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Why math, why? [How do I calculate portfolio gain/loss from asset allocation?]

Posted: Thu May 28, 2020 12:41 pm
by jvini
I apologize if this is a simple question. I was very much not a math major.

Can someone tell me the correlation between a drop or rise in the S&P index and a 60/40 S&P/intermediate treasury bond index. I'm trying to make the 60/40 portfolio as basic as possible for math purposes.

I think I read somewhere that a 4% drop in the S&P equalled a 1% drop for a 50/50 portfolio, but that may be wrong too and I'm looking at a 60/40 portfolio.

It seems like there should be a calculator for this, but I can't seem to find it. I hope this question makes sense.

Thanks all!

Re: Why math, why?

Posted: Thu May 28, 2020 12:43 pm
by TropikThunder
jvini wrote: Thu May 28, 2020 12:41 pm I apologize if this is a simple question. I was very much not a math major.

Can someone tell me the correlation between a drop or rise in the S&P index and a 60/40 S&P/intermediate treasury bond index. I'm trying to make the 60/40 portfolio as basic as possible for math purposes.

I think I read somewhere that a 4% drop in the S&P equalled a 1% drop for a 50/50 portfolio, but that may be wrong too and I'm looking at a 60/40 portfolio.

It seems like there should be a calculator for this, but I can't seem to find it. I hope this question makes sense.

Thanks all!
[60% of the stock change] + [40% of the bond change].

Re: Why math, why?

Posted: Thu May 28, 2020 12:44 pm
by Cheez-It Guy
Are you assuming for this exercise that the bond portion remains static? Is this a single fund that automatically rebalanced, or a two-funder that must be manually re-balanced?

Re: Why math, why?

Posted: Thu May 28, 2020 12:48 pm
by bryanm
I may be missing something, but from a math perspective, a 60/40 portfolio will get 60% of whatever happens to the S&P side. So a 4% drop in S&P--assuming bonds stay steady--means 60% of 4% in the 60/40: 2.4%. Of course, bonds probably won't stay steady, but at that point you're trying to put precise numbers on something inherently unpredictable.

I have no idea why someone would say that 4% drop in the S&P results in a 1% drop in a 50/50 portfolio. To get that number, bonds would need to rise 2%, which is possible but not guaranteed.

Re: Why math, why?

Posted: Thu May 28, 2020 12:59 pm
by Elric
As others have said, you can calculate the impact of the stock change on the total portfolio (0.04 x 0.60), but the total change depends on what happens to bonds. There is very little correlation between bonds and stocks. Sometimes they go up or down together, sometimes one changes and the other doesn't, and sometimes they move in opposite directions. Here's the correlation: https://www.portfoliovisualizer.com/ass ... &months=36

Re: Why math, why?

Posted: Thu May 28, 2020 1:50 pm
by jello_nailer
How about this to make it extra easy?:
Run a 50/50 allocation and every month (or whenever) make the balances the same. Subtract one from the other and take 1/2 of that and give it to the smaller allocation. You could do the same for 60/40 but I'm trying to make the maths easier.
Not sure re-balancing can be done with less or easier arithmetic.

I'm interested to hear comments on this approach.

Re: Why math, why?

Posted: Thu May 28, 2020 3:03 pm
by dratkinson
jvini wrote: Thu May 28, 2020 12:41 pm...
I think I read somewhere that a 4% drop in the S&P equalled a 1% drop for a 50/50 portfolio, but that may be wrong too and I'm looking at a 60/40 portfolio.
...
A 50/50 portfolio will drop 2% if either term drops 4%... while the other term remains constant (1).

98% = (50%*(1-.04) + 50%*1)

98% is 2% less than the 100% you would have had if the first term had not been reduced by 4% (0.04).

Lather, rinse, repeat for a 60/40 portfolio.

Add the second term moving independently of the first, and things become less certain.

Add in government intervention and it's anybody's guess as to the outcome.


Stocks move based on investors' perception of the economy. (The economy and the stock market are separate animals. A temporarily poor economy does not translate into a poor stock market.)

Bonds. Suggest reading Swedroe's bond book, The Only Guide to a Winning Bond Strategy You'll Ever Need, to help understand bonds.

Maybe read a few books on "behavioral economics".

After you've done as much as you can do to understand, the only thing left is to follow your IPS.

Re: Why math, why?

Posted: Thu May 28, 2020 3:39 pm
by arcticpineapplecorp.
As others have said an easy way to think about it is multiply the loss on the stock by the percentage stocks you own, to get a rough idea of what your loss might be. So if stocks fall 30% and you have 50% in stocks, you're looking at "around" a 15% loss (.30 x .50 = .15). Could be better/worse than that depending on how bonds do (if good or bad).

To give you an example...in 2008 stocks roughly fell around 40% (to make the math easy). US fell 38% and I think total international was down 42% so let's just take the average between the two.

Bonds (US bonds) in 2008 went up 5% I believe.

These returns were for the entire year 2008.

If you simplify, you say you lost 40% on whatever percentage you have in stocks:
100% stock = 40% loss
90% stock = 36% loss
80% stock = 32% loss..

...50% stock = 20% loss.

See?

Now this is not exact. You have to look at what bonds did on the other side (the bond side) of the portfolio.

So if in 2008 bonds went UP 5%, then you'd earn 5% on the percentage of bonds owned.

So a 50/50 portfolio in 2008:
(.50 x -.40) + (.50 x +.05) =
-.20 + .025 =
-17.5% loss

that's the more exact math. does that help?

here's how to think roughly about large 50% declines (source: https://investingroadmap.wordpress.com/)
A Look at Historical Market Losses – Downside Risk You can get a fair perspective on risk by looking at actual stock market losses compared to how much money was allocated to stocks. The table below is based on actual market losses (price) encountered in the brutal 1973-74 bear market. A bear market is normally defined as a market decline of 20% or more. Drops of 10% to 15% are called corrections. Note in the table that a 100% stock portfolio lost nearly 50% of its value in two years (46% actual). If you had 50% stocks and 50% bonds, your loss would have been limited to 20%.

Equity Exposure……. Max loss
20%…………………………………5%
30%……………………………….10%
40%……………………………….15%
50%……………………………….20%
60%……………………………….25%
70%……………………………….30%
80%……………………………… 35%
90%……………………………… 40%
100%……………………… …… 50%
Data provided by Author Larry Swedroe on Morningstar’s ‘Bogleheads Unite’ Forum

Re: Why math, why?

Posted: Thu May 28, 2020 3:53 pm
by jvini
Thank you all. I suppose they took the average rise and fall over a certain period to get that number. I really don't know. I'm happy sticking with my 60/40 ratio and rebalancing but was curious how that correlation was arrived at.

Re: Why math, why?

Posted: Thu May 28, 2020 4:00 pm
by arcticpineapplecorp.
jvini wrote: Thu May 28, 2020 3:53 pm Thank you all. I suppose they took the average rise and fall over a certain period to get that number. I really don't know. I'm happy sticking with my 60/40 ratio and rebalancing but was curious how that correlation was arrived at.
here's something else to think about mathwise. People think they'll lose less money as they take a more conservative approach with age. This may be true in percentage terms, but is not likely in dollar amounts. I.E. the reality is, as one's portfolio grows over time, the dollars lost will increase even though the percentage losses (amount of risk) would decrease if you choose less risk over time.

Failing to understand that, will result in panicking each and every time declines come. Because the dollar losses will be larger as the portfolio grows (even if you take less risk).

And I think people panic more because they say "I lost X dollars!!", not "I lost 20%!"

So, understanding that you're risking losing more dollars later on, even though the percentage will be less (if you take less risk) is crucial to be a good investor.

For those who don't understand this, here's the scenario:

You're in your 20s, contributing to 401k, getting employer match, reinvesting dividends, getting growth on investment.
You're 100% in stock because you have the need, ability and willingness to take risk.
You're portfolio has grown to $50,000.
The stock market falls 30%
What's the dollar loss? $15,000.
(.30 X $50,000 = $15,000)

You stay the course and keep investing over the next few decades.
You're in retirement now.
Your portfolio has grown over the decades to $500,000 (10X the size it was in your 20s. Yes, you hopefully would have a million or more, but just go with me)
But since you're in retirement you can't take the risk you did in your 20s. So now you have 30% of your money in stock. The rest in fixed income.
The stock market now falls 30% (just like it did in your 20s).
What is your percentage loss now? (it's not 30%, you'd have to have 100% invested like in your 20s to lose 30%. But since you only have 30% in stocks, if the 30% in stocks loses 30%...)
You'd have a 9% loss (.30 X .30 = .09 or 9%)
But what's the dollar loss?
$45,000 (.09 X $500,000 = $45,000).


You now lost 3 times the amount (of dollars) you did as when you were in your 20s.
But you only took a third of the risk as you did in your 20s. (30% stock is a third roughly of 100% stock)

See how you will have to get used to larger and larger dollar losses...even if your portfolio gets less and less risky as time goes on?

Of course, you'd rather have 91% of your portfolio remaining (in the example above) than 70% (in the example in your 20s). But do people really say, "Whew. I've got 91% of my portfolio left!" No, they say, "I lost $45,000!!"

Do you think the average investor is really, truly aware of this? I don't.

Re: Why math, why?

Posted: Thu May 28, 2020 4:23 pm
by JimInIllinois
jvini wrote: Thu May 28, 2020 12:41 pm I think I read somewhere that a 4% drop in the S&P equalled a 1% drop for a 50/50 portfolio, but that may be wrong too and I'm looking at a 60/40 portfolio.
A 4% drop in stocks equals a 1% change in the allocation ratio of a 50/50 portfolio, turning it into a 49/51 portfolio.

Re: Why math, why? [How do I calculate portfolio gain/loss from asset allocation?]

Posted: Thu May 28, 2020 4:38 pm
by LadyGeek
This thread is now in the Investing - Theory, News & General forum (theory).

Don't get confused about percentages. A 20% loss followed by a 20% gain will not get you back to the starting point.

See the wiki: Percentage gain and loss

jvini - If you don't understand an answer, please let us know and we'll try again.

Re: Why math, why?

Posted: Thu May 28, 2020 4:55 pm
by bryanm
JimInIllinois wrote: Thu May 28, 2020 4:23 pm
jvini wrote: Thu May 28, 2020 12:41 pm I think I read somewhere that a 4% drop in the S&P equalled a 1% drop for a 50/50 portfolio, but that may be wrong too and I'm looking at a 60/40 portfolio.
A 4% drop in stocks equals a 1% change in the allocation ratio of a 50/50 portfolio, turning it into a 49/51 portfolio.
That's an interesting way to look at it! Though, being a pedant, I would argue that this is a 2% change in allocation (0 point spread to 2 point spread). :D

Re: Why math, why?

Posted: Thu May 28, 2020 5:09 pm
by JimInIllinois
bryanm wrote: Thu May 28, 2020 4:55 pm
JimInIllinois wrote: Thu May 28, 2020 4:23 pm
jvini wrote: Thu May 28, 2020 12:41 pm I think I read somewhere that a 4% drop in the S&P equalled a 1% drop for a 50/50 portfolio, but that may be wrong too and I'm looking at a 60/40 portfolio.
A 4% drop in stocks equals a 1% change in the allocation ratio of a 50/50 portfolio, turning it into a 49/51 portfolio.
That's an interesting way to look at it! Though, being a pedant, I would argue that this is a 2% change in allocation (0 point spread to 2 point spread). :D
OK, but being argumentative, I would point out that when people have "5% rebalancing bands" that means they plan to rebalance when 50/50 becomes 45/55. It is not intuitive to most people that it takes a 20% change in the stock market to hit a 5% band.

Re: Why math, why?

Posted: Thu May 28, 2020 5:42 pm
by bryanm
JimInIllinois wrote: Thu May 28, 2020 5:09 pm OK, but being argumentative, I would point out that when people have "5% rebalancing bands" that means they plan to rebalance when 50/50 becomes 45/55. It is not intuitive to most people that it takes a 20% change in the stock market to hit a 5% band.
Fair point! I am 100% stocks, so I hadn't come across that situation before, but it sounds persuasive to me.

Re: Why math, why? [How do I calculate portfolio gain/loss from asset allocation?]

Posted: Fri May 29, 2020 6:51 pm
by FactualFran
jvini wrote: Thu May 28, 2020 12:41 pm Can someone tell me the correlation between a drop or rise in the S&P index and a 60/40 S&P/intermediate treasury bond index. I'm trying to make the 60/40 portfolio as basic as possible for math purposes.

I think I read somewhere that a 4% drop in the S&P equalled a 1% drop for a 50/50 portfolio, but that may be wrong too and I'm looking at a 60/40 portfolio.
Look at the results in the Metrics tab of the analysis by Portfolio Visualizer of the Vanguard Balanced Income fund. The fund is 60% total US stock market and 40% aggregate bond index, rather than S&P 500 and intermediate Treasurys. The R-square correlation is 95.86% (out of a maximum of 100%).

However, I suspect that instead of correlation you are interested in volatility relative to the S&P 500. That is given by the Beta value on the Metric tab of the Portfolio Visualizer results. It is 0.61 for the Vanguard Balanced Index fund relative to the S&P 500. That web page also reports a beta of 1.00 for the Vanguard 500 Index fund relative to the S&P 500.