When you're building your own portfolio, it's really useful to have some examples you can look at. These are off-the-shelf portfolios which have been back-tested based on historical returns, in this blog I'll look at two in particular: the Golden Butterfly portfolio and the Pinwheel portfolio. I'll show you exactly which measures to look at, not just measures for risk but also measures for return and what happens after you retire. If you want to learn more about investing then why not sign up to our free weekly market roundup.

Let's look at those two portfolios in a bit more detail.

**Golden Butterfly Portfolio**

The Golden Butterfly portfolio is very simple and consists of one-fifth of each of these different types of assets. The golden part comes from the fact that a fifth of the portfolio is invested in gold so when you plot it on a pie chart it looks a bit like a butterfly.

There are two equity wings: one is the total stock market and the other one is a factor fund which is small-cap value. This is a combination of small stocks that tend to outperform over the long term but have a subset of those small caps where the valuation of the stocks is low.

Value has certainly been out of favour recently but historically it's been a very good performer and the combination of small-cap and value has outperformed even more.

On the fixed income side of the portfolio, we've got long-term and also short-term US Treasuries, and each of these assets has an equal weighting in the portfolio, we'll be comparing that with a Pinwheel portfolio.

**Pinwheel Portfolio**

In addition to the total stock market and small-cap value, this portfolio has: international and emerging market stocks, real estate investment trusts, cash (which is equivalent to very short-term government bonds) and it has intermediate bonds so there are no long-term bonds in this portfolio.

The weights alternate between 10% and 15% all the way through which is why the pie chart of the allocations looks like a child's toy called a pinwheel.

**Return Distribution**

When you invest your money in any financial asset, what you're actually buying is a return distribution.

On the x-axis below is the daily return going from -10% on the far left up to +10% on the far right and it is broken up into one percent buckets. Due to 0.2% return being slightly positive, and between zero and 1%, that's going to go into this zero to one bucket which now has a count of one.

- June 23rd 2009 - you can see the value of this developed world equity fund (issued by Vanguard) was worth exactly a hundred
- June 24th - the value of that fund went up by 0.2%. If we look at the distribution of returns so far, we've only got one return
- June 25th - the return now is 2.2%, a positive return which is larger than the first one that's between the two percent and three percent bucket
- June 26th - we got a return of -0.8%, our first negative return, which goes into the -1% to zero percent bucket

If we fast forward for 10 days of returns you can see that the distribution is starting to look a like a bell shape where most of the returns are close to zero. Sometimes you get big positive or negative returns but mostly the returns are small and clustered around zero.

Below is about 5 months of data and you can really start to see that bell shape starts to emerge. It's not a normal distribution, the wings are wider and the middle is pointier or as statisticians like to say it's a little bit 'leptokurtic’.

If we now look at a decade’s worth of daily returns, you can see that the counts become really smooth and we're starting to see some of the detail in the upper and lower tail on the downside. This is where you make your losses, on this very long downside tail and just a slight positive asymmetry is enough to give you this gentle drift upwards in the value of the index.

We can now compare different assets to see what kind of return distributions they tend to give us.

In the left left-hand panel, we've got developed world equity and the buckets are narrower so you can see the finer structure in these daily returns. You can now make out these very large positive one-day returns of more than 5% but also these very large negative returns in excess of 5%.

Most of the time you'll see the returns are clustered around zero and if we compare the daily returns with UK investment-grade corporate bonds you can see that the return distribution is much more narrow which means we have less uncertainty about the returns we get but it's also the case that we generally get smaller returns for investment-grade corporate bonds than we do for equity.

The average or mean return that we get for investment-grade corporate bonds is just 2.7% whereas the mean return for equity is 5.4 percent and that makes a big difference over a long period of time.

Another thing that these return distributions let us look at is how often the return is negative and here the two assets are quite similar. The returns are negative about 44% of the time for the developed world equity (excluding the UK) and about 45% of the time for UK investment-grade corporate bonds.

If we look at the width of the distribution, which is what those arrows signify, you can see that uncertainty characterized as a wider distribution. We're more uncertain about the exact level of return we're going to get for equity.

Volatility is a very commonly used measure of risk, in statistical terms it's called the standard deviation which is 14.9% for equity but only 5.4% for UK investment-grade bonds.

### Comparing Distribution of Returns

In the distribution above, the average return is 6.4% and the standard deviation (volatility) is 7.9%, which is fairly low, much lower than for equity on its own. The loss frequency (annual data rather than daily) is just 18%, in other words, 1 year out of 5 you're going to get a negative return with this portfolio.

If we compare that with a Pinwheel portfolio the average return is higher which we like as it means the upward drift will be a slightly faster but what we don't like is that the standard deviation (the volatility or uncertainty of returns) is 10.9%, which is quite a bit higher than it is for the Golden Butterfly portfolio.

The probability of a loss in any given year is also higher at 26% which is another unfavorable characteristic.

**Time To $1 Million**

Another question we can ask is if we have a fixed level of investing every year, how long will it take to accumulate a million dollars?

In the case of the Golden Butterfly, what you can see is lots of green lines, each starting off at a different year from 1970 to 1971 to 1972 and so on, looking at the period of time it took in order to build up a million dollars where you're contributing ten thousand every year.

The bigger the returns that you get with the portfolio the less time it takes to accrue a million dollars.

In the worst case, it took about 27 years and in the best case, it took 21 years when there was a stretch of very strong returns for the portfolio.

In the Pinwheel portfolio, the actual best and worst cases are very similar, 20 years is the best case and 27 years was the worst case for building up that million-dollar threshold.

## Drawdowns - **The Ulcer Index**

One beautiful measure which the portfolio charts website has is called the ulcer index.

The blue line in this graph is the level of the S&P 500 where dividends are reinvested and I've also adjusted for the rate of inflation. Overall, here you can see about 70 years of data on the graph, so you'll see I've had to use a logarithmic scale here on the y-axis. The red lines that show the peak up to that point in time show we get these very frustrating periods when the value of the S&P falls below that peak and takes a long time to return to it. So this period was about a decade which spanned the end of the 70s and reached the mid-80s.

Then we had another one of these lost decades just after the beginning of 2000 when the dot-com bubble burst, then when equity markets were just about to recover we got the global financial crisis which knocked us down again and meant that it took over 10 years to get back to where we were.

You can imagine the frustration during these drawdown periods as they're called and that's what the ulcer index tries to measure.

The definition of the ulcer index, which was co-created by Peter Martin and is described here on this **tangotools.com** website, is a number which measures not just the size of these drawdowns but also the duration and how long the portfolio takes to recover to earlier highs such that a deeper and a longer drawdown increases the number of ulcers.

### Comparing The Ulcer Index

The horizontal line below shows the deepest, in other words, the biggest peak to trough fall for the Golden Butterfly portfolio, and that was 11% which isn't too bad bear in mind that for equity you sometimes get falls of 50% or more.

The longest drawdown is the vertical line which was a period of two years, which although painful is a lot better than those decade long periods which we saw for the S&P 500.

This ulcer index really shows up the Pinwheel portfolio as being not as good as the Golden Butterfly because the deepest drawdowns are much deeper, they go down to 26% versus the 11% with the Golden Butterfly and the longest one at about 5 years, yet still not as bad as equity alone.

Combining those factors we get an ulcer index of 6.3 for the Pinwheel and just 2.7 for the Golden Butterfly.

**Long Term Returns**

We're always hearing that it's good to be a long-term investor, you'll almost certainly get more out of investment if that's your approach. Let's look at the long-term returns of these two portfolios.

The way Portfolio Charts does this is to let you adjust the horizon over which you invest and then look at those returns.

In the upper panel, you can see the value of $1 invested in the Golden Butterfly portfolio in 1969, it's gradually drifting upwards over time. In the lower panel you can see the rolling year returns what we mean by that is you invest a dollar in 1969 you shut your eyes and then two years later you can see that you've got a return of just over 5%.

In 1970 we do exactly the same thing, we invest a dollar we shut our eyes and two years later we look to see what we've got and this time we've got a return of over 10% but because this is a short investment period of just 2 years we often get negative returns over this very short period which is why we usually invest over a longer period of time.

### Comparing Long Term Returns

Now let's look at the 10-year periods.

If we invest a dollar in 1969 and then look to see what it's worth in 1979 the compound return over that period was pretty good at 6.4%, then we'd roll one year forward and do the next 10 year period and there the return was a bit better.

On Portfolio Charts you can set the investing time frame which is here 10 years and you can see that the typical investment return over that period is 6.3%.

The highest 10-year rolling period gave a return of 8.3% and the worst period was 4.1% but notice the return is never negative.

Looking at the Pinwheel portfolio you can see the typical return is higher at 6.8% but you can see that sometimes it disappoints so the minimum return here is 2.4% but the upside was also higher at 9.8% so by this measure I'd say Pinwheel is ahead but the downside tail is letting it down slightly.

**Withdrawal Rates**

The last thing we'll compare is the safe withdrawal rates for the two portfolios.

If we were to plot the value of most people's retirement savings over time, it would look like the graph below.

Let's say you start work at the age of 20 and you start saving into your pension and very gradually over time you see the value of the savings build up. Now at that point, you start to withdraw money from your savings, and the name of the game is that you don't want to outlive your money.

In this case, the money only reaches zero after you die which is a good thing because if your savings reach zero before you die then you're going to be asking your family to support you which nobody wants.

If we have the same level of savings when we retire but draw a larger income, you can see that the value of the portfolio depletes much more rapidly over time and now you can see that the money's run out before we die which is what we're trying to avoid.

### Increasing Capital

Another thing we can look at is what level of withdrawal never depletes our portfolio, in other words, the level of capital remains constant or even increases.

So let's say that we retired in 1990 and we'd built up a million dollars in our pension pot and let's say that our living expenses are about $50,000 a year in 1990 which is 5% of our total savings. Once we take out that $50,000 we'll be left with $950,000 from our original million.

The investment returns of the Golden Butterfly portfolio were -8.6% in 1991 so the value of our $950,000 investment actually fell for that year but we still need to survive so we still need to withdraw at least $50,000 but because of inflation, we're going to need more than that.

We'll need another $1,500 in 1991 and because inflation is generally positive we'll find that year on year we're taking out more than $50,000 and that's going to leave us with $817,000.

In 1992 the returns were very good we got 15.6% with our portfolio which is great and that took the value of our portfolio back up to about $945 000.

Our cost of living has increased again due to inflation, it's now over $53,000 so that's what Portfolio Charts is going to be doing in the graphs we're about to see.

The actual withdrawal rate percentage is only for the first year in the period, the amount actually withdrawn in the following years will be increased by the rate of inflation.

In fact, if we track that $50,000 over time it increases very significantly over that 30 year period, such that by the end of the period it's almost doubled to $100,000 a year.

That's why inflation is so important in retirement, if the rate of inflation gets out of control it can have terrible consequences for retirees.

If we look at the value of that million-dollar pot starting with that $50,000 withdrawal, for the first decade the value of the portfolio is stable but after about 2,000 inflation starts to eat away at our pot as we need ever-larger amounts of money to pay our bills. After the 30 year period ends we just about reach zero.

### Comparing Withdrawal Rates

The chart below from Portfolio Charts looks very complicated but in fact, it's really elegant.

First of all, you enter your retirement length which is 30 years in this case and this SWR number is the safe withdrawal rate which looks at the worst 30 year period since 1970 and sets the safe withdrawal rate such that your money exactly runs out at the end of the 30-year period.

In the case of the Golden Butterfly you can start off by withdrawing 6.4% of your investment pot and then each year after that you'd increase the withdrawal by the rate of inflation.

The safe withdrawal rate is considerably lower for the Pinwheel portfolio at just 5.8% and if when you pass away you do want to leave some money for your family you may be interested in the perpetual withdrawal rate.

In other words, the size of your pot remains constant and for the Golden Butterfly that's 5.3% whereas for the Pinwheel portfolio that's considerably lower, it's 4.7%.

Looking at the withdrawal rates both the safe withdrawal rate and the perpetual withdrawal rate the Golden Butterfly based on historic returns certainly looks better.

**Which portfolio is best?**

On balance I think I probably prefer the Golden Butterfly to the Pinwheel portfolio because the safe withdrawal rate is higher and the ulcer index was lower. Nobody wants too much risk in their portfolio and nobody wants ulcers.

I wasn't just trying to show you those two portfolios in this blog but how to use **portfoliocharts.com** which is an excellent website to compare other portfolios. They have lots of other stock portfolios but you can also create your own based on their returns and you can adjust it for your country's currency which is great.

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