August 21, 2014 11:20 AM Subscribe

I'm trying to compare the compounded losses due to inflation to non-compounded profits. E.g. if inflation makes me lose 3% of an investment each year, what % would I need to profit each year in order to exactly keep up? This is tricky because the profit is not compounded the way inflation is. I'm not looking for an answer to this specific question, but rather, looking for a tool or method to do this kind of math.

Ok, that's convoluted, so here's an example.

Say I can spend $100 to build a machine that prints $3 per year. In other words, it returns 3% of the initial investment each year.

Assuming inflation stays at 3%, how do I compare profits to inflation loss? Initially I thought that 3% gain would match up exactly with inflation each year, since your money would lose $3 of value (due to inflation) but you'd print $3.

But then I realized you'd lose less than $3 of value after the first year, since the inflation works on money that's already lost value. In other words, it compounds, so you would lose a little less than $3 of value the second year, and even less the next year. So then I figured that printing the $3 would put you above inflation.

But then I realize that the $3 you print is subject to inflation as well. It loses $0.03 of value, so you're not adding $3 per year, you're adding $2.91, which means the first year your value would go down from $100 to $99.91. So then I thought that you would be doing worse than inflation.

The more I think of this, the more confused I get. How do I do this math??

Obviously, this isn't a realistic scenario. A more real scenario, I'd be selling something, and the sale price would go up as inflation went up, so it would be easier to match inflation. But I can't even figure this out in a vacuum, I don't know how I'm going to figure it out with the added complexity of being realistic.
posted by brenton to Work & Money (12 answers total) 2 users marked this as a favorite

Ok, that's convoluted, so here's an example.

Say I can spend $100 to build a machine that prints $3 per year. In other words, it returns 3% of the initial investment each year.

Assuming inflation stays at 3%, how do I compare profits to inflation loss? Initially I thought that 3% gain would match up exactly with inflation each year, since your money would lose $3 of value (due to inflation) but you'd print $3.

But then I realized you'd lose less than $3 of value after the first year, since the inflation works on money that's already lost value. In other words, it compounds, so you would lose a little less than $3 of value the second year, and even less the next year. So then I figured that printing the $3 would put you above inflation.

But then I realize that the $3 you print is subject to inflation as well. It loses $0.03 of value, so you're not adding $3 per year, you're adding $2.91, which means the first year your value would go down from $100 to $99.91. So then I thought that you would be doing worse than inflation.

The more I think of this, the more confused I get. How do I do this math??

Obviously, this isn't a realistic scenario. A more real scenario, I'd be selling something, and the sale price would go up as inflation went up, so it would be easier to match inflation. But I can't even figure this out in a vacuum, I don't know how I'm going to figure it out with the added complexity of being realistic.

So let's for a minute talk about a less contrived example example.

In 2014, you buy a machine that makes widgets for $100. The widgets sell for $3.

In 2015 that same machine now would cost $103. If we ignore depreciation, that's how much yours is now worth, in 2015 dollars. You also have $3 from the widget you sold in 2014. Your net worth is $106 in 2015 dollars. And your next widget will sell for for $3.09, and your machine next year will be worth $106.09, so you'll be worth $112.18 in 2016.

If you just kept the money your net worth in 2016 would still be $100. $100 in 2016 is only $94.26 in 2014 dollars. Your $112.18 in 2016 is $105.74 in 2014 dollars. So great! You're beating inflation!

The fact that your hypothetical machine's output is a fixed dollar value makes things more like an investment instrument, like a CD or something. The value of its output is going down, so*the machine's* value should go down at the same rate. Meaning its value will stay at $100. Which means your analysis is basically right that it's not quite keeping up with inflation. But it's still better than just hanging on to the cash.

posted by aubilenon at 12:09 PM on August 21

In 2014, you buy a machine that makes widgets for $100. The widgets sell for $3.

In 2015 that same machine now would cost $103. If we ignore depreciation, that's how much yours is now worth, in 2015 dollars. You also have $3 from the widget you sold in 2014. Your net worth is $106 in 2015 dollars. And your next widget will sell for for $3.09, and your machine next year will be worth $106.09, so you'll be worth $112.18 in 2016.

If you just kept the money your net worth in 2016 would still be $100. $100 in 2016 is only $94.26 in 2014 dollars. Your $112.18 in 2016 is $105.74 in 2014 dollars. So great! You're beating inflation!

The fact that your hypothetical machine's output is a fixed dollar value makes things more like an investment instrument, like a CD or something. The value of its output is going down, so

posted by aubilenon at 12:09 PM on August 21

I'm tracking with this. It's going to take me a few re-reads, but I'm getting it. Let me digest leopard's post first. Based on his answer, I'm thinking I can generalize some equations.

s =**s**tarting money ($100) that is used to buy or not buy the money machine

y =**y**ears

k_{n} = worth after **y** years if I keep the money (nominal)

k_{i} = worth after **y** years if I keep the money (inflation-adjusted)

b_{n} = worth after **y** years if I buy the machine (nominal)

b_{i} = worth after **y** years if I buy the machine (inflation-adjusted)

k_{n} = s

b_{n} = s + (s · 1.03 · y)

k_{i} = k_{n} · ( 1 / (1.03^{y}))

b_{i} = b_{n} · ( 1 / (1.03^{y}))

Basically, just do the math nominally first, and then run the inflation on that number at the very end? Does it really work like that to just calculate inflation at the end like that? Seems like you'd need to take it into account as you go.

posted by brenton at 1:03 PM on August 21

s =

y =

k

k

b

b

k

b

k

b

Basically, just do the math nominally first, and then run the inflation on that number at the very end? Does it really work like that to just calculate inflation at the end like that? Seems like you'd need to take it into account as you go.

posted by brenton at 1:03 PM on August 21

No, if you combine dollar values across years, you inflation-adjust everything first.

This is easier if you don't use equations and variables.

If you buy a machine for $100 in 2014, then get $3 every year for 10 years from 2015 to 2024, and then sell the machine for $90 in 2024, what's your profit?

Nominally, you make $40 -- this number is economically meaningless.

In 2014 dollars, you make -$100 + $3 /1.03 + $3 / 1.03^2 + ... + $3/1.03^10 + $90/1.03^10 (whatever that turns out to be). This is the "real" value of your investment in 2014 dollars.

This will be smaller than $40 since your expenditures come up front and your revenues come later (with a weaker dollar).

posted by leopard at 1:09 PM on August 21

This is easier if you don't use equations and variables.

If you buy a machine for $100 in 2014, then get $3 every year for 10 years from 2015 to 2024, and then sell the machine for $90 in 2024, what's your profit?

Nominally, you make $40 -- this number is economically meaningless.

In 2014 dollars, you make -$100 + $3 /1.03 + $3 / 1.03^2 + ... + $3/1.03^10 + $90/1.03^10 (whatever that turns out to be). This is the "real" value of your investment in 2014 dollars.

This will be smaller than $40 since your expenditures come up front and your revenues come later (with a weaker dollar).

posted by leopard at 1:09 PM on August 21

The problem is much simpler if you assume that the profit you make each year, $3, is also reinvested at 3%. That is, each year you take your $3 and use it to buy more machines, so your profit is compounded.

In this case, your real return is just the nominal return minus the inflation rate. So in your case, the nominal return is 3% and the inflation rate is 3% so your real return is exactly zero.

In other words, if you have a compounded 3% return, you will have 1.03^10 * $100 = $134 at the end of 10 years. But your $134 will be worth exactly $100 due to inflation, so your real return was zero.

If you don't reinvest your annual $3 in new machines, you actually have a negative return.

posted by JackFlash at 1:56 PM on August 21

In this case, your real return is just the nominal return minus the inflation rate. So in your case, the nominal return is 3% and the inflation rate is 3% so your real return is exactly zero.

In other words, if you have a compounded 3% return, you will have 1.03^10 * $100 = $134 at the end of 10 years. But your $134 will be worth exactly $100 due to inflation, so your real return was zero.

If you don't reinvest your annual $3 in new machines, you actually have a negative return.

posted by JackFlash at 1:56 PM on August 21

Wait a minute. Is my calculator broken, or is my head?

100 / 1.03 = 97.08737

... shouldn't it just be 97 a.k.a 97%? I was thinking dividing by 1.03 was the same as getting 97% of the original, but to do that you'd do:

100 · .97 = 97

So what does dividing by 1.03 do?

posted by brenton at 1:59 PM on August 21

100 / 1.03 = 97.08737

... shouldn't it just be 97 a.k.a 97%? I was thinking dividing by 1.03 was the same as getting 97% of the original, but to do that you'd do:

100 · .97 = 97

So what does dividing by 1.03 do?

posted by brenton at 1:59 PM on August 21

OMG. 100 is 3% more than **97.087...** but **97** is 3% less than 100. I am so confused.

posted by brenton at 2:05 PM on August 21

posted by brenton at 2:05 PM on August 21

Dividing by 1.03 and multiplying by 0.97 are different operations (although the results are pretty close).

3% inflation typically means that the price level rises by 3%. So something that costs $100 in 2014 would cost $103 in 2015.

To convert from 2015 dollars to 2014 dollars, you have to multiply by 100/103, or equivalently, by 1/1.03.

This isn't the same as multiplying by 97%. You can see this more clearly when you have a bigger rate. If inflation is 50%, then something that costs $100 in 2014 will cost $150 in 2015. You have to multiply by 1/1.5 to go from 2015 dollars to 2014 dollars; cutting the 2015 dollars in half would give you a clearly wrong answer.

JackFlash is correct above about reinvestment, but I figure that's a somewhat more advanced topic for now.

posted by leopard at 2:08 PM on August 21

3% inflation typically means that the price level rises by 3%. So something that costs $100 in 2014 would cost $103 in 2015.

To convert from 2015 dollars to 2014 dollars, you have to multiply by 100/103, or equivalently, by 1/1.03.

This isn't the same as multiplying by 97%. You can see this more clearly when you have a bigger rate. If inflation is 50%, then something that costs $100 in 2014 will cost $150 in 2015. You have to multiply by 1/1.5 to go from 2015 dollars to 2014 dollars; cutting the 2015 dollars in half would give you a clearly wrong answer.

JackFlash is correct above about reinvestment, but I figure that's a somewhat more advanced topic for now.

posted by leopard at 2:08 PM on August 21

Wow, ok thanks so much, that was really tripping me up. I think I'm going to write a little program that does these calculations for me, because creating equations is becoming really confusing.

I think I know how to go about what I was trying to do now.

posted by brenton at 2:13 PM on August 21

I think I know how to go about what I was trying to do now.

posted by brenton at 2:13 PM on August 21

leopard, in your example you said the nominal profit was $40. But it should be $20 (-100 + 30 + 90). Due to inflation (plus depreciation), the real profit is much less than $20, and in fact, it is negative.

If interest and inflation are equal, you lose money if you don't reinvest (compound) your interest. You've also thrown in equipment depreciation, but that is a different complication.

posted by JackFlash at 2:35 PM on August 21

If interest and inflation are equal, you lose money if you don't reinvest (compound) your interest. You've also thrown in equipment depreciation, but that is a different complication.

posted by JackFlash at 2:35 PM on August 21

You might want to read about internal rate of return. Your spreadsheet of choice probably has a function to calculate it.

posted by emilyw at 11:51 AM on August 22

posted by emilyw at 11:51 AM on August 22

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Nominal is easy, that's just the dollar amounts that are observed in reality, with no consideration for inflation.

For the inflation-adjusted dollars, you convert everything into, say, 2014 dollars. So if inflation is 3% per year, you take payments/receipts in 2015 and multiply them by 1/1.03 to convert to 2014 dollars. You take 2020 payments/receipts and multiply them by 1/(1.03)^6 to convert them to 2014 dollars. And so on.

Economically speaking, the "inflation-adjusted" results are "real" results, while the "nominal" results are what your results look like on paper.

posted by leopard at 11:36 AM on August 21