to many cash flow 100 00:06:09,480 --> 00:06:11,810 streams we'll come across in practice. 101 00:06:11,810 --> 00:06:17,521 And like our annuity stream, we can represent the present value of this cash 102 00:06:17,521 --> 00:06:22,860 flow stream with a simple formula as follows. 103 00:06:22,860 --> 00:06:27,610 We take the cash flow [COUGH] as of the first period divided by 104 00:06:27,610 --> 00:06:33,390 the discount rate less the growth rate, times this factor here. 105 00:06:33,390 --> 00:06:39,540 That will give us the present value of this cash flow stream right here. 106 00:06:43,130 --> 00:06:44,320 But remember, 107 00:06:44,320 --> 00:06:49,940 a critical assumption is that the first cash flow arrives one period from today. 108 00:06:51,000 --> 00:06:51,500 Okay. 109 00:06:53,640 --> 00:06:55,050 Let's do an example. 110 00:06:55,050 --> 00:06:58,336 How much do we have to save today to withdraw $100 at the end of this year 111 00:06:58,336 --> 00:07:02,820 $102.50 next year, $105.06 the year after, and so 112 00:07:02,820 --> 00:07:07,130 on for the next 19 years if we can earn 5% per annum? 113 00:07:07,130 --> 00:07:11,570 Well, let's draw a timeline, and what we see is that our 114 00:07:11,570 --> 00:07:16,405 first withdrawl of $100 occurs one year out, 115 00:07:16,405 --> 00:07:20,880 then $102.5, and on and on and on. 116 00:07:20,880 --> 00:07:24,980 What we can discern from this problem is that these cash flows are growing at 117 00:07:24,980 --> 00:07:29,650 a constant rate, g, equal to two and a half percent per annum. 118 00:07:30,980 --> 00:07:36,080 So this cash flow stream satisfies all of the requirements 119 00:07:36,080 --> 00:07:39,130 needed to use the present value of a growing annuity formula. 120 00:07:41,160 --> 00:07:43,120 So, our first cash flow of $100. 121 00:07:43,120 --> 00:07:47,100 Our discount rate of 5%. 122 00:07:47,100 --> 00:07:50,628 And, here's our growth rate of 2 1/2%. 123 00:07:50,628 --> 00:07:55,852 That's going to get a present value of $1,529.69. 124 00:07:55,852 --> 00:08:02,072 That's how much we would need to withdraw $100 growing at 2 1/2% every year. 125 00:08:02,072 --> 00:08:05,100 Now let's talk about a perpetuity. 126 00:08:05,100 --> 00:08:09,230 So a perpetuity is just like an annuity except the cash flows go on forever. 127 00:08:09,230 --> 00:08:15,190 We get the same amount of money equally spaced in time forever. 128 00:08:17,610 --> 00:08:20,200 So where does this thing come up in practice? 129 00:08:20,200 --> 00:08:22,930 Well, oddly enough it actually does. 130 00:08:22,930 --> 00:08:26,990 And something called perpetuity or consol bonds which exists over in the UK. 131 00:08:29,860 --> 00:08:35,160 And interestingly enough the formula for this cash flow stream is very simple. 132 00:08:35,160 --> 00:08:38,240 It's just the cash flow divided by the discount rate. 133 00:08:38,240 --> 00:08:38,746 CF over R. 134 00:08:41,484 --> 00:08:42,990 Let's do an example. 135 00:08:42,990 --> 00:08:46,490 How much do you have to save today to withdraw $100 at the end of each year 136 00:08:46,490 --> 00:08:50,050 forever if you can earn 5% per annum? 137 00:08:50,050 --> 00:08:53,075 Well, the timeline looks as follows, all right? 138 00:08:53,075 --> 00:08:57,370 $100 every year, forever, and clearly the brute 139 00:08:57,370 --> 00:09:01,270 force method of discounting each cash flow one at a time is never going to work. 140 00:09:01,270 --> 00:09:02,350 It's just impossible. 141 00:09:03,450 --> 00:09:04,680 So we have to use our formula. 142 00:09:06,190 --> 00:09:10,030 We take the $100, divide it by the discount rate of 5%, and 143 00:09:10,030 --> 00:09:12,304 that gives us $2000. 144 00:09:12,304 --> 00:09:17,244 We need $2000 to be able to withdraw $100 a year forever, 145 00:09:17,244 --> 00:09:20,880 assuming that money can earn 5% annum. 146 00:09:20,880 --> 00:09:25,330 And intuitively what's going on is once we get out hundred years, 147 00:09:25,330 --> 00:09:26,880 two hundred years, whatever. 148 00:09:26,880 --> 00:09:30,276 The present value of that money is so small, it's very close to zero, 149 00:09:30,276 --> 00:09:33,051 which is why you don't need an infinite amount of money. 150 00:09:35,312 --> 00:09:41,410 And the perpetuity's cousin, a growing perpetuity, is just that. 151 00:09:41,410 --> 00:09:46,260 It's an infinite stream of cash flow that grows at a constant rate, g. 152 00:09:46,260 --> 00:09:48,410 That are evenly spaced out through time. 153 00:09:50,080 --> 00:09:52,350 So here's a visual representation. 154 00:09:52,350 --> 00:09:55,740 Here's our timeline of a growing perpetuity. 155 00:09:55,740 --> 00:09:58,400 What's an example of a growing perpetuity in practice, 156 00:09:58,400 --> 00:10:01,540 well, dividend streams are much like a growing perpetuity. 157 00:10:01,540 --> 00:10:02,890 They're useful approximation. 158 00:10:02,890 --> 00:10:03,850 Don't take it literally. 159 00:10:03,850 --> 00:10:08,250 Companies don't last forever, but we can treat them as such Because there is no 160 00:10:08,250 --> 00:10:13,710 finite end date to most companies, absent some event such as a bankruptcy or 161 00:10:13,710 --> 00:10:15,880 an acquisition or a takeover, something like that. 162 00:10:17,460 --> 00:10:19,960 So, what's the formula for a growing perpetuity? 163 00:10:19,960 --> 00:10:24,510 Well, it's just the cash flow that we're going to receive in the first year divided 164 00:10:24,510 --> 00:10:28,007 by the discount rate minus the growth rate of that cash flow. 165 00:10:30,110 --> 00:10:33,170 Again we're going to have to assume that the first cash flow 166 00:10:33,170 --> 00:10:35,530 arrives one year from today to use this formula, 167 00:10:36,800 --> 00:10:40,840 as well as having all the other requirements being satisfied. 168 00:10:40,840 --> 00:10:44,060 The cash flow's being evenly spaced and then growing at a constant rate. 169 00:10:45,420 --> 00:10:47,500 So let's do a little example now. 170 00:10:47,500 --> 00:10:53,270 How much do you have to save today to withdraw $100 at the end 171 00:10:53,270 --> 00:10:57,240 of this year, 102.5 next year and 105.06 the year after and so 172 00:10:57,240 --> 00:11:00,300 on forever if you can earn 5% per annum? 173 00:11:00,300 --> 00:11:02,970 Well, let's draw our timeline. 174 00:11:02,970 --> 00:11:05,245 So we're going to get $100 here in year one. 175 00:11:05,245 --> 00:11:08,840 102.5 in year two, that's growing at 2.5%, 176 00:11:08,840 --> 00:11:13,270 as is the 105.06, if I had written it here. 177 00:11:16,500 --> 00:11:18,400 So our g is 2.5%. 178 00:11:18,400 --> 00:11:23,900 The first cash flow comes one year from today, there is our first cash flow. 179 00:11:23,900 --> 00:11:27,840 This goes on forever, and the spacing is equal. 180 00:11:27,840 --> 00:11:31,460 So this is a growing perpetuity to which we can apply our 181 00:11:31,460 --> 00:11:33,560 growing perpetuity present value formula. 182 00:11:34,840 --> 00:11:37,695 So we take the first cash flow. 183 00:11:37,695 --> 00:11:41,860 $100 divided by the difference between the discount rate and 184 00:11:41,860 --> 00:11:45,350 the growth rate to get $4000. 185 00:11:45,350 --> 00:11:48,940 In other words, we have $4,000 dollars today, and 186 00:11:48,940 --> 00:11:55,100 it's earning 5% interest per annum, every year thereafter. 187 00:11:55,100 --> 00:11:57,710 We can pull out $100 next year and 188 00:11:57,710 --> 00:12:00,420 have that amount grow by 2.5% every year thereafter. 189 00:12:04,580 --> 00:12:05,220 Let's summarize. 190 00:12:06,440 --> 00:12:07,934 We learned a couple of useful short cuts today. 191 00:12:07,934 --> 00:12:11,060 We talked about an annuity and its present value formula. 192 00:12:11,060 --> 00:12:14,286 We talked about perpetuity and its present value formula. 193 00:12:14,286 --> 00:12:17,413 We talked about their growing cousins, the growing annuity, 194 00:12:17,413 --> 00:12:21,137 the growing perpetuity, and the present value of formula for those guys. 195 00:12:21,137 --> 00:12:25,066 And while that might seem somewhat esoteric and bland or 196 00:12:25,066 --> 00:12:28,578 boring, we also discuss some of the applications 197 00:12:28,578 --> 00:12:33,110 that you might see in practice these cash flow streams arising. 198 00:12:34,940 --> 00:12:38,200 And where these short cuts are really useful, 199 00:12:38,200 --> 00:12:42,160 more than just finding the present value or the future value, 200 00:12:42,160 --> 00:12:46,660 is in finding the cash flow associated with the stream. 201 00:12:46,660 --> 00:12:50,320 So being able to manipulate these formulas is very important. 202 00:12:50,320 --> 00:12:55,920 And so in the problems you are going to spend some time in real life context, 203 00:12:55,920 --> 00:13:00,532 or at least as close to real life as we can get, manipulating these formulas to 204 00:13:00,532 --> 00:13:05,880 derive certain aspects of interest, whether 205 00:13:05,880 --> 00:13:10,290 it's the cash flow or the amount of time, or the discount rate or the growth rate. 206 00:13:11,440 --> 00:13:15,390 So tackle the problem set, it really brings the material to life. 207 00:13:15,390 --> 00:13:19,610 But be very careful that applying these formulas takes care. 208 00:13:19,610 --> 00:13:23,450 Don't blindly apply them to any setting, because as we discussed, 209 00:13:23,450 --> 00:13:29,110 certain characteristics of cash flows have to be met in order to, 210 00:13:29,110 --> 00:13:33,580 or certain requirements of the formula have to be met, in order to use it. 211 00:13:33,580 --> 00:13:34,630 So good luck with the problems.