Very simple problem on % - explanation

IrinaK.

Homework Statement

Hello!
For example, there is a 'cash flow' of 80 000. Discount rate is 13%. Then to find discounted cash flow I need to do the following: 80 000 / 1.13 = 70 796.46

If, for example, the same 80 000 is a gross amount and I need to find net one with the same rate of 13%, then 80 000 * 0.87 = 69 600

Why are these numbers different? What is the meaning behind the first and second equation? I realize that it is very simple but somehow I have blanked out on this one.

Thank you!

Homework Helper
Gold Member

Homework Statement

Hello!
For example, there is a 'cash flow' of 80 000. Discount rate is 13%. Then to find discounted cash flow I need to do the following: 80 000 / 1.13 = 70 796.46

If, for example, the same 80 000 is a gross amount and I need to find net one with the same rate of 13%, then 80 000 * 0.87 = 69 600

Why are these numbers different? What is the meaning behind the first and second equation? I realize that it is very simple but somehow I have blanked out on this one.

Thank you!

The Attempt at a Solution

If you take the rate as a decimal ##r=.13##, in one equation you are dividing by ##1+r## and in the other you are multiplying by ##1-r##. They are not the same because$$\frac 1 {1+r} \ne 1-r$$In fact$$\frac 1 {1+r} = 1 - r + r^2 - r^3...$$which is an infinite geometric series. Notice that if ##r## is very small, the higher order terms are really small ##\frac 1 {1+r}## is close to ##1-r## in that case.

Mentor
For example, there is a 'cash flow' of 80 000. Discount rate is 13%. Then to find discounted cash flow I need to do the following: 80 000 / 1.13 = 70 796.46

If, for example, the same 80 000 is a gross amount and I need to find net one with the same rate of 13%, then 80 000 * 0.87 = 69 600
In the first case, the 13% refers to the discounted cash flow. 70 796.46 plus 13% of 70 796.46 equals 80 000. In other words, the question asked is: how much cash flow do you need such that the cash flow growing by 13% will give you 80 000.

In the second case, the 13% refers to the 80 000. 80 000 minus 13% of 80 000 equals 69 600.

Homework Helper
Dearly Missed

Homework Statement

Hello!
For example, there is a 'cash flow' of 80 000. Discount rate is 13%. Then to find discounted cash flow I need to do the following: 80 000 / 1.13 = 70 796.46

If, for example, the same 80 000 is a gross amount and I need to find net one with the same rate of 13%, then 80 000 * 0.87 = 69 600

Why are these numbers different? What is the meaning behind the first and second equation? I realize that it is very simple but somehow I have blanked out on this one.

Thank you!

The Attempt at a Solution

As others have indicated, the mathematical reason is that $$\frac{1}{1.13} \neq 1 - 0.13$$
In other words, ##0.8849557522 \neq 0.87##.

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IrinaK.
In the first case, the 13% refers to the discounted cash flow. 70 796.46 plus 13% of 70 796.46 equals 80 000. In other words, the question asked is: how much cash flow do you need such that the cash flow growing by 13% will give you 80 000.

In the second case, the 13% refers to the 80 000. 80 000 minus 13% of 80 000 equals 69 600.
yes, exactly, I understand that, but I don't understand why is this so :) it seems that there are some basics which I miss.

IrinaK.
If you take the rate as a decimal ##r=.13##, in one equation you are dividing by ##1+r## and in the other you are multiplying by ##1-r##. They are not the same because$$\frac 1 {1+r} \ne 1-r$$In fact$$\frac 1 {1+r} = 1 - r + r^2 - r^3...$$which is an infinite geometric series. Notice that if ##r## is very small, the higher order terms are really small ##\frac 1 {1+r}## is close to ##1-r## in that case.
thank you, yes, it's true, I understand this with the use of these formulas, but I don't understand the reason

Mentor
yes, exactly, I understand that, but I don't understand why is this so :) it seems that there are some basics which I miss.
Do you mean that the problem is not with the math, but with the economics?

IrinaK.
I've tried to delete this particular massage to repost it as a new thread because it refers to a bit different topic, but couldn't, so I have deleted the text

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IrinaK.
Do you mean that the problem is not with the math, but with the economics?
no no - no problems with economics, but a huge one with basic math ;)

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Homework Helper
Gold Member
2022 Award
no no - no problems with economics, but a huge one with basic math ;)

I guess your problem is as follows:

You start with 100 and take 10% off. This gives you 90. Then you add 10% on. This gives you 99.

Why, when you take 10% off, then add it back on again, do you not get back to where you started?

The simple answer is that the first 10% was 10% of 100 and the second 10% was 10% of only 90. And they are different.

Does that help?

IrinaK.
I guess your problem is as follows:

You start with 100 and take 10% off. This gives you 90. Then you add 10% on. This gives you 99.

Why, when you take 10% off, then add it back on again, do you not get back to where you started?

The simple answer is that the first 10% was 10% of 100 and the second 10% was 10% of only 90. And they are different.

Does that help?

simple as it is! :) Thank you so much! Right! kindergarten issue - sometimes it happens :) Please, take a look at the following:

1) in case 80 000 is 100%, then 80 000 less 13% is 0.87 of 80 000 and equals 69 600

to "restore" back to 80 000 we have to multiply 69 600 by 1.1494 which is higher then initial 13% - why in this case is this one higher, if we are not changing the percentage base?

2) if 80 000 is 113%, then to find 100% value (present value), I have to 80 000 / 1.13.

Mentor
simple as it is! :) Thank you so much! Right! kindergarten issue - sometimes it happens :) Please, take a look at the following:

1) in case 80 000 is 100%, then 80 000 less 13% is 0.87 of 80 000 and equals 69 600

to "restore" back to 80 000 we have to multiply 69 600 by 1.1494 which is higher then initial 13% - why in this case is this one higher, if we are not changing the percentage base?

2) if 80 000 is 113%, then to find 100% value (present value), I have to 80 000 / 1.13.
If you write it as an equation it might be easier to understand. I'm using PV for present value and FV for future value (= 80,000).
80,000 = 1.13PV
So PV = 80,000/1.13 = 70,796.46 (rounded to nearest cent)