this is a lecture from open tuition to benefit from the lecture you should download the free lecture notes from open tuition calm the last bit of the arithmetic on discounted cash flow is annuities and perpetuate ease and so in this lecture let me go through them first of all an annuity what we mean by an annuity is an equal amount and equal cash flow each year and so to explain exactly what I mean and how we deal with it look an example for a machine will cost forty five thousand and it's expected to generate eight thousand a year for each of the following eight years the cost of companies fifteen percent a year calculate Li and PV now you could actually do it exactly the same way as we have been doing in the previous lectures you know there's an outflow of 45,000 the cost of the machine now we've then got an inflow of eight thousand a year for eight years so eight thousand in one year eight thousand in two years in three years and so on and just like before you could use the tables of discount each flow individually and so that my present value tables here the cost of capital is fifteen percent so the discount factors of fifteen percent forty five thousand its forty five thousand but from then on from fifty percent for one year it's point eight seven for two years 0.75 six three years 0.65 eight and so on now you could do that but it's not hard but it obviously takes time and because it's an equal amount each year instead of multiplying 8,000 by 0.87 I've got so far in each case you multiply the disco factor by 8,000 it'll be somewhat more efficient simply to multiply 8,000 by the total of those discount factors well fine unless we have to sit down and list of all about the low and then because it's something that was reasonably commonly an equal amount each year we are given a second set of discount tables they're annuity tables which give you the total discount factor for any number of years from one well on the tables up to a maximum of 15 so instead of giving the discount factors for each year separately which is what the present value tables do their new tea tables are giving you the totals of the factors and so now we may as well make use of it's a much quicker for example four will be to say okay that'll help flow a cost of forty five thousand and then instead of writing eight thousand a year down eight times we say well it's eight thousand per annum thing is one to eight that's a sort of short term for it that's an a short way of writing it's a thousand in one a two-year three-year and so on when we come to discount at fifteen percent forty five thousand discounts to forty five thousand but the eight thousand a year again instead of multiplying each year separately by each year's discount factor the tables the annuity table gives us the total of the discount factors here we want fifty percent for the fifteen percent column we want the total for eight years you look at the area row it's four point four eight seven now you can't check that yourself if you want if you turn to the present value table where you've got the individual factors on year two year three year up to eight years and the mode and you'll get 4.4 eighty-seven as subject to rounding because the tables on they go to three places but as a result the present value of the inflows the annuity and I get to be thirty five eight nine six and so the net present value is minus nine one zero four and of course because it's negative will reject having positive we accept so make things quicker now you can only use these factors when it's an equal amount each year if the flows are changing each year then you have to discount each one separately as we did before there's just one thing to be careful of that the annuity factors are giving you the total from year one to anything look at example five example five the cost of capital is twelve percent per year once the present value of twenty thousand first receivable in four years time and there are each year for a total of ten years so that's a slight problem here there's a sorry we don't do that baby justice what's the present value there is an annuity in that we're receiving 20,000 for a total of 10 years but the first receipt is in four years time and when's the last receipt gonna be now be careful here and I know it sounds silly but I always use math fingers because it says we're going to be a total of 10 years so we receive it in four years time for yes time five years time 6 7 8 9 10 11 12 13 the last receipt will be in 13 years time you know a lot right for 240 but it's 4 to 13 4 5 6 counter to 13 is 10 years in total its 20,000 per year so again we have an annuity and we want to discount it at 12% but as a set of anything oh the problem is in the tables for the annuity factors are only giving you the total factor for want to anything and so what we have to do is this we can get the total for 1 to 13 that's the 13 year annuity factor over the tables 13 years at 12% it's six point four to four but that's giving you the total have one two three four or five up to thirteen we only want the total from 4 to 13 while 6 by 4 to 4 is the total for 1 to 13 to get the total from 4 to 13 when he subtract the total for one two three if we remove one two three we're left with the total for 4 to 13 and how can we get the total for three years it's the three-year annuity and again from the tables 12% of three years is 2.40 to take that out and what are we left with two to four point zero to two that is the total for 4 to 13 and so the present value 18 for 40 so watch for that again if you do have n equal cash flow make use of the air and beauty tables but what for a flow starting late as it does here the other big though is what we call a perpetuity a modern perpetuity is it's my call the ultimate annuity it's where you've got an equal cash flow each year but instead of being for five years for eight years for ten years or whatever it's an equal cash flow each year for ever now I've got nothing in this world is forever but what we're talking about is an equal cash flow for so long as you know it mess will be forever now look at example six a machine cost 1030 a hundred thousand and is expected to generate twelve thousand in perpetuity so again the way we write it the cost was now a hundred thousand we then get inflows of twelve thousand per year but as I've just said instead of it being one to five one to ten it's forever it's one to infinity and the present value well a hundred thousand this comes two hundred thousand but the perpetuity but of course you could do each flow individually do for one year two year three year for you and you'll find you know when you're getting up to 20 years 30 years 40 years lettuce come father gets so small it effectively disappears well then that would be ridiculous you can turn that in it is an annuity but of course the tables only go up to a maximum of 15 years that doesn't help this is the one place you need a formula I'll justify the formula after but the discount factor for a perpetuity we multiply by 1 over r where r is the interest rate and what is it the interest the cost of capital is 10% we multiply by 1 divided by 0.1 as always we write it as x the factor but multiplying by 1 over PI - the same as dividing by 0.1 the present value 120,000 the net present value 20,000 we'd accept is positive so that's the one bit of learning here that you must know maturity is actually our rare in the exam this certainly can be asked annexed in one place you can't use tables you must learn that the factor is 1 over R now the logic behind it is suppose you and I who ever is what she placed up married but we've just got divorced and the courts have said I've got to pay you twelve thousand a year as part of the settlement well I don't want to have to keep contacting you to give you twelve thousand a year maybe though I've checked with the bank the bank are paying ten percent interest so instead of keep having to think about new each year and pay you twelve thousand I am going to put a hundred and twenty thousand in the bank you know better than twenty thousand of deposit it will at ten percent it will generate twelve thousand a year and I can then forget you completely you're giving the twelve thousand a year I just paid one amount of 120 that's the logic we just turning backwards I want to have twelve thousand a year we know interest is ten percent that would mean 1320 in the back finally for exactly the same exact the same problem as we had a minute ago with annuities 1 over r is the discount factor but only when it's warm to infinity it's the total factor the years 1 through to infinity look at the 1 number 7 the rate of interest is 55% this one wants the present value of 18,000 first receivable in five years time and thereafter annually in perpetuity so it's 5 to infinity its 18,000 a year we want the present value and 5% now factor there are two ways you can get the same answer I'll do it both ways so both give the same answer apart from rounding and that one way it's exactly the same way as we did with annuities to say well together total from 5 to infinity we need the total from 1 to infinity less the total for the first 4 years if I take off the total for the first 4 we're left with the top of Phi 2 infinity now 1 to infinity is 1 over R which have 5% 1 over point 0 5 which is 20 what's the top for the first four years it's the 4 year annuity which from the tables for years at 5% the annuity factor 3 point 5 for 6 which taking that out leaves us a 5-2 in thin see of 16 at 0.45 for which therefore gives the present value of 296 1 and 72 I said I'll show you two ways it's whichever you find most obvious for most people I think what we've done there is the quickest they alternative though is this is to say if it was 1 to infinity we multiplied by 1 over R which is 1 over point zero five five percent which is 20 so if I multiply by 20 have you been one to infinity that would give me the present value however ours isn't one to infinity it's 4 to infinity it's starting three years late instead of starting in one year was it fortunately sorry yes instead of starting in one year sorry I'll entertain I beg your pardon have it mean one to infinity multiplied by 1 over R but this one is not one to infinity it's 5 to infinity the perpetuity starts four years later it starts in five years time instead of in one year's time and so whereas and he'd been one to infinity multiplied by 20 we get the present value if he starts four years late multiplied by 1 over R will give us a value 4 years late and so to get two times zero with then need to multiply by the ordinary four year present value I'm sorry for you discount factor to get back those extra four years because it started four years late time five instead of time one and so the ordinary factor from the present value tables four years five percent is point eight two three giving total discount factor sixteen point four six so what did we get before sixteen point four five four and the difference is irrelevant for the exam but arises simply because all the factors are rounded to three decimal places so it doesn't matter which approach you take as I say for most people I think they found the first approach more obvious if you're happy with that fine on the around if you find the second approach more obvious do it that way it doesn't matter all right well as far as discounted cash flow calculations are concerned that is all what you might call the mechanics I swear you get perpetuity z-- most of the time it's either individual flows we use our present value table to discount occasionally annuities use your annuity table and of course making sure you can calculate intermarry return you