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0:04
Hi and welcome to this finance lecture for Stockholm Business School. My name is Anders, and I'll talk about the time value of money and more specifically, I will talk about the present value of an annuity,
0:17
First of all, however, we start with some repetition.
0:20
Johan talked about financial decisions and said that it was about giving up resources today, in the hope of obtaining something good in the future
0:31
In order to be able to compare the initial outlay, or the resources we give up today with the good stuff we get in the future, we had something called the time value of money,
0:44
We said that money today is better than money in the future. Why is that?
0:50
Well, because I will have inflation. For the same amount of money I can buy more stuff today than I can in two years'time
1:00
We also have the interest, the interest rate If I get the money today I can deposit it to a bank account and get some interest and have more money in the future
1:13
Last of all, we have some uncertainty. If I get the money today, I obviously know that I have the money in my hand If someone tells me that I will get the money in the future You never know. The company might go bankrupt, your friend might disappear,
1:30
might not be very likely, but you never know. So of these three reasons, money today is better than money in the future.
1:41
In order to compare money today with money in the future, we had some formulas that Johan showed you before
1:50
We said that to get the future value, or
1:55
the equivalent future value, so to speak, we took the cash flow today, which would be this one, this one. here,
2:04
we multiplied it with the factor one plus r
2:08
where r here is the interest rate, or, the discount rate as we often call it
2:16
and raised to the power of n, which is the amount of years we want to move this specific value
2:24
so we said that
2:27
obtaining
2:29
this C today,
2:32
is equally good as obtaining
2:36
this hear, C times one plus the interest rate raised to the power of N in N year's time.
2:47
In your formula sheet it will look like this here is the future value.
2:52
so the future value of some cash flow, C, is this cash flow multiplied by one plus the discount rate raised to the power of N, which is the amount of years you want to move it.
3:07
However we also, or Johan also talked about
3:12
present value, which is basically just the inverse So instead of saying that equivalent future value of some cash flow we said this is the equivalent present value of some cash flow
3:31
if we take this cash flow
3:34
that we will receive
3:38
in N years from now,
3:40
and if we divide it with one plus, r where, r is the discount
3:47
raised to the power of N, which is the amount of years until we get this cash flow
3:53
we get the present value of this cash flow
3:57
The present value of this cash flow is the equivalent cash flow today, compared to the cash flow C in N years
4:06
So, it is equally good for me to get this here the present value
4:13
today as getting C in N years
4:18
If you for one moment, forget about inflation and the uncertainty, you can think of it as if I get the present value today I can deposit the money on a bank account and get
4:35
r or some interest rate for N years,
4:41
So basically, if I deposit C here I will have this in N years
4:48
or the other way around
4:51
if I would deposit the present value here
4:56
today and get C, or get the interest rate for N years, I will have C in the future. That is why they are equivalent, so to speak,
5:08
Now we can move on, because you have already learned about this
5:18
We can go on to the interest rates or the discount rates
5:23
They should reflect the inflation, other investment opportunities, and risk. So, as we talked about earlier with the time value of money.
5:35
Ah, the interested rate used or the discount rate should reflect the inflation it should reflect what other investment opportunities you have. Could you deposit your money in another bank account? Could you
5:49
perhaps purchase a car and gain some money on that? Could you, whatever, you can buy stocks? You name it.
5:59
Last one is risk. With this specific investment decision,
6:05
what risk do you have? Is it a risky investment, or is it a less risky investment?
6:11
Generally, deposited money in the bank account is not so risky it's less risky
6:18
Perhaps investing in a big factory is a bit more risky, obviously, since you have a great or a big outlay You never know what the future will show you.
6:33
This interest rate is referred to as discount rate, or cost of capital,
6:41
or hurdle rate It is all the same thing. Don't get confused if we use different words for the same thing.
6:54
Next up is the net present value
6:57
Johan talked a bit about the net present value, and that it was the present value of the positive cash flows, minus the present value of the negative cash flows
7:08
Generally, when you take investment decisions, you give up resources today,
7:15
which is then the present value of the negative cash flows,
7:20
in the hope of obtaining some positive cash flows.
7:25
Which is here
7:27
So, if you take, or if you calculate the present value of the positive cash flows and deduct the present value of the negative cash flows, you get the net present value.
7:37
You can say that it is the present value of all cash flows
7:43
and it is, as I said, net present value It's called that because it is the net present value obviously
7:54
It could look like this It could be that we invest three hundred today
8:03
and we get eighty every year, in five years time, for example
8:11
In order to decide if this investment is a good investment we need to calculate the present value of these cash flows
8:23
These cash flows, since they are of equally big size and since it is exactly one period between every cash flow it is called an annuity We get to that later
8:38
As I said, in order to compare their positive cash flows to the negative cash flows, we discount it till today,
8:47
by using the formula that we have
8:53
We use this formula down here
8:57
in order to decide if it is a good investment opportunity or not,
9:03
so we can start with discounting, as we say, all the positive cash flows
9:09
and we got eighty
9:13
after one year,
9:16
so we take eighty and divide by one, plus the interest rate, which is fifteen percent in this case
9:25
raised to the power of one, since it is one year
9:33
and if we calculate this
9:36
it is
9:38
approximately Sixty-Nine point
9:43
five six five
9:48
so then we take the next year and do the same thing
9:53
take eighty divided by
9:56
one plus zero point fifteen, since the interest rate was fifteen percent
10:03
raised to the power of to, since it is in two years time
10:10
that is
10:12
sixty point forty nine
10:19
and we take the next cash flow
10:23
which is in three years time also eighty
10:29
the same interest rate of fifteen percent raised to the power of three, since it is in three years
10:39
and that is
10:41
Fifty-Two point sixty
10:45
all right
10:49
and will take the fourth year,
10:54
we get eighty
10:57
and we divide it by one, plus zero point fifteen. once again, and raised to the power of four.
11:07
and this is
11:09
forty five point seventy four
11:18
and lastly, the last cash flow, which is eighty
11:22
and it is in five years. so we divide by one plus zero point fifteen.
11:28
raised the power of five,
11:31
that is approximately thirty-nine point Seventy-Seven,
11:39
so
11:42
if we sum this up all the positive cash flows
11:47
I denote it by
11:50
sigma
11:52
we get.
11:55
so we get sixty nine point five hundred sixty five plus sixty point forty nine plus fifty two point sixty plus forty five point seventy, four plus thirty, nine point seventy, seven, which is approximately
12:15
two hundred and sixty eight,
12:23
if we want to get the
12:26
net present value of this investment opportunity,
12:30
we take the present value of the positive cash flows, which is two hundred and sixty eight
12:38
and deduct present value of the negative, which is then initial outlay.
12:45
that was three hundred. so if I write here net present value,
12:53
is equal to two hundred and sixty eight,
12:59
minus three hundred
13:02
is equal to minus thirty-two
13:09
so we can conclude that this investment,
13:13
got us a net present value of minus thirty-two which is not good
13:21
the company will decrease in value. if we do this investment why we obviously will not do it. and however,
13:30
I think this was quite boring. and I think you also thought it was quite boring, you perhaps imagine but what if it is twenty years or fifty years
13:40
I mean I can't sit there and calculate for all day
13:44
that is completely correct and that is why we will go to a model expansion, or we will expand the model
13:54
we will have the present value of an annuity
13:58
which allows us to compute the present value of an annuity with a single formula so we don't need to calculate the present value of every cash flow by itself,
14:11
and for this, we have some requirements
14:14
first of all, all cash flows, except the initial outlay needs to be of equal size,
14:22
so as in our example, every flow was of the size eighty. so that is fine
14:29
and it has to be equally along between every cash flow in our example, it was one year between every cash flow. so it is also fine this an annuity, as I said before
14:42
So if it is like this. If all cash flows are of equal size, and it is equally long, between every cash flow we can calculate a present value of this annuity with one single formula
14:58
It transform every future cash flow to a single value one period before the first cash flow
15:07
and I highlight this because it is very important that you remember.
15:12
If the first cash flow comes in one year that is now a previous example we move all these cash flows till today one year before the first cash flow but if the first cash flow comes in year five we move all the cash close to year four
15:33
it is just good to remember it seems you might have to
15:40
you might not be eh done with the calculations when you have done the present value of an annuity formula, so to speak
15:51
now example, we move all the cash flows at the same time, which is a lot better, or it takes less job
16:01
this is the formula
16:04
so in order to get the present value of this annuity, this stream of equally big cash flows,
16:12
we take the first cash flow
16:16
and it is important that is the first
16:24
Sorry
16:25
It''s not important that it is the first. That's later on. Since all cash flows are equally big,
16:33
however you take the cash flow in our case, eighty you divide by the interest rate, or the discount rate
16:43
and multiply this with this whole factor, one minus one divided by one plus or r, the discount rate
16:55
and this raised the power of N, which is the amount of cash flows I write that down
17:13
So, it is not the amount of years you want to move it. It is the amount of cash flows so in our case, we had five cash flows the n is five, in our case
17:24
and in the formula sheet, it will look something like this where P V is the present value, or short, for the present value. Once again, you take the cash flow divided by the interest rate and multiply by this factor, one minus one divided by one plus r raised the power of n
17:50
If we apply this formula to our example
17:54
we just get the present value of the positive cash flows to start with
17:59
and it was eighty
18:04
divided by one plus the interest rate, which was fifteen percent so zero point fifteen.
18:14
sorry
18:17
not one plus, it should be only the interest rates
18:23
so fifteen percent, zero point fifteen,
18:29
multiplied with one minus
18:32
one divided by
18:36
now it is one plus the interest rates so zero point fifteen
18:41
raised to the power of five since it was five cash flows
18:48
so if we type this into the calculator
18:54
we will get
18:56
approximately
18:59
two hundred and sixty eight,
19:03
which is, of course, the same as we got before
19:07
so in order to get the net present value
19:14
we take, present value of the positive cash flows, two hundred and sixty eight,
19:20
minus present value of the negative cash flows which is the initial outlay, or three hundred,
19:28
and we get minus thirty-two
19:32
as before
19:38
so we can now continue to another example
19:42
and here it says that an old teaching assistant is looking to sell solutions to old exams this demands her to refrain work that would have given her a present value salary of one hundred thousand
19:56
she estimate that she will be able to sell solutions worth ten thousand each year in twenty years, time,
20:04
and she uses a discount rate of ten percent when evaluating investments
20:10
is this a good investment? or, in other words, what is the net present value?
20:19
so just as a reminder, it is always good to start off by typing down all the information so we don't get lost
20:29
so the annual net cash flow, the positive cash flows, we start by calculating the present value of these, using the present value of an annuity formula
20:44
so take the cash flow, which is ten thousand
20:53
divide by r, so zero point ten, in this case
20:59
multiplied with one
21:03
one divided by
21:06
one plus r, which is ten percent
21:11
and raised to twenty
21:14
since it is twenty cash flows
21:21
and if we type this in
21:25
we should get approximately
21:30
eighty-five thousand
21:34
one hundred and thirty six,
21:39
so if we want to know the net present value of this investment opportunity,
21:45
or this business idea
21:49
we take the present value of the positive cash flows, which is eighty five thousand one hundred and thirty six
21:57
minus the present value of the outlay, or
22:02
what she couldn't work for. so minus hundred
22:15
and we get minus fourteen thousand eight hundred and sixty four
22:25
well you wouldn't work for minus fourteen thousand so obviously this is not a good investment, or this is not a good business idea, as it looks at the moment, it is better that this teaching assistant does her regular job and earns the net present, the present value, salary of one hundred thousand
22:48
so lastly we can have some final example
22:54
and here a guy called Jon Lindgren
22:58
have been offered,
23:00
fourteen million U S dollars for the lead role in the coming three Larry Boy movies which is paid out when the contract is signed
23:10
if Jan accept this offer, he must refrain other roles that would have given him five million years dollars in the end of every upcoming three years
23:22
also assume that Jan's cost of capital is ten percent
23:26
and the question is, what is the net present value of taking the lead role in the three Larry Boy movies
23:36
so we start with typing down what we know
23:40
and I think it's good to also make a timeline If it is twenty years it could be, or
23:48
feels as a lot of work to do. However, I do recommend you to do it, because it is sometimes easy to move the cash flows one year earlier or later than you want to or something.
24:03
it is always good to do and here it is three years, so it is definitely worth
24:10
so we said that, it
24:15
looks something like this.
24:19
the sign on bonus or basically it is better to
24:26
make it like this perhaps
24:30
the sign on bonus was fourteen million US dollars
24:34
and this made him refrain work for five million US dollars
24:41
in the end of every three years
24:44
so this is pretty much what the investment looked like, or the lead role in the movies. you either get fourteen million dollars today, or you do some other work and get five million dollars each year, with the start in the end of the first year. so here, if this is the first year. this is the second year. this is the third year
25:08
then you can think that you get five million
25:11
last December in the first year and get five million
25:16
the last December in the second year
25:20
and you get five million in the last of December the third year.
25:26
As usual, in order to compare these two, you need to calculate the present value of this three year salary
25:36
so if we calculate the present value of these we used to formula
25:42
and the cash flow is five million
25:46
divide by r, which is zero point fifteen,
25:53
sorry it was ten,
26:00
you multiply with one minus one divided by one plus r, once again, ten percent, not fifteen. as I said and it is three cash flows
26:17
if you calculate this present value, you will get twelve million
26:23
four hundred and thirty four thousand US dollars
26:29
and you can now compare the present value of the positive cash flows,
26:34
with the present value of the negative cash flows. in this case, it is perhaps not a negative cash flow, but its cash flows that you would have given if you didn't take a lead role
26:45
so it is the alternative
26:48
to take, net present value,
26:53
which is equal to
26:55
fourteen million
26:58
minus twelve million four hundred and thirty four thousand
27:08
and you get,
27:10
one million
27:12
five hundred and
27:15
Sixty-Five thousand approximately
27:25
actually you get sixty six
27:32
so
27:34
this is basically the net present value of taking the lead role,
27:39
you get a bit over one point five million US dollars in present value terms
27:45
Ah which, of course, we see as a good investment. and Jan, in this case, should take the lead role in the three Larry Boy movies
27:56
so to sum up
27:58
a series of equally big cash flows are called an annuity
28:03
the present value of such an annuity is calculated as follows
28:09
to get the present value you take the cash flow here
28:13
you are divide by the interest rate, or the discount rate,
28:19
and you multiply this by the whole factor, one minus one
28:24
divided by one plus r, once again the discount rate,
28:28
raised to the power of N, which is the amount of cash flows in the annuity. so if the annuity have five cash flows N is five, if it is twenty cash flows, N is twenty and so on
28:45
and last of all, you need to remember that all cash flows, except for the initial outlay, needs to be of equal size
28:55
and it has to be equally along between every cash flow
28:59
and the value received is in one period before the first cash flow
29:07
and that was all for today thank you very much good luck
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