Consumer Price Index and assorted thoughts

I was thinking ...
>A big mistake!
... that one reads a lot about the real return on investments, taking inflation into account.
That is, if Sam currently has $A invested in the S&P 500 and the price of butter is $B per kg, then Sam can buy A/B kg of butter.
During the following year his portolio goes from $A to $A(1+R), so R is the nominal return.
It's how much the S&P increased. But if inflation for that one year has increased the price of butter from $B to $B(1+i)
then his new portfolio will buy [A(1+R)] / [B(1+i)] kg of butter.
>I take it that i is the inflation rate.
Well ... for the price of butter.
>What's your point?
My point is that Sam's portfolio, in terms of buying power, has only increased by a factor (1+R) / (1+i).
That's equivalent to a real return of r, where 1 + r = (1+R) / (1+i).
>That's a real return of ... uh ...
That's a real return of r = (Ri) / (1+i).
If the S&P goes up 8% (that's R = 0.08) but inflation
is 2% (that's i = 0.02), then the real return is (0.08  0.02) / (1+0.02) = .0588 or about 5.9%.
>Are we still talking about butter?
No. We're now talking about the increase in prices in general, the increase in the cost of living,
the increase in the consumer price index.
I did the butter thing so we could generate that formula

If R is the nominal return and
i is the inflation rate,
then the real return is:
r = (Ri) / (1+i)


>For the real rate of return, is there a really big difference?
Yes. Check out Figure 1 where the S&P annualized gain (or Compound Annual Growth Rate CAGR) was 10.7% nominal,
but just 7.3% real.
>And the CPI grew at an annual rate of 3.2%?
Yes, over that period. A $1.00 investment growing at the nominal return would wind up as $1,687 (that's the
green curve), whereas its buying power would only
be $167 (the red curve).
The curious thing ... and that's what got me thinking ... is that everybuddy who invests in the S&P (or any stock or mutual fund) has
the very same value for the nominal return (that's R). However, when one talks about the real return, one used an inflation rate
(that's i) which is some kind of national average.
>You're ignoring taxes, right?
Yes, and foreign exchange rates ... if you're a Canadian or German or Italian, investing in the S&P.

Figure 1

>So what's your point?
Look at Figure 2. Are 4.5% of your expenditures on alcohol and tobacco?
>More like 20% because ...
And 27.9% on housing?
>That's 1992, Canadian, eh?
The U.S. figure was about 36.7% for housing, in 19978.
>You're trying to tell me something, right?
I'm telling you that Sam lives in Smallville, doesn't pay rent, owns his house free and clear, has no debts, buy a new car
every ten years, doesn't drink or smoke, never travels more than ten miles a day, grows his own vegetables ...
>So he can ignore some of those items?
His "weights" are different than Sally who ...
>Don't tell me! Sally lives in Metropolis, pays rent, is in debt up to her ...
Uh ... yes, and she buys a new car every two years and eats out every night.

Figure 2

Figure 3

>So, if Sally invests in the S&P then her real return is less than Sam's.
Yes. Inflation affects her much more than it does Sam.
>But when I read about real returns, are they talking about Sam or Sally?
Aah, that's the big question!
>And what's the big answer?
Okay, we need some sort of "personalized real return". For example, if Sally had the national average inflation but Sam's
"personal" inflation was half of Sally's, he'd wind up with $517 in buying power ... the magenta
curve in Figure 3.
>Sam is investing from year 1928 to year 2000? How old is Sam?
Pay attention. We're going to compute a "personal" inflation rate ...

Let's suppose that our annual expenses are divided as indicated in Figure 1 and ...
>Alcohol & Tobacco at just 3%?
... the inflation rates for each of the eight aggregates are i_{1}, i_{2}, ... i_{8}.
>Aggregates?
That's what we call them ... elementary aggregates. When the govenment calculates the "national" CPI, they first divide living costs into a variety of
elementary aggregates such as Food, Housing, etc. and determine the increases, year to year, in these aggregates. Then they combine these
aggregates into the CPI by assigning a weight to each. Our Figure 1 gives the weights we'll use for our "personal CPI".
Now, for each annual $100 expenditure, we'd spend 30% of shelter (that's $30) and 20% for transportation (that's $20) and so on.
 Figure 1

That means that our $100 is divided like so:
[0] CPI(0) = 100 = 30(for shelter)+20(for transportation)+19(for food)+...+3(for alcohol)
That 100 is our CPI for the current year.
Now, as we assumed above, the aggregates increase so that, next year, the CPI is:
[1] CPI(1) = 30(1+i_{1})+20(1+i_{2})+19(1+i_{3})+...+3(1+i_{8})
>So if the cost of shelter goes up by 2%, then, next year, we'd spend $30*1.02 = $30.60 on shelter.
Exactly. That's 30(1+i_{1}) with i_{1} = 0.02 (meaning a 2% increase).
For example, we might have:
CP(0) = 30.0 + 20.0 + 19.0 + 9.0 + 8.0 + 7.0 + 4.0 + 3.0 = 100.0
CPI(1) = 30.0 (1.014)+20.0 (1.019)+19.0 (1.041)+9.0 (1.043)+8.0 (1.027)+7.0 (1.027)+4.0 (1.009)+3.0 (1.015) = 102.5
so our CPI has increased by 2.5%.
So, for the following year, we'd be spending $102.50 made up of shelter at $30(1.014) = $30.42
and transportation at 20.0 (1.019) = $20.38 etc. etc.
We can still write, as we did in [0] above:
[1a] CPI(1) = 102.5 = 30(for shelter)+20(for transportation)+19(for food)+...+3(for alcohol)
>You're assuming I spend the same proportions on shelter and ... uh, beer.
Yes. 30% on shelter, 20% on transportation and so on ... except that
the various expenses are now $30.42 for shelter and $20.38 for transportation etc., so:
[1b]
CPI(1) = 102.5 = 30.42+20.38+19.78+9.39+8.22+7.19+4.04+3.05
Now, for the following year we again incorporate the increases in shelter, transportation etc., using something like [1], getting:
[2] CPI(2) = 30.42(1+i_{1})+20.38(1+i_{2})+19.78(1+i_{3})+...+3.05(1+i_{8})
>With the SAME increases?
Well, no. We'd use different values for i_{1}, i_{2}, etc.
However, we now have a prescription for calculating our "personal CPI".
>But when steak goes up in price, I'd spend less on food and more on ... uh, beer.
Yes, so your weights would change from year to year. Further, even in the food category (which, for the "national" CPI, may be the
aggregate of a thousand food prices from steak to prunes, you may buy less steak and more prunes and ...
>Yeah, so my food bill may go DOWN, even when steak goes up!
Yes. That's always a problem in calculating CPIs, even for the "national" CPI. To see the problems involved, take a peek at
this PDF paper.
Nevertheless, I still feel that I should calculate my real returns based upon my own personal inflation rates. For example, the method of calculating
the "national" CPI changes from time to time, so, IMHO, it makes little sense to talk about real returns over the past hundred years, using some government
generated CPI.
>IMHO?
In My Humble Opinion.
>I HATE it when people use that!
GMTA, but that's NetSpeak.
AAMOF you see IMHO everywhere but OTOH it's so common that FWIW I often find myself ROFL and BTW prefer IMCO to IMNSHO and IAC ...
>Can we continue?
Okay ...
Let's talk about the elementary aggregates ...
>Like food or housing or ...
Yes. If there are, say, 1000 items in the food category ... steak, bread, prunes etc. ... then we can calculate a number which represents the "average" consumer's
cost by talking to 10,000 households and asking how much they spend on each. If, on average, these households spend 7% of their food bill
on steak and 5% on bread ... we then come up with weights for each of our 1000 food items hence can allocate a $100 food bill like so:
COST(0) = 100 = 7(for steak) + 5(for bread) + ... + 0.1(for prunes)
>Ten cents on prunes?
Yes, and $7 for steak and $5 for bread, and so on.
Now we suppose that these 1000 items increase by i_{1}, i_{2}, ... i_{1000}.
The new costs would then be:
steak = $7(1+i_{1})
bread = $5(1+i_{2})
...
prunes = $0.1(1+i_{1000})
So what's the increase in cost, for this food bill?
We could just add them to get a new cost:
[3] COST(1) = 7(1+i_{1})+5(1+i_{2})+...+0.1(1+i_{1000})
in which case the increase in cost is COST(1)/COST(0).
We might also look at the individual price increases 1+i_{1}, 1+i_{2}, ...
and calculate NOT the Arithmetic Mean of the increases (as in [3]) but a Geometric Mean, like so:
[3a] COST(1) = (1+i_{1})^{7}(1+i_{2})^{5}...(1+i_{1000})^{0.1}
