## Mathematics of international asset allocation

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gordoni2
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### Mathematics of international asset allocation

I'm searching for a simple mathematical model to help understand how much to allocate to unhedged international versus domestic. This is what I came up with.

Model

RD = domestic expected annual arithmetic real total return
SD = domestic volatility
Ri = international expected annual arithmetic real total return calculated in foreign currency
Si = international volatility calculated in foreign currency
RE = expected real exchange rate return
SE = real exchange rate volatility
coviE = expected annual covariance between international return and exchange rate return
corrDI = expected correlation between real domestic and real international returns
Rrf = real risk-free return

First estimate the expected return and volatility of international when expressed in real domestic currency:

RI = international expected annual arithmetic real total return calculated in domestic currency
RI = Ri - RE

SI = international volatility calculated in domestic currency
SI^2 = Si^2 + SE^2 - 2.coviE

Note the minus signs infront of the first RE and 2.coviE. They exist because the domestic value of international investments moves in the opposite direction to the exchange rate.

Now perform mean-variance optimization on RD, SD and RI, SD and compute the tangency portfolio to determine the relative allocations to international and domestic.

Example

Here is a rough example of applying the model assuming a world in which the same amount of risk buys the same amount of return when expressed in local currency, and using historical data for future expectations.

RD = Ri = 7% -- Real, arithmetic not geometric; Dimson et al. World returns 1900-2000.
SD = Si = 17% -- Dimson et al. World returns 1900-2000.
RE = 0%
SE = 6% -- Fred: Real trade weighted U.S. dollar index: broad (TWEXBPA) 1974-2016; I am currently uncertain this is the correct data series to use; it would be nice if this was an investment weighted rather than a trade weighted index.
coviE = -0.0055 -- covariance of TWEXBPA against CPI-U adjusted MSCI EAFE total return 1974-2015; corriE = -0.47.
corrDI = 0.92 -- portfoliovisualizer.com: correlation of VTI to VEU 2008-2016.
Rrf = -2% -- Fed: 1 month t-bill rate 0.52% less Survey of Professional Forecasters 2017Q1 CPI-U 2.2% 2017-01-13.

Gives:

RI = 7%
SI = 20.9%

and MVO then gives:

100% domestic, 0% international

Questions and commentary

What is wrong with this model? I understand valuations matter, but is it just a case of incorrect inputs, or is this simple model itself seriously flawed. Note that boosting Ri, to say 9%, gives a 57% domestic, 43% international breakdown, so the model doesn't always predict the corner case.

Is there a better model for incorporating the impact of unhedged exchange rate volatility? I'm reluctant to just use MVO on historical returns directly because of the uncertainty regarding the underlying returns. Even with 100 years of data, a 17% standard deviation implies a 6.8% wide 95% confidence interval on the underlying returns.

corrDI is high at 0.92, reflecting the domestic/international equity correlations over just the last 9 years. If I reduce it to 0.69 (the correlation between Shiller U.S. large and MSCI EAFE 1970-2015), I get 81% domestic, 19% international.

I read Vanguard's paper Considerations for investing in non-US equities. They mention exchange rates increase international volatility, but it is not incorporated explicitly into their asset allocation analysis, appearing possibly only indirectly through the data series they employ. They mention "the most likely scenario is one in which correlations remain elevated but in which volatility for non-U.S. markets more closely resembles that of U.S. markets", but it seems to me the volatility of international should always be expected to be higher than domestic due to exchange rate volatility.

If the model is OK, what inputs might you use?

rob
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### Re: Mathematics of international asset allocatiion

What's wrong with the market weight..... 50/50 ish
| Rob | Its a dangerous business going out your front door. - J.R.R.Tolkien

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### Re: Mathematics of international asset allocation

Good luck with that, invest a bunch of money based on it and let us know the results.

Seriously ... if it works let us know.

International stocks are friggen hard for average folk like me.

lack_ey
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### Re: Mathematics of international asset allocation

On a quick glance the model and analysis looks okay.

I was going to note corrDI = 0.92 being high, reflecting a relatively modern phenomenon (correlations used to be lower) but you did address that below. To be honest I don't really know what to estimate for the future. I guess correlations could come back down some or maybe there's been a secular shift as global economies and particularly markets with the flow of capital around the globe being more interconnected.

I agree that with the exchange rate vol riding on top, it is likely that ex-US has higher vol from USD perspective than US stocks. You do see that US market vol in local currency terms is lower than some other markets in local currency terms or similar, but the other markets when combined have some diversification benefit and so local currency vol for ex-US overall is I think similar by market cap weighting. I'd need to check the stats. But I doubt it could get low enough in local currency terms that including the exchange rate, ex-US vol drops down to US vol.

Anyway, what you get with MVO using this approach and these inputs attempts to maximize risk/return given these inputs for an investment of one year. In practice you are probably interested in returns for decades and not necessarily trying to optimize for yearly Sharpe ratio or similar.

And the problem is primarily in not knowing the inputs. The main argument for diversification is in case RD is significantly different from RI in the future for let's say at least a span of some decades you care about.

in_reality
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### Re: Mathematics of international asset allocation

gordoni2 wrote:I'm searching for a simple mathematical model to help understand how much to allocate to unhedged international versus domestic. This is what I came up with.
For reference, you might look at Research Affiliates 10 yr expected returns.
That have estimated returns for:
Yield
Growth
Valuation
FX

They also give a probabilities for a range of returns ... for example a 75% chance of 4% returns and a 5% chance of 10.2% returns for EAFE equities.

https://www.researchaffiliates.com/en_u ... ation.html

I take it with a huge grain of salt though and aren't sure you can calculate the optimal performance. And you model is too difficult for me without spending a fair amount of time trying to figure it out....

Dirghatamas
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### Re: Mathematics of international asset allocation

gordoni2

Your model basically says in words: " if the returns of US/International stocks are similar and their volatility in local currency is similar AND long term returns on currency movements are 0, then holding unhedged International stocks exposes you to uncompensated currency risk. As such, the optimal risk/reward portfolio will be very highly US portion(local currency)"

This is more or less consistent with what Bogle says. However, in my view this type of analysis completely misses the point of unbiased international diversification: tail risk or insurance.

Consider this thought experiment. We divide all US based companies as east or west based on whether their headquarters are east or west of the Mississippi river. Further, if you as the investor are located east, your ER for east companies is slightly less and vice versa. Lets say expected returns for both east and west are the same. What should you do? This type of model will concentrate your portfolio east or west. If nothing drastic happens, this indeed is a good choice.

However, lets say there is a giant earthquake on the west coast and it destroys Most of the Tech heavy coast. Or lets say there is a Tsunami which destroys the North east Financial Sector? A person who is diversified will still suffer a lot but not as much as the concentrated one.

Thats the way I think about global investing and why always invest by global cap (which coincidentally happens to be close to 50/50). I am not looking for better returns, just lower tail risks for future.

Your model can't model any of this because its expected returns are backward looking. Given a choice of 10,000 companies from US vs. 20,000 companies from the world, I would always go for diversification.

There is also the tail risk of this happening EVEN without any natural or man made disaster. Consider a Japanese investor running a similar model to yours in 1990. The future returns in ANY country could be very different from the past. Diversification always reduces this type of risk. Your model doesn't have any parameters to model deep/tail risks.

*3!4!/5!
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### Re: Mathematics of international asset allocation

Just skimming. I'd be concerned about arithmetic vs geometric.

As a test case, what does your model tell you to do if you will only invest in domestic cash and foreign cash. Assume that "cash" has zero return, zero volatility, in local currency, so the only factor is fluctuation in exchange rate.

It would seem that foreign cash is "return-free risk" that is dominated by "return-free risk-free" domestic cash, but it could be that some rebalanced allocation of both domestic cash and foreign cash, may beat pure domestic cash.

Think through this simplified scenario as a "sanity check" of the model.

MIretired
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### Re: Mathematics of international asset allocation

I just read this short article from Michael Edesess, 1-9-17:
I guess he concludes that maximum diversification is not really definable (or something) in real world investing.

afan
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### Re: Mathematics of international asset allocation

MVO produces extreme weights whether you use realized returns to find what would have been optimal in retrospect or you input similar values to estimate optimal going forward.

In retrospect you know your inputs are correct. For the future you don't know what the realized values will be. One could determine the optimal weights over a series of past periods and hope that would guide what the future optimum might be. Of course, changes in international development, correlations and returns could mean that past optimal allocations will not be optimal going forward. MVO does not help here.

With high correlations between the asset classes, the value of diversification declines. When this happens there is not much room for it to matter. This may be part of Bogle's argument.

The primary unknown then becomes the currency effect. The world market weights reflect decisions of investors around the world, not just those who live in the US. It is not obvious how to convert international concensus bets about expected risk and return to values for a US investor.

MVO requires an answer to the expected risk, including exchange rates and covariances, and return. Then it gives you an answer.

One could do a sensitivity analysis with MVO and hope varying inputs across a reasonable range converges optimum to some reasonable result. But that might be a simple average of 100%/0 and 0/100% allocations.

Whatever happens it is highly unlikely a portfolio a prudent investor would hold will turn out to have been optimal.
We don't know how to beat the market on a risk-adjusted basis, and we don't know anyone that does know either | --Swedroe | We assume that markets are efficient, that prices are right | --Fama

galeno
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### Re: Mathematics of international asset allocation

Just buy and hold VT if you're a USA person and VWRD/VWRL (LSE) for the USA-NRA and be done with this unnecessary worry forever.

BTW, did you include the dividend taxes that USA, non-USA developed, and EM stock markets levy on your dividends BEFORE they're sent to your brokerage account? They are an important cost for a Boglehead.

If you're a USA person investing thru a taxable account, you may get some of it back as "qualified dividends". If you invest thru a (USA) retirement account or are an USA-NRA those taxes are an unrecoverable portfolio cost. We lose 0.11% of port per year to these taxes.

Basing your analysis on back testing is wrong.

There's a far easier way to do what you want to do. Especially if you are a USA person.
AA = 40/55/5. Expected CAGR = 3.8%. GSD (5y) = 6.2%. USD inflation (10 y) = 1.8%. AWR = 4.0%. TER = 0.4%. Port Yield = 2.82%. Term = 33 yr. FI Duration = 6.0 yr. Portfolio survival probability = 95%.

*3!4!/5!
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### Re: Mathematics of international asset allocation

The theoretical discussion is interesting.

It may be that the OP's parameters are not exactly knowable and one just has to decide an allocation and go with it. But that's not the point.

Some people argue you should go with market weight, while others say there are reasons to put more weight in your domestic market. It seems to me that when you want to measure the value of your portfolio in local currency, then that might be one of the factors that could give a solid theoretical justification to "overweighting" domestic. I'd like to see that theory. (There's probably decades old academic papers addressing exactly this.)

alex_686
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### Re: Mathematics of international asset allocation

I see you have a Risk Free rate. Where do you use it?

I am not sure what you mean by "expected real exchange rate return" but the nominal exchange rate return is not zero. The US has a higher risk free rate then the EU and a higher rate of inflation. We expect the US dollar to fall to the EUR. To the extent that it differs is a return on currency. I think this is the number you are looking for.

Your model is fine, but it is only as good as the inputs. In short, more data often leads to worse results. I would recommend using shorter data series - say 3 years. When doing correlations you have to make a few assumptions.

You have to assume that there have been no secular changes over the period. That is, things more or less worked the same over the period being tested. Things have changed over the past 115 years. Things worked differently from 1900 - 2000, 1974-2016, 2008-2016. Markets have integrated for one thing.

You have to assume that volatility has reminded constant over the period being tested. If you change the volatility you will increase the correlation between your two data sets generating false positives.

nisiprius
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### Re: Mathematics of international asset allocation

afan wrote:...MVO produces extreme weights whether you use realized returns to find what would have been optimal in retrospect or you input similar values to estimate optimal going forward...
And that's not the fault of MVO, it's the fault of financial data. MVO produces extreme weights because even if there were no fundamental changes in the nature of the underlying assets, the sampling error is big, and therefore the numbers you calculate for return, standard deviation, and correlation for any period of time are very different from those for any other period of time. The available periods of time are, at the same time, not long enough to reduce sampling error, and yet so long that the question of whether "something changed" always arises.

International stocks used to have a correlation of around 0.66 with US stocks--by no means truly low, but low enough to have some effect. Of late it's been about 0.9. Nobody knows which number is "right." Nobody knows whether the 0.9 value is just a fluke from sampling error, or whether something has changed. if something has changed, nobody knows whether it could change back.

If you had a portfolio of US and international stocks, and someone promised you that 2017 returns would be the same as that of a randomly-selected year from anywhere within the twenty-year period 1970-1989 inclusive, your optimum bet would be to make your portfolio 11% US, 89% international. If someone promised you that it would be the same as a randomly-selected year from 1996-2015, your optimum bet would be to make it 100% US, 0% international. That's what the data tells you. 1970-1989 were that different from 1996-2015.

What you're seeing is that, in the real world, the difference in return between international and US stocks in each of these period was so large that it basically dominated the picture. For 1970-1989 the best mix was almost pure international, because international had considerably higher return. It's as simple as that. The sophisticated effect of the 0.640 correlation was enough that despite lower return, a tiny allocation to US stock did product a tiny improvement--the theory worked. For 1996-2015, you didn't even see that.

So the problem with your mathematics is that there's no way to get input data that's trustworthy enough to give you trustworthy output. Unless someone descends from on high with tablets declaring what the actual, true, intrinsic input numbers for your model are, you are a prisoner of "past [anything] is no guarantee of future [anything]." (The usual way out, which I think is totally bogus, is to use your guesses (aka "analyst views") as inputs to the "Black-Litterman" method).
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

nisiprius
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### Re: Mathematics of international asset allocation

P.S. Where did you get the actual Dimson & Marsh data? I'd love to have it...

If you're using 1900-2000, the elephant in the room is that world-ex-US returns were just much lower than US returns over... well, 1900-2015, anyway. Something like 6.4% for the US and 4.4% for the world-ex-US. So as long as you use all available data, the lower returns of ex-US data will tell you to allocate almost 100% to the US. You're faced with the completely impossible choice: if you throw out, say, data before 1950, you're biasing your results--why shouldn't you use all of it? But if you don't throw it out, you're pretending that World War I and World War II didn't matter.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

Rodc
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### Re: Mathematics of international asset allocation

nisiprius wrote:
So the problem with your mathematics is that there's no way to get input data that's trustworthy enough to give you trustworthy output.
I think it is actually even worse. While it is true that the mathematics are very sensitive to errors in inputs and past data are noisy at best, saying that really good data would really help implies that the underlying market forces and dynamics never change and so the statistical properties of the past (if only we could estimate them accurately) can accurately represent the statistical properties of the future.

If the OP wants something useful Bayesian statistics is likely are better area to explore.
We live a world with knowledge of the future markets has less than one significant figure. And people will still and always demand answers to three significant digits.

*3!4!/5!
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### Re: Mathematics of international asset allocation

OP?OP?

Northern Flicker
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### Re: Mathematics of international asset allocation

Nisiprius is correct. The data used are highly biased datasets. There is no reason to think that sample means of return, sample variances, or sample covariances are good estimates of the actual means, variances, and covariances. A consecutive sequence of returns is especially problematic.

Moreover, even if you knew the actual distributions of future returns, the period of future returns you care about may not be long enough for the mesn of the future trial to converge on the mean of the distributions. To clarify this point, if you flip a coin repeatedly, the number of tails will converge on 50% of the number of flips with a sufficiently long sequence of flips. If we new the mean of the future return of equities, how long a trial period would it take to get similar convergence of equity returns?

Clive
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### Re: Mathematics of international asset allocation

Rodc wrote:
nisiprius wrote:
So the problem with your mathematics is that there's no way to get input data that's trustworthy enough to give you trustworthy output.
I think it is actually even worse. While it is true that the mathematics are very sensitive to errors in inputs and past data are noisy at best, saying that really good data would really help implies that the underlying market forces and dynamics never change and so the statistical properties of the past (if only we could estimate them accurately) can accurately represent the statistical properties of the future.
Much of stock gains are a consequence of a few unpredictable outlier instances.

If for instance you compare average yearly total gains with the median then typically those choices that have performed relatively well have a significantly higher average value compared to the median; And those that performed relatively poorly have a significantly lower average compared to the median. Indicative that for instance small cap value had more than its 'fair' share of right tail large up cases whilst small cap growth had more left tail deep down cases. Which could be a consequence of pure (bad)luck arising out of a single/few holdings.

As a example, IIRC historically consumer goods was one of the best performing sectors and within that Coca-Cola (KO) was a primary driver of that. Depending upon whether KO policy was to pay dividends or perhaps not (prefer share repurchase instead) might sway measures of low versus high dividend based partitioning to include or exclude that outlier and sway you to perhaps believe that generally one choice of partitioning was better than another.

Rodc
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### Re: Mathematics of international asset allocation

Clive wrote:
Rodc wrote:
nisiprius wrote:
So the problem with your mathematics is that there's no way to get input data that's trustworthy enough to give you trustworthy output.
I think it is actually even worse. While it is true that the mathematics are very sensitive to errors in inputs and past data are noisy at best, saying that really good data would really help implies that the underlying market forces and dynamics never change and so the statistical properties of the past (if only we could estimate them accurately) can accurately represent the statistical properties of the future.
Much of stock gains are a consequence of a few unpredictable outlier instances.

If for instance you compare average yearly total gains with the median then typically those choices that have performed relatively well have a significantly higher average value compared to the median; And those that performed relatively poorly have a significantly lower average compared to the median. Indicative that for instance small cap value had more than its 'fair' share of right tail large up cases whilst small cap growth had more left tail deep down cases. Which could be a consequence of pure (bad)luck arising out of a single/few holdings.

As a example, IIRC historically consumer goods was one of the best performing sectors and within that Coca-Cola (KO) was a primary driver of that. Depending upon whether KO policy was to pay dividends or perhaps not (prefer share repurchase instead) might sway measures of low versus high dividend based partitioning to include or exclude that outlier and sway you to perhaps believe that generally one choice of partitioning was better than another.

Good points. Not only are the parameters like mean and std not well known, we know the data are not particularly Gaussian, or necessarily smooth, but we don't know what distribution one should use.
We live a world with knowledge of the future markets has less than one significant figure. And people will still and always demand answers to three significant digits.

*3!4!/5!
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### Re: Mathematics of international asset allocation

So there are uncertainties in the model parameters. But so what? You still have to decide what to do.

Some people just decide to go by cap weight. Their argument presumably doesn't rely on historical data.

But regardless of what setup you use, what I want to know is, how should your allocation depend on what currency you will use? Intuitively, forex uncertainty should justify some home bias. But what is the theoretical argument for this?

Clive
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### Re: Mathematics of international asset allocation

*3!4!/5! wrote:what I want to know is, how should your allocation depend on what currency you will use? Intuitively, forex uncertainty should justify some home bias.
Observationally :

Initially equal weight across a bunch of countries/currencies, diversifying equally across sectors, bought and held ... will be low ongoing cost once bought and will find its own 'cap weightings' over time. A few will do exceptionally well, others poorly, where the winners more than compensate for the losers (max downside -100%, max upside unlimited).

By contrast tilting to particular sectors or countries/currencies may do well, may lag according to luck.

Home bias is a tilt. OK provided the home economy doesn't turn out to have been a left tail over the period across which you invested. Broader diversification might lag the home economy, but will still tend to have been acceptable (and more assured) rewards (middle road is OK). Diversified forex will tend to wash (middle road). Home bias is a tilt. The upside of that tilt is nice (above broader average), the downside from having tilted could be dire. Why take the risk if you don't need to.

In 1900 Railroads accounted for 63% of sector weightings, by 2000 that had declined to just 0.2%. In 1900 the UK was around 30% of global weight, Germany around 10%, US 15%. By mid 1940's Germany was near 0%. In the mid 1960's the US had risen to around 75% of weight. In 1990 Japan had risen to 30% from near 0% in the mid 1940's and compared to the US 30%. By 2000 both UK and Japan were down at 10% weights each (US 50%).

patrick013
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### Re: Mathematics of international asset allocation

I think it's hard to get the info needed for the average investor.
Where's a chronological weighted average of earnings by index
or region, for example. Apply a PE to each country to get a resulting
price. Then an average exchange rate to apply to the result. Then
a +or- error factor to see an expected interval.

I'm always expecting negative correlations within the index between
countries unless all countries have superb years. It can become
very subjective then. Low growth, low valuations, plenty of diversification,
but average price appreciation. A lower asset allocation would result
whether objective or subjective, or wait-and-see methods are utilized.

Without better info wait-and-see seems best keeping the allocation 10-20%
until a definite upturn which is about the 10% allocation for value or defensive
stocks I always like to have.

Rodc
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### Re: Mathematics of international asset allocation

*3!4!/5! wrote:So there are uncertainties in the model parameters. But so what? You still have to decide what to do.
But you do not need to use models that you know are nonsensical.
We live a world with knowledge of the future markets has less than one significant figure. And people will still and always demand answers to three significant digits.

Northern Flicker
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### Re: Mathematics of international asset allocation

The model also used short-term variances which are of most concern for folks in or near their withdrawal phase. This also biases toward home bias because currency fluctuations have the most variance over short time frames, so one would expect an MVO to be biased against the extra short-term variance of forex rates.

alex_686
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### Re: Mathematics of international asset allocation

patrick013 wrote:I think it's hard to get the info needed for the average investor.
Where's a chronological weighted average of earnings by index
or region, for example. Apply a PE to each country to get a resulting
price. Then an average exchange rate to apply to the result. Then
a +or- error factor to see an expected interval.
I would think that real returns in USD would be sufficient. I would also think that looking into the PE ratio would be either be a a distraction or worthless. PE ratios are worthless in of themselves - they only have a meaning in relationship to something else. For example, one would expect a a low PE ratio in high inflation countries verse a low inflation country. A stock price would anticipate the effect that inflation would have.
patrick013 wrote:I'm always expecting negative correlations within the index between
countries unless all countries have superb years.
I am not sure what point you are trying to make here but IIRC almost all of the stock markets have a positive correlation. "Low" is anything below .8. I can't think of any major market that is currently below .6.

patrick013
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### Re: Mathematics of international asset allocation

alex_686 wrote:
patrick013 wrote:I'm always expecting negative correlations within the index between
countries unless all countries have superb years.
I am not sure what point you are trying to make here but IIRC almost all of the stock markets have a positive correlation. "Low" is anything below .8. I can't think of any major market that is currently below .6.
There's always the situation where, for example, Brazil could be inversely
correlated with Mexico, and as a result an Emerging Market Index would
have a lousy return.

alex_686
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### Re: Mathematics of international asset allocation

patrick013 wrote:There's always the situation where, for example, Brazil could be inversely
correlated with Mexico, and as a result an Emerging Market Index would
have a lousy return.
20 years ago the MSCI Emerging Market Index had a correlation of about .6 to the S&P 500. Now it is about .8. Markets are becoming more integrated. Brazil, one of the largest economies in the index, would only have a minor effect on the index if it blew up. Negative correlations within the index won't have much of a effect on the overall correlation to a outside investor.

patrick013
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### Re: Mathematics of international asset allocation

alex_686 wrote:
patrick013 wrote:I think it's hard to get the info needed for the average investor.
Where's a chronological weighted average of earnings by index
or region, for example. Apply a PE to each country to get a resulting
price. Then an average exchange rate to apply to the result. Then
a +or- error factor to see an expected interval.
I would think that real returns in USD would be sufficient. I would also think that looking into the PE ratio would be either be a a distraction or worthless. PE ratios are worthless in of themselves - they only have a meaning in relationship to something else. For example, one would expect a a low PE ratio in high inflation countries verse a low inflation country. A stock price would anticipate the effect that inflation would have.
Before an AA is made you have to know what you're allocating. More detailed
numbers are needed IMO. For each country in an index : earnings history and
estimated, average PE, year-end exchange rates to US dollars. Then you could
do a basic projection of price appreciation for that index. PE is just a way to
change earnings to stock price in a range of low-high PE's for a set time period.
Real returns for short term projections are oversummarized. But, if total returns
are low the underlying numbers are also low.

If total returns are increasing the "casino" is paying out more winning hands. The
Intl Index has more countries with higher profits, figuratively. So the index would
look better in a diversified portfolio then at a higher AA.

Topic Author
gordoni2
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### Re: Mathematics of international asset allocation

OP here. I realized using MVO for 2 asset classes is over-kill. It is sufficient to simply plot the combined returns and standard deviations, and then determine the point of tangency to the capital market line.
lack_ey wrote:what you get with MVO using this approach and these inputs attempts to maximize risk/return given these inputs for an investment of one year. In practice you are probably interested in returns for decades
This is a true. MVO is a poor tool for retirement planning. You care about consumption not wealth. For retirement planning you would have to use the matrix version of Merton's portfolio problem solution (and be prepared to accept leverage and short positions), or stochastic dynamic programming (which is far from simple). But if all you care about is teasing out the ratio of domestic to international in a risk-free/domestic/international portfolio, then I think under the simplifying assumption that returns are normally distributed, all three methods would yield the same answer.
afan wrote:One could do a sensitivity analysis
Excellent idea:

International return sensitivity: I found the expected international return had to be 1.2% greater than the domestic return before there is any allocation to international, and if it is 3.1% higher than domestic then the allocation to international becomes 100%.

Domestic/international correlation sensitivity: The correlation has to drop from 0.92 to 0.81 for there to be an allocation to international. At a correlation of 0.5, the allocation is 30% international.

And finally, forcing the allocation from 0% international to 50% changes the expected standard deviation from 17% to 18.5%. So the ex-ante cost of getting the international allocation wrong isn't great. In other words, this debate about how much to allocate to international doesn't matter very much.
alex_686 wrote:I see you have a Risk Free rate. Where do you use it?
The risk-free rate helps determine the capital market line that determines the recommended tangency portfolio.
I am not sure what you mean by "expected real exchange rate return"
This is the expected real return of foreign currency. It has an estimated value is zero.
nisiprius wrote:If you're using 1900-2000, the elephant in the room is that ...
I am using 1900-2000, but just to get a rough handle on return and volatility. I used the same values when expressed in local currency for both domestic and international returns and volatility, not the separate Dimson et al. values. In this case the model suggested you should be 100% domestic.

An alternative viewpoint might be that domestic is overvalued by, say, 30% at present, and let's say it will correct by loosing an average of 1.5% per year for the next 20 years. If that is the case the model would suggest being at 68% domestic and 32% international. So judgment calls about valuations and corrections definitely matter.
jalbert wrote:Nisiprius is correct. The data used are highly biased datasets. There is no reason to think that sample means of return, sample variances, or sample covariances are good estimates of the actual means, variances, and covariances.
Unfortunately, the past is all we have to predict the future. The best we can do is judicially use it decide how to act, taking into account sensitivities, and various alternative possibilities. The model I suggested may, or may not be reasonable, but if you don't use mathematics as a guide, what do you use?

The model I suggested makes several hopefully useful predictions: between 0 and 30% international might be reasonable depending upon your world view about valuations and corrections; using the market weighting of about 50% would appear harder to justify; and the whole debate about how much to allocate within the range 0-50% doesn't matter very much but this isn't to say it doesn't matter at all.

No matter what you choose to do there will be a big difference between ex-ante expectations and ex-post outcomes, but that doesn't detract from the need to attempt to maximize the ex-ante expectations. This difference between expectations and outcomes is especially true for short time periods.
The model also used short-term variances which are of most concern for folks in or near their withdrawal phase.
The work of Merton suggests short-term variances effect asset allocations equally independent of age, or are you are making the argument that the 20 year variance is substantially less than 20 times the 1 year variance?
This also biases toward home bias because currency fluctuations have the most variance over short time frames
I'm a little out of my depth when it comes to exchange rates. Are you saying real exchange rates exhibit some sort of predictability / reversion to the mean? If this is substantial, this would require changes to the model.

Rodc
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### Re: Mathematics of international asset allocation

patrick013 wrote:
alex_686 wrote:
patrick013 wrote:I think it's hard to get the info needed for the average investor.
Where's a chronological weighted average of earnings by index
or region, for example. Apply a PE to each country to get a resulting
price. Then an average exchange rate to apply to the result. Then
a +or- error factor to see an expected interval.
I would think that real returns in USD would be sufficient. I would also think that looking into the PE ratio would be either be a a distraction or worthless. PE ratios are worthless in of themselves - they only have a meaning in relationship to something else. For example, one would expect a a low PE ratio in high inflation countries verse a low inflation country. A stock price would anticipate the effect that inflation would have.
Before an AA is made you have to know what you're allocating. More detailed
numbers are needed IMO. For each country in an index : earnings history and
estimated, average PE, year-end exchange rates to US dollars. Then you could
do a basic projection of price appreciation for that index. PE is just a way to
change earnings to stock price in a range of low-high PE's for a set time period.
Real returns for short term projections are oversummarized. But, if total returns
are low the underlying numbers are also low.

If total returns are increasing the "casino" is paying out more winning hands. The
Intl Index has more countries with higher profits, figuratively. So the index would
look better in a diversified portfolio then at a higher AA.
In my humble opinion you are just playing with noise. I think few if any professional investors actually manage to pull things like this off successfully with any consistency.

99.44% of investors will only make things worse by getting this complicated.

If you are in the (1-.9944) my hat is off to you!
We live a world with knowledge of the future markets has less than one significant figure. And people will still and always demand answers to three significant digits.

lack_ey
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### Re: Mathematics of international asset allocation

gordoni2 wrote:
lack_ey wrote:what you get with MVO using this approach and these inputs attempts to maximize risk/return given these inputs for an investment of one year. In practice you are probably interested in returns for decades
This is a true. MVO is a poor tool for retirement planning. You care about consumption not wealth. For retirement planning you would have to use the matrix version of Merton's portfolio problem solution (and be prepared to accept leverage and short positions), or stochastic dynamic programming (which is far from simple). But if all you care about is teasing out the ratio of domestic to international in a risk-free/domestic/international portfolio, then I think under the simplifying assumption that returns are normally distributed, all three methods would yield the same answer.
Yes, one problem being that there are probably issues with that simplifying assumption.

That said, there is still some value in determining the appropriate answer with those assumptions or stipulations in place.
gordoni2 wrote:
The model also used short-term variances which are of most concern for folks in or near their withdrawal phase.
The work of Merton suggests short-term variances effect asset allocations equally independent of age, or are you are making the argument that the 20 year variance is substantially less than 20 times the 1 year variance?
Perhaps not substantially and honestly given the data limitations it's hard to make a case for this or even quantify the effect, but yeah the variance at N years tends to be less than N times the 1-year variance.
gordoni2 wrote:
This also biases toward home bias because currency fluctuations have the most variance over short time frames
I'm a little out of my depth when it comes to exchange rates. Are you saying real exchange rates exhibit some sort of predictability / reversion to the mean? If this is substantial, this would require changes to the model.
Again, likely, can be seen somewhat in the data. And here there is some intuitive economic justification at least for exchange rates between developed economies not to be some kind of untethered random walk. See purchasing power parity, the Big Mac Index, etc. (like all things this is probably not particularly a strong effect)

But these are relatively small issues compared to the fact that the underlying return distributions probably aren't stationary in any way.

Northern Flicker
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### Re: Mathematics of international asset allocation

Unfortunately, the past is all we have to predict the future. The best we can do is judicially use it decide how to act, taking into account sensitivities, and various alternative possibilities. The model I suggested may, or may not be reasonable, but if you don't use mathematics as a guide, what do you use?
Using mathematics as a guide would inform the modeling with inferences from central limit theorems regarding the convergence of sample means and sample variances to distribution means and variances that you need a sufficiently large random sample of independent data points for sample means, sample variances, and sample covariances to be reliable estimators for those properties of the actual probability distributions of future returns.

Portfoliovisualizer data for emerging markets only goes back to 1995. There's a good reason for that. Before 1994, there were not really widespread investment opportunities in EM equities for US residents, although stock exchanges in Brazil, Mexico, and Argentina have been in existence a long time . It is always an interesting question what comprises any given multi-country equity dataset that runs from 1900 to the present. For example, Russian equities went to zero in 1917, and there was no Russian stock exchange again until 1995. Does the data reflect it going to zero, and a new allocation being made with funds taken from allocations from other countries in the portfolio or does it just use annual returns for all countries (incorporating a survivorship bias).

afan
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### Re: Mathematics of international asset allocation

Keep in mind that the reason the market portfolio is ex ante efficient is that no one has a way to predict a better risk adjusted return from an alternative. Everyone understands that the ex post efficient portfolio will almost always be far from the market. But no one knows what that different portfolio will be.

That same issue arises for international vs domestic. One needs a reason to pick an allocation other than market.

Because of foreign exchange and taxes, there can be volatility and cost penalties for the international holdings that do not apply to domestic stocks. Vanguard says there is no long term benefit to the exchange rate variability, so they hedge it away. They seem to dismiss the theoretical possibility that the exchange rate volatility may balance other portfolio volatility.

If Vanguard is right about this, then the advantages of international would be return and underlying performance of the investments. If the international has a high correlation with domestic, as has been the case for a while now, then the only advantage left is better performance. If you think there is reason to expect higher risk adjusted return from international than domestic, then you would want more international.

If you think international will have the same long term risk adjusted returns as domestic, after hedging the forex risk and will be highly correlated with domestic, then it will not matter much how much you put in international. At that point, just look at tax issues and cost.
We don't know how to beat the market on a risk-adjusted basis, and we don't know anyone that does know either | --Swedroe | We assume that markets are efficient, that prices are right | --Fama

Northern Flicker
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### Re: Mathematics of international asset allocation

Because of foreign exchange and taxes, there can be volatility and cost penalties for the international holdings that do not apply to domestic stocks. Vanguard says there is no long term benefit to the exchange rate variability, so they hedge it away. They seem to dismiss the theoretical possibility that the exchange rate volatility may balance other portfolio volatility.
Vanguard generally doesn't hedge currency in int'l equity funds.

Doc
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### Re: Mathematics of international asset allocation

gordoni2 wrote:What is wrong with this model?
gordoni2 wrote:coviE = expected annual covariance between international return and exchange rate return
corrDI = expected correlation between real domestic and real international returns
US large cap and foreign large cap are mostly made up of international large cap. They all swim in the same pool. What is the covariance of Shell and Exxonmobil or GM and Toyota or Merck and Novartis?

If you want your foreign holdings to have low covariance with their domestic counterparts it seem to me that you want your foreign holdings to be small size companies which are more likely to swim in their local pond not the world's oceans.

So I would suggest that you need to add that to your analysis. Of course the data on the covariance would be hard to come by but so is the availability of attractive foreign small mutual funds at a reasonable cost.

Your other option might be emerging markets except that while they may have low covariance with domestic they may also have a high variance all by themselves.

I just wing it myself. I breakup my 25 to 30% into several foreign funds emphasizing small or value and part hedged and unhedged. Ask my grandchildren how well I did when they reach retirement age.
A scientist looks for THE answer to a problem, an engineer looks for AN answer and lawyers ONLY have opinions. Investing is not a science.

dharrythomas
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### Re: Mathematics of international asset allocation

The biggest thing wrong with the model is that you're using expected values for 5 variables. You're guessing and need to be right 5 times.

You're covering up the imprecision of the process and hiding it behind a precise answer to your equation.

Rodc
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### Re: Mathematics of international asset allocation

dharrythomas wrote:The biggest thing wrong with the model is that you're using expected values for 5 variables. You're guessing and need to be right 5 times.

You're covering up the imprecision of the process and hiding it behind a precise answer to your equation.
I might just say I would phrase it as "You're covering up the inaccuracy of the process and hiding it behind a precise answer to your equation"

I want a fountain of youth, but just because I want it really badly, does not mean one exists.

Models are fun. Some are even useful. This one not not so much.
We live a world with knowledge of the future markets has less than one significant figure. And people will still and always demand answers to three significant digits.

Rodc
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### Re: Mathematics of international asset allocation

if you don't use mathematics as a guide, what do you use?
Mathematics in inexpert hands can be more harmful than helpful. Many learn to move symbols around on a page and yet never really understand either the power or the limitations of mathematics.

You can use history. You can use theory. You can use common sense. You can use a blend off all these things. You can even use mathematical models if you first actually understand their limitations.
We live a world with knowledge of the future markets has less than one significant figure. And people will still and always demand answers to three significant digits.

AlohaJoe
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### Re: Mathematics of international asset allocation

Another way to stress test any allocation advice is to move it from the US to somewhere else; that removes the large amount of US bias in data, correlations, and success. For instance, would the algorithm tell a South African or Swedish investor to be 100% invested in local equities?

afan
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### Re: Mathematics of international asset allocation

Besides the extreme portfolios produced by MVO there is the fundamental assumption that volatility is the sole appropriate measure of risk.

Since there are ample data proving that investors care about and price higher moments of the return distribution, one should not consider the mean/variance optimal portfolio to correspond to actual optimal. There are plenty of examples of portfolios that maximize the Sharpe ratio but no one would consider optimal.

See the paper "Sharpening Sharpe Ratios".
We don't know how to beat the market on a risk-adjusted basis, and we don't know anyone that does know either | --Swedroe | We assume that markets are efficient, that prices are right | --Fama

nisiprius
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### Re: Mathematics of international asset allocation

Doc wrote:US large cap and foreign large cap are mostly made up of international large cap. They all swim in the same pool. What is the covariance of Shell and Exxonmobil or GM and Toyota or Merck and Novartis?

If you want your foreign holdings to have low covariance with their domestic counterparts it seem to me that you want your foreign holdings to be small size companies which are more likely to swim in their local pond not the world's oceans.

So I would suggest that you need to add that to your analysis. Of course the data on the covariance would be hard to come by but so is the availability of attractive foreign small mutual funds at a reasonable cost.
That's an interesting and convincing thought, Doc.

Hmmm... I can test it. If you're right, the correlation between Vanguard FTSE All-World Ex-US Small-Caps, VFSVX and Vanguard [US] Small-Cap index, NAESX, should be believably lower than the correlation between Total US, VTSMX and Total International, VGTSX.

Unfortunately PortfolioVisualizer is only showing available data for 03/20/2009 - 01/19/2017 for VFSVX. But maybe that's still kind of interesting because this is an era of high correlations so it's a time when a lower correlation would be especially nice to have.

It checks out!

Unfortunately, O.80 isn't really "low," and over that time period, the Sharpe ratios were 0.37 for VFSVX and 0.86 for NAESX, and 0.37/0.86 = 0.43, so the correlation would need to be less than 0.43 for VFVFX to improve NAESX. And indeed a portfolio of 80% NAESX, 20% VFSVX has a lower Sharpe ratio than 100% NAESX.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

afan
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### Re: Mathematics of international asset allocation

Well the Sharpe ratio just tells you the higher US than international return over that period. But I would not count on small cap portfolios showing lower correlation. After all, these are portfolios of many stocks. Individual small caps likely have more idiosyncratic risk than do individual large caps. But that diversifies away when you form the fund.
We don't know how to beat the market on a risk-adjusted basis, and we don't know anyone that does know either | --Swedroe | We assume that markets are efficient, that prices are right | --Fama

Topic Author
gordoni2
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### Re: Mathematics of international asset allocation

OP here.
lack_ey wrote:But these are relatively small issues compared to the fact that the underlying return distributions probably aren't stationary in any way.
This is a good point. In the example I gave it would have improved things if I had employed more recent correlations and covariances, not long run historical values.
Doc wrote:If you want your foreign holdings to have low covariance with their domestic counterparts it seem to me that you want your foreign holdings to be small size companies which are more likely to swim in their local pond not the world's oceans.
This is a neat idea. Unfortunately, I don't have the necessary data to explore it further.
dharrythomas wrote:The biggest thing wrong with the model is that you're using expected values for 5 variables. You're guessing and need to be right 5 times.
Actually eight variables! But let's look at that closely.

Four of the variables are the real returns and volatilities of domestic and international. I think it is quite reasonable to assume that on a risk adjusted basis domestic and international have the same ex-ante expected return. Given that assumption, over a broad range of real expected returns (0% - 20%) and a broad range of volatilities (5% - 30%) the model predicts 100% domestic.

Next is the ex-ante expected real exchange rate return. Over short time periods this can be reasonably be expected to be close to zero.

Volatility of the real exchange rate: Eyeballing this on Fred from 1974-2016, the real exchange rate volatility seems to lack any significant trends and appears reasonably constant.

Correlation of domestic and international: I previously used a value of 0.92. Using portfoliovisualizer for the past 5 years on a daily basis this has ranged from 0.83 to 0.89, so a guess of a value in this range next year doesn't seem unreasonable, and would result in 100% domestic.

Correlation of real exchange rate and international stocks: I previously used a value of -0.47. Splitting 1976-2015 into four decade long periods this has ranged from -0.45 to -0.63. A value of -0.07 or smaller would be required of the example before there was any allocation to international. This seems unlikely.

It is possible that exchange rate volatility is smaller than historically, the correlation of domestic and international comes in lower than it has over the past 5 years, and also the ex-ante expected correlation of the real exchange rate and international is more positive, resulting in the optimal strategy involving a significant allocation to international, but I am not sure you should set asset allocations assuming all of this.

Yes, I am making educated guesses, and they may not be what the future produces, but that wouldn't make the predictions wrong. Don't confuse ex-ante strategy with ex-post outcomes. The predictions made by the example are intended to be the best strategy if the same situation was encountered hundreds or thousands of times, not a single time.

Another set of reasonable assumptions would be that domestic is currently overvalued, and that a correction is likely. In that case it seems some allocation to international would be reasonable. This is a form of market timing, but I see nothing wrong with that. The important thing is stating the assumptions that determine your international allocation.
AlohaJoe wrote:Another way to stress test any allocation advice is to move it from the US to somewhere else; that removes the large amount of US bias in data, correlations, and success. For instance, would the algorithm tell a South African or Swedish investor to be 100% invested in local equities?
This is a good idea. I can't find any real exchange rate data, but iShares produces both Swedish and South African country indexes. If I make the assumption that ex-ante returns are the same for domestic and international on a risk adjusted basis, and that for the sake of an academic exercise the real currency volatility and correlation are the same. Sweden has had a slightly lower domestic/international correlation and a higher volatility, resulting in a 7% international allocation. South Africa has had a significantly lower domestic/international correlation and a high volatility, resulting in a 21% international allocation.

The precise numbers don't matter, but the model does show the impact of real currency volatility on unhedged international allocations is quite substantial. I believe the impact of this is greater than most people imagine.

Doc
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### Re: Mathematics of international asset allocation

gordoni2 wrote:I think it is quite reasonable to assume that on a risk adjusted basis domestic and international have the same ex-ante expected return.
Given the differences in governmental/investment policies between US and foreign I'm not sure this is a a good assumption. Thinking about Eurozone for example you have a big difference in the social welfare area (private vs government pension/healthcare) and tax system (income vs. consumption). I would expect that the risk/return would not have the same expected returns except perhaps in the large international area which has already been addressed.

Another question about a projection model for the purpose being discussed is the long term nature of the data needed. I recall reading that currency exchange differences net over long periods of time but that may not be true over a shorter even several year periods. Another time period problem is illustrated with the period since Lehman. The US is recovering albeit slowly, the Eurozone is still getting large infusions of monetary stimulus and EM acts like "who's this Lehman guy". So you may get little correlation over a ten year period you might get a lot during shorter periods. Whether this can translate into some kind of rebalancing bonus is another question.

I don't think there are any good quantitative answers here.
A scientist looks for THE answer to a problem, an engineer looks for AN answer and lawyers ONLY have opinions. Investing is not a science.

Northern Flicker
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### Re: Mathematics of international asset allocation

This is a good point. In the example I gave it would have improved things if I had employed more recent correlations and covariances, not long run historical values.
Why would that be an improvement? Doesn't it have the same theoretical flaw of not being a random sample of independent data points, and just have different biases in the sample?