Ladder Investing

What’s the best way to invest?

That question is too broad. It’s too big of a question to address in one post. That’s a book question, not an article question.

One way to constrict this question to make it more manageable is to specify what type of vehicle you’re investing inside of.

Here, we consider a long-term, fixed scenario, in which your capital is essentially locked, preventing you from liquidating prematurely.

How can we maximise the expected terminal value of our portfolio, as well as the value of the worst-case scenarios, under such conditions?

Theory

What we have now

There are two main schools of thought.

Some propose investing religiously in a collection of stocks. The rationale being that stocks are likely to out-perform other assets in the long-term.

The major issue with this approach is return sequence risk. Stocks are volatile, and you are likely to experience several nasty negative periods of return over, say, a 50-year period. If these arrive at the end of this period, you will eat painful losses right before you need the money.

My other criticism of this approach is that it is predicated on the fact that market X (or Z) will continue its out-performance over the next 50 years. There are reasons to believe why this might be the case, but relying on empirical arguments, which disciples often do, along the lines of “market X has historically at least doubled in 99% of 50-year periods”, or shoving charts comparing £1,000 invested in 1870 in market X vs. gold vs. gov. bonds in your face with the observation that your humble £1,000 invested in stocks in the 19th century would be worth £1,390,299,142.00 today but the hypothetical investment gold would be worth a pathetic £42.50. This is a dangerous line of reasoning.

So because of these flaws, others think “hmmm, how can I still capture the superior returns that stocks give you, whilst reducing the reliance on them and removing return sequence risk?”.

Their answer is Life-Cycle Investing.

LC is built on two ideas.

First, that diversification can reduce risk (defined as volatility of returns…) whilst maintaining an acceptably-high level of return. So if the JSE FTSE All Share index is tanking, your portfolio won’t tumble off a cliff because das bunds are doing rather well.

Second – gradually rebalancing from high-risk assets to low-risk assets as time progresses. This increases the reliability of your returns as the investment vehicle matures and reduces return sequence risk. There is also the added benefit of buying high and selling low; rebalancing to set % levels means buying under-performing assets and selling over-performing ones.

But stock-only investors would rally back that LC is time-consuming, costly (transaction fees), and ultimately less effective than their approach. Why invest in worse-performing assets? It also usually involves investing in a fund, hence handing over control of your finances to someone else.

Just climb the ladder

There is another way: Ladder Investing.

LAD is the act of purchasing progressively less volatile assets as you approach the end of the investment period. You line up all your possible investments from least volatile to most volatile, and map this line-up to your investment period, investing in the most-volatile assets first and the least-volatile assets last.

This sounds very similar to both the Buy-and-Hold strategy and the Life-Cycle strategy, doesn’t it? That’s because the core of these approaches is the same: invest passively, regularly, and unconsciously. Any investment strategy for the majority of retail investors should be centred on these principles.

Initially, it seems more like BAH; you purchase relatively volatile assets (mainly stocks) and hold them in perpetuity. A key difference is the flexibility in what you can buy (which is a differentiator to LC, too). Whereas with a BAH approach one usually purchases some big and famous collection of stocks, using the LAD method allows you to invest in a wide array of assets, enjoying the benefits of diversification. This also allows you to take on higher risk by investing in assets with more upside potential.

The other obvious differentiator is the stability that one enjoys from investing in lower-volatility assets closer to the terminal date. LAD mitigates some of the return-sequence risk by purchasing these stable assets.

So on reflection, it seems that LAD might be more similar to LC. In both techniques you purchase a collection of assets, putting more emphasis on stability over upside the closer you get to the expiration date. However, there are subtle reasons why LAD might be better:

  1. We don’t need diversification. One main point of diversification à la LC is to try and provide the investor with stable, consistent returns. But that’s irrelevant in a fixed portfolio. I only care about terminal value, not how I got there. So having a beautifully balanced portfolio to smooth the return line doesn’t matter to me. By doing this you invest in more-stable-and-worse-performing assets early in the investment period. This comes with massive opportunity cost and lost returns because this capital could have been allocated to higher-performing assets.
  2. More risk. Early risk.
    • Start compounding earlier. LAD mandates that we commit all of our capital to assets that will benefit most from a long investment horizon. This is because they 1) are more volatile so need more time to realise their true rate of return and 2) profit most from the magic of compounding.
    • The stakes are lower. The start of the investment period is likely to align with the start of your career. At the start of your career, you are likely to earn less than the middle and later parts. You probably have less capital to invest, therefore the cost of making poor investments is lower. This is why you can try some, ahem, “experimental” investments that might not come off. LC does not allow for these types of investments.
  3. An expanded investment universe. This is something that I touched on before, but it’s an important point that needs to be repeated; LAD affords you the opportunity to invest in anything, including off-the-beaten-track things that have high upside potential. When you start investing, you want to be investing in maximum-risk assets.
  4. The cost of rebalancing.
    • Transaction costs. Rebalancing your portfolio to some theoretically-optimal % mix of assets requires buying and selling. This costs money. Even if you are using a fund, these transaction costs are priced-in to the fees of the fund. See for yourself – go on Vanguard and look at the fund fees for an LCF vs. a purely passive fund. With LAD, we buy-and-hold assets, never selling and never incurring the transaction costs of doing so.
    • Upside costs. In LC, when assets are performing really well, you sell them to free up capital to invest in assets that aren’t performing so well in an attempt to stick to a predefined ideal allocation. If asset returns were purely stochastic (and therefore independent), this would be a sensible strategy. However, because there is some persistence in performance, selling well-performing assets may cost you some juicy return as these assets are more likely to perform well in the future.
    • Time and cognitive costs. Let’s say you have 10 assets that all have an ideal % weighting in your LC portfolio. Now, these assets will perform differently over any rebalancing period. They will also have different theoretical allocations at the end of each rebalancing period. This leaves the portfolio manager with the rather nasty job of solving 10 simultaneous equations; to buy and sell each asset in the correct proportions to arrive at the optimal portfolio allocations. This is complex and takes time and effort and aaahh I don’t want to do it.
    • But a caveat which solves most of the pain caused by the above 3 points. A better way is to take the lazy approach of just buying more of what assets are under-weight according to your ideal allocation. This eliminates most of the costs of a, b, and c. The one issue is that you might get into a situation in which assets are performing in such a way that it is impossible to get anywhere near your ideal split. In reality, you usually have to sell at some point.

What Does Monte Say?

Makes sense, right?

Now let’s run some numbers.

We simulate returns for 2 assets: a low-risk asset and a high-risk asset. The low-risk returns are drawn from a normal distribution with mean 0.05 and standard deviation 0.025. The high-risk returns have a mean of 0.1 and standard deviation of 0.2. We observe a 50-year investment period, assuming annual investment of £1,000 and a starting portfolio value of £0.00. The value of each asset is equal to the (return in period t * the value in period t-1) + 1000.

The distribution of returns for our low-risk portfolio.
The distribution of returns for our high-risk portfolio.

When we run LAD, we invest in HR for the first half of the period then switch to LR for the second half. For LC, we gradually adjust the weighting of LR from 0.2 at the start of the period to 0.8 at the end of the period. These ratios were selected as they seem to be the rough industry standard (see chart below). The value of each asset is initially (return in period t * the value in period t-1) + 500. Then rebalancing occurs via buying and selling occurs to reach the desired allocation for that period (this is done numerically).

A typical asset mix for a Vanguard LC fund. Source: Vanguard.

Some descriptive statistics for the terminal value of the portfolio for 100,000 trials:

MeanMedian5%1%
Bond Portfolio200200160150
Stock Portfolio1,00060013060
LC Portfolio400360180140
LAD Portfolio1,00060013090
Descriptive statistics of the terminal values of our simulated portfolios (with constant investment) using 100,000 trials. Value in £000s.

Here, we have mean, which is the arithmetic average of all the terminal values, median, which is the middle value, “5%” which is the bottom 5th percentile, and “1%”, which is the bottom percentile. 

A few things to note.

First, if we look at upside, there is a clear hierarchy. By considering the mean, investing in LR is worse than LC, which is in turn worse than just investing in HR, and LAD. The same is also true for the median return (which is what you might expect your portfolio value to be by using these methods).

LAD in general is strangely similar to our purely HR portfolio. It seems that the real return generator is the early investments in high-risk assets that are allowed to compound. The major difference is the performance of the bottom 1% portfolio. This is where those investments in the low-risk asset are beneficial: they prevent disastrous downside scenarios.

But LAD is not as good at doing that as investing purely in LR – or LC. And this is important – we want to invest to minimise the probability of an unacceptable outcome. Expected performance is important, sure, but so are downside scenarios.

It looks like LC is superior to the purely LR portfolio and LAD is better than the HR method. LC vs. LAD seems like a trade-off between expected values and downside scenario protection.

Income side-note

Some of you may take exception to the use of constant income in these simulations. After all, I did highlight it as a reason for LAD’s preeminence.

But if we include increases in income in our simulations, the results are very similar. The following table is a replication of the above but now investment amount gradually increases from 500 to 1,500:

MeanMedian5%1%
Bond Portfolio160160140130
Stock Portfolio75045010060
LC Portfolio300290150120
LAD Portfolio70040011080
Descriptive statistics of the terminal values of our simulated portfolios (with increasing investment) using 100,000 trials. Value in £000s.

Note the lack of difference in relative numbers.

Add this to the fact that when you look at actual income figures you don’t see that much change on an aggregated level. People are also more likely to have higher expenses as they get older. That mortgage and those school trousers and them holidays for 4 cost money. These things add up. Investment amount might not necessarily increase that much as a result of this phenomenon. Although, given that the most common application of these ideas is pension-pot investing, which is entirely income-driven, maybe increasing expenses don’t matter that much after all. And people do have more total savings as they get older, but this seems to be due to accumulation rather than increases in net income.

Mapping to Reality

The results of these simulations may be a little misleading.

Because they vary depending on what mean and standard deviation we are using for our returns. We have assumed that stock and bond returns are independent. And then there’s the classic problems associated with approximating the distribution of returns; they have fat tails, there is autocorrelation, blah blah blah, you should know this by now, etc. etc. etc.

We can check that our theory and simulations resemble reality by looking at real-life data.

Her Majesty’s United Kingdom

Let’s take stock market total return and long-dated gov. bond YTM and compare the investment strategies when restricting our investment universe to these two assets.

For the UK we use data on Consols, provided by FRED and the BoE, and UK Share Price data provided by again the BoE. One thing to note here is that this is price-return stock market data, not total return. The lack of dividends makes a significant difference. We must estimate dividend yield. Estimates for dividends for the UK range from 4%7%. In the US, over a similar-but-shorter time period, they averaged roughly 4.5%. Given that dividends have been slightly lower in recent history, we take 4% as a conservative estimate for dividend yield and apply this to each period.

We restrict our data set to where we have both stock return and bond return available, which for the UK is the period 1753 to 2016. Let’s have a look at these returns:

UK stock and bond return. Note: stock return is estimated total return based on a constant dividend yield of 4%. Source: BoE, FRED.

As you can see, stock returns are more volatile but, importantly, more likely to be greater than bond returns. This mirrors our low-risk and high-risk assets from our simulations.

Let’s use the same strategies, implemented in the same way as in our simulations. For an example period, the value of stocks and bonds for LAD and LC looks something like this:

LC return path for an example period. See above for investment technique. Note: stock return is estimated total return based on a constant dividend yield of 4%.
Source: BoE, FRED.
LAD return path for an example period. See above for investment technique. Note: stock return is estimated total return based on a constant dividend yield of 4%.
Source: BoE, FRED.

We look at sequential 50-year periods from 1753 to 2015 and examine the terminal values from each strategy. The first period is 1753-1802, the second period is 1754-1803, etc.

MeanMedian5%1%
Bond Portfolio14012010090
Stock Portfolio640300150120
LC Portfolio360180140140
LAD Portfolio610280140130
Descriptive statistics of the terminal values of our portfolios for sequential investment periods. See above for investment technique. Note: stock return is estimated total return based on a constant dividend yield of 4%. Value in £000s. Source: BoE, FRED.

We see broadly similar results as in the simulations, with a few key differences. Stocks perform better than the HR portfolio in the worst-case scenarios. Better than any other strategy in the 5% scenario, in fact. They are only beaten narrowly in the 1% scenario by LAD and LC. Another thing to note is just how consistent LC is in these downside scenarios: the 5% and 1% cases are essentially the same.

There are problems with this type of methodology, though. A better way may be to select the returns from a year at random and create a synthetic 50-year return period by stacking these returns from randomly chosen years on top of each other.

Let’s try it. We sample with replacement from all available stock and bond returns, selecting the return of both assets for each generated year. Again, we do this 100,000 times and observe some descriptive statistics of the terminal values:

MeanMedian5%1%
Bond Portfolio33013010090
Stock Portfolio400360150100
LC Portfolio250240160140
LAD Portfolio400340150110
Descriptive statistics of the terminal values of our portfolios whilst sampling with replacement using 100,000 trials. See above for investment technique. Note: stock return is estimated total return based on a constant dividend yield of 4%. Value in £000s.
Source: BoE, FRED.

This might give us a fairer picture of the returns one might expect from each of the two strategies, because we are taking many samples. And although this gives us broadly lower terminal values, the patterns observed thus far seem to be the same.

With one exception. Bond returns become more skewed. My suspicion is that this is caused by a handful of ‘perfect draws’, in which returns were sampled from the high-yield period 1960-90. This is one problem with this method – you can get unrealistic results if you sample mainly from an outlier period. These effects should wash out over 100,000 trials, but sometimes if the distribution is sufficiently kurtotic, they don’t.

There are other issues, too. These observations are not independent; we are not drawing balls from an urn. Return in year t has an influence on return in year t+1. So it might be unfair to create these synthetic periods by combining randomly-chosen years.

There is also the question of how relevant the returns of the 1700s are to the markets of today. The UK looked a little bit different back then, and we don’t really know to what extent assets from this period should influence how we invest right now.

Geometric average return for UK stocks and bonds. Note: stock return is estimated total return based on a constant dividend yield of 4%. Source: BoE, FRED.
Across the pond

Let’s repeat this exercise for the good ol’ U-S-of-A. This time our sample period is a little shorter (1872-2015), and we don’t have to estimate dividends as they are already provided for us. And this time we invest in $$$.

Again, we first consider sequential samples:

MeanMedian5%1%
Bond Portfolio1301309090
Stock Portfolio1,0001,000330290
LC Portfolio460360230220
LAD Portfolio1,0001,000310280
Descriptive statistics of the terminal values of our portfolios for sequential investment periods. See above for investment technique. Value in $000s. Source: http://www.econ.yale.edu/~shiller/data.htm.

Ok, so now investing in stocks looks like a very good idea. This is partly due to exceptional stock performance in the US and partly due to the reduced time period (stocks have performed better in recent history in both the US and the UK). Such is the extent of this out-performance that even in the worst-case scenarios, the stock-based portfolios out-perform. BAH even out-performs LAD across all lower percentiles.

And then drawing samples with replacement:

MeanMedian5%1%
Bond Portfolio2801409080
Stock Portfolio1,20078017090
LC Portfolio400370190150
LAD Portfolio1,100720170110
Descriptive statistics of the terminal values of our portfolios whilst sampling with replacement using 100,000 trials. See above for investment technique. Value in $000s. Source: http://www.econ.yale.edu/~shiller/data.htm.

This seems a tad more realistic. LAD slightly underperforms BAH in the mean and median, but out-performs in the downside scenarios.

LC’s downside-protection crown has also been restored. The question here is, is it worth it? Because to enjoy this protection you have to relinquish upside. And here, that upside is large. Is it worth an extra 20k in the 5% scenario to halve your median return? Or cut your mean return by two-thirds? I’m not sure.

What Does It All Mean?

Whatever method of investigation you use, the answer is clearly not to invest all of your capital in low-risk assets. Although the terminal values will be less variable, the expected values are substantially lower, even for the worst-case scenarios.

But we already kind of sort of knew that. What is far less clear is which of BAH-LC-LAD is the best, and when.

The argument for investing purely in stocks using some type of BAH method rests on the assumption that they out-perform. This is why we see (slightly) higher expected terminal values for the BAH strategies. Sometimes, when the gap in performance between stocks and bonds is large, this can spill over into the worst-case scenarios; it doesn’t matter that stocks are waaay more volatile; even the unlucky paths produce solid returns.

But this is largely a function of contemporary stock performance, particularly in the US. How relevant is this to the UK market? Or should we simply be investing in the S&P 500 instead – is there something intrinsically unique about the US that leads to out-performance? And will this continue into the future?

I add to these questions one from above: how relevant is the historical data? If the answer is not very, we can expect stocks to perform more like they have done in recent(ish) history. If we take this ancient data seriously, we must dial back our expectations.

It’s under such empirical doubt that other methods of finding answers become useful.

Are there logical reasons why stocks out-perform? Yes. Are there logical reasons why a specific market will return X% over a set period of time? Not really.

This also helps us to explain why LAD might be superior to BAH. This method gives very similar results under simulation and empirical analysis. It also out-performs in the 1% scenarios. So conditioning on these facts, it’s superior to BAH because of the logical reasons expressed above.

What becomes tricky is when there is no clear answer using any method of investigation.

LC seems to dramatically reduce the variability of returns and give us more downside protection. But this gap is eclipsed both in absolute and relative terms by the gap between expected and upside terminal values of LAD vs. LC. There are also practical reasons to prefer LAD. But then some of these reasons are made pretty much irrelevant by implementing lazy LC – by only purchasing assets and in rough quantities.

And if you think about it, these two methods – LAD and lazy LC – are pretty similar. As you move closer to the terminal date, you purchase fewer high-risk assets and more low-risk ones.

Which is better for Y.O.U? Depends on situation-specific details that couldn’t be covered in an already-too-long post.

It will of course hinge on your financial situation and disposition. Would you prefer – financially and otherwise – the kind of return distribution you are likely to receive from LC, or from LAD? It also depends on the returns of assets – all of the assets – moving forward into the next 50 years and beyond. If high-risk assets, defined today, perform well, LAD will do better. A practical factor is what assets you actually have access to. Your choices inside of, say, a pension fund are rather limited. Investing in some set, managed LC fund or strategy might be better, for a number of reasons, than trying to construct your own little mangled LAD approach. I could ramble on and on and on (and on).

General personal finance and investing information works at a high level because everyone’s finances look roughly the same. But ultimately, you have to make some choices when it comes to some details and the actual implementation. You can delegate these choices to someone else, sure. But there are still choices to be made, whether you are the one making them, or not.

What do you think?

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