Investment Decision Rules 3 - Equivalent Annual Annuity - YouTube

Hello and welcome back. Today we are going to look  at a investment decision rule called equivalent  

annual annuity. You've actually already seen  this when Anders did it in the time value of  

money. He computed an equivalent annual annuity  but now we're going to apply it on projects. We  

are still in this series where we look at  different investment decision rules. Today,  

we're going to look at equivalent annual  annuity. Before we start it could be useful  

to have a little bit of terminology. When you  have two investments and you cannot choose both,  

so if you choose one it means that  you cannot do the other. For example,  

let's say that I'm going to build a house on the  ground that I own. Either I build a big house or  

a small house. If I choose one I cannot also do  the other because then the space is already used.  

If you have two projects, or investments,  which are like this, so we cannot do both,  

then they're said to be mutually exclusive. I'm  going to use the term later on. Let's take an  

example. We have two different investments here.  One has cash flows of 50 first year, 70 next year,  

80 the third year, 100 in the fourth and fifth  year. The second one has 80 every year and they  

both cost 300. If these are mutually exclusive,  which one should we choose? Well, they have the  

same lifespan and that means that we should use  the net present value. We simply compute the net  

present value both and we take the one with a  higher NPV. That's what we did earlier. Here,  

in this case, I've done it. The NPV for investment  B is much higher than for the investment A,  

which means that we should choose investment B  of these two. What happens if it looks like this  

instead? Well, now it might still be so that  B have a higher NPV but as you see here; after  

the fifth year the investment B still goes on and  investment A doesn't, so the question then comes;

what are we going to do in this later period?  What happens here? Because then we already get  

all the money from A so maybe we can reinvest  it at something else with a positive NPV. How  

can we then say whether or not B is better than  A? We need something else. So, in this case it  

turns out that B has a slightly higher NPV but  as I said they have different lifespans. So,  

it's not so easy to distinguish which one is  actually better just by looking at the NPV. Let's  

say that they can be reinvested. So, we can do the  same investment over and over again. That would  

mean that; for investment A, we could again do  this here. So, we get the same cash flows again.  

It would be very nice if we had a way to convert  this present value, or the net present value,  

into an annual cash flow. So, we would like to  convert this cash flows in the future, one set  

per year for the entire life of the investment,  which corresponds to this NPV. In fact, you  

already done that. You did that in the in one of  the videos of time value of money with Anders. So,  

just to remember. We always consider net cash  flows and we want it to be in comparable terms.  

In this case, comparable could mean that we have  an annual net cash flow instead of this NPV. So,  

if we can change this so it's something positive  every year. then these two can be compared. So,  

remember one of the disadvantages with NPV.  When we compare a mutually exclusive investment,  

that have unequal lifespans and they can be  replaced by positive NPV investment at the end  

of the investment, then just NPV is not enough  to say which one is better. But we're going to  

make it slightly simpler than your say that some  positive. We're going to look at investments  

that can be repeated so, we repeat the exact same  investment or at least that exact same terms so,  

then we can simply add them onto each other  after the end of the investment. Again,  

NPV is not enough to say which one is better but  we can compute the equivalent annual annuity,  

which we already know how to do, and we do it  like this. We take the NPV and then we divide  

it by this big sum here. Or not a sum, but with  a big factor. Then, we get what is called an  

equivalent annual annuity. I want to stress  it's very important that you remember to have  

parentheses on the sides here. If you don't,  it might be wrong when you hit it into your  

calculator or your computer. What happens then if  you do? Let's take this example that we had. So,  

we had the NPV on both investments and we have the  formula for the equivalent annual annuity. Again,  

remember the parentheses. So, we can simply do it  this time. We have a cost of capital of 8%. So,  

we take for A. For example, we take 11.38 and  then we should divide this by a big parenthesis,  

1 divided by the interest rate, 0.08, times 1  minus 1 divided by 1 plus interest rate, 0.08,  

to the power of n. The n here is the life span  of the projects. So, for A this is 5 years. Then,  

we close the parenthesis. For B we can do the same  thing. We take the NPV of 12.38 and we divide it  

by 1 divided by 0.08 times 1 minus 1 divided by  1 plus 0.08. Now, this n should be the number of  

years it exists, which in this case is 7. We close  the parentheses. So, the only difference here

in fact the five here and the seven here and that  corresponds then to the different lifespan. So,  

we compute something where the only difference in  the computation is how long the investment keep  

going. If we do this, we find that the equivalent  annual annuity of A is 2.85 while B is only 2.38.  

Equivalent annual annuity. So, you take the  NPV and you find an annuity, that is a cash  

flow every year with all the same size, which  is equivalent in present value to this NPV. So,  

what this says is that instead of receiving  an NPV of 11.38, we would be equally happy to  

receive 2.85 every year for five years then. For  B, instead of receiving an NPV of 12.38, we will  

be equally happy to receive 2.38 each year. Since  we said that A could be repeated it means that we  

can do this one more time or many more times. So,  we could get this 2.85 every year for a longer  

period and then we have suddenly converted it to  something which can be compared although they have  

different lengths. Because A would give us 2.85  every year and B would give us 2.38 every year or  

something which is equivalent to that. Therefore  A is the better investment. The general investment  

decision rule for the equivalent annual annuity is  just like NPV. If the equivalent annual annuity is  

greater than zero, then you should invest. If  it's not greater than zero, if it's negative,  

you should not invest. If you have two mutually  exclusive investments, which can be repeated,  

then or replace at equal terms, then we should  choose the one with the highest equivalent  

annual annuity. Now, we can suddenly compare  investment with different lifespans when they can  

be repeated. It has many names. Sometimes it's  called equivalent annual benefit or equivalent  

annual cost. They all refer to the same technique  of how you do it. It could be good to know with  

you encounter it some other time okay. Remember,  an annuity is a regular constant cash flow for a  

fixed number of periods. Like this. So, what  we did know was that we took the NPV, here,  

and then we converted it to cash flows that are  of equal size every year, which has a present  

value corresponding to this NPV. So, it's a way  of taking the NPV and making it into instead  

annual positive cash flows if it's a positive NPV  then. Therefore, we could compare them when they  

have different lifespans. As I said earlier,  if we find the EAA, equivalent annual annuity,  

getting that sum every year for the rest of the  life, or for the entire life of the investment,  

would make us equally happy to receiving the NPV  today. That is why we can compare then, different  

investments. Advantages of the equivalent annual  annuity; it accounts for the time value of money.  

It accounts for the risk of the project because  it has an interest rate. It accounts for all  

cash flows caused by the project. It enabled us to  rank investments of unequal length, if they can be  

replaced. You need to remember that if we have to  choose between two mutually exclusive investments,  

and they have unequal lifespans but you cannot  repeat them or you cannot replace them in any way  

with a positive NPV, then you should not look at  equivalent annual annuity. In that case, then NPV  

is the decision criteria you should use. Let's say  that you have a once-in-a-lifetime opportunity,  

you can only choose this one or that investment,  then there's no replacement so there's no need to  

think what happens after this point. You should  take the one with highest NPV. Let's take a final  

example. Here A and B are mutually exclusive  and they can be repeated. Which investment is  

the better one? You can see we have an initial  investment of 150 for A, 100 of B. We have a net  

annual cash flow of 40 for A and of 35 for B. This  means, 40 every year for five years and this means  

35 every year for four years. We can also say that  A has an end value, a salvage value of 20. So,  

in the fifth year we can sell on maybe this  machine and receive 20. We have a cost of capital  

about 10%. So, what do we need to do? We first  need to compute the net present value of both  

so we start with A. So, the net present value of  A is then equal to minus 150 plus 40. Since this  

is a net annual cash flow, we can use what we  learned in time value of money if we take this  

times 1 divided by the interest rate, which  is ten percent, times one minus one divided by  

one plus zero point 10, to the power of five,  because it's five years. We also need to add  

salvage value here, which is in the last year.  So, we add plus 20 times one divided by one plus  

zero point 10 to the power of five. If you would  draw this in a time line. A would cost 150 today

You would get 40 every year for five  years and then in the last year we would,  

except for this 40, we would also get an extra  income of twenty. This was very sloppy writing

If we compute this net present value, we find  that it's 14.05. We can make the similar thing  

for B. So, the NPV of B would then be minus  hundred plus the annual cash flow here is  

thirty five times one divided by zero point  ten times one minus one divided by zero point  

ten. Now we have four years, so we take it  to the power 4. We find that this NPV is  

equal to ten point 95. What we should do now  is to take this NPV and divide it by this,  

to get the equivalent annual annuity. I already  showed you how to do that so you can take it  

as a practice at home. You find that the  equivalent annual annuity of A is three  

point seventy one and the equivalent annual  annuity of B is three point forty five. So,  

which investment should we then choose, if we  assume that this can be repeated? We should  

focus on the highest equivalent annual annuity. A  has three point seventy one. B has 3.45. So here,  

A is clearly the better investment. That was also  what NPV would have told us but that's not always  

the case. In this case they agreed but it doesn't  have to be like that. We have looked at NPV and  

NPV is always like the benchmark that we measure  well methods to. You calculate the present value  

of all future cash flows and the investment is  profitable if NPV is greater than zero. Today  

we looked at the equivalent annual annuity. We  take the NPV and you convert it to an equivalent  

annual annuity and then you have like a yearly  surplus which you can compare between investment.  

For a single investment it's profitable if  the equivalent annual annuity is greater  

than zero. If you compare mutually exclusive  investments that can be repeated, you take the  

one with the highest equivalent annual annuity.  That's it for this video. Thank you very much.