Any financial planner worth his or her salt will tell you that the amount of money that you have to spend in retirement will depend mainly on the amount of money you start with, how much you contribute to your savings, the rate of return on your assets and the taxes you pay.

But let’s get concrete. As an example, suppose we wanted to find out how much we have to put aside each year to live comfortably in retirement. I will cover the opportunities and issues with pension plans and insurance-related savings products like annuities later, but for now assume you are saving up and have the luxury of ignoring tax.

Building your savings

Suppose you are 30 years old and have savings of £10,000 after finally paying off your student loan. Let’s say you expect to work until you are 67, and plan to be able to put aside £10,000 a year in today’s money (so real numbers) for your retirement. To be comfortable in retirement you think you’ll need £20,000 a year in today’s money (inflation will make the future number bigger but be equivalent to £20,000 of purchasing power today) on top of other pensions (e.g. the state pension) you have and you expect to live to 90.

Because you are a prudent person you have decided that the way forward is to put half your savings into the minimal risk asset and the other half into world equities. To ensure that the ratio between the minimal risk asset and equities does not get too out of kilter you plan to rebalance the portfolio every year to ensure that the split is 50/50 at the start of the year. The plan is then to move everything (i.e. 100% of your portfolio) to minimal risk assets when you are 67 and will start needing the money.

You may expect to make 0.5% on the minimal risk asset and 5% on the equities (0.5% + 4.5% equity risk premium) before minor fees. While the risk of the equity return has more frequent extreme outcomes than suggested by standard statistics – called fat tails – the annual standard deviation (SD) of that equity return is probably in the range of 20%. Taking into account these fat tails we increase our standard deviation assumption to 25% (a simplification).

To get an idea of the probabilities of having enough money in retirement here is a relatively simple excel model. The inputs are summarised as follows:

You expect to live to age 90 and wish to withdraw £20,000 in today’s money each year after you turn 67, so you will be withdrawing a total of £460,000 for your 23 years of retirement spending. Taking into account that you will make 0.5% on your investment in the minimal risk asset even in retirement you ‘only’ need approximately £434,000 in assets when you turn 67 to have enough money. This is of course excluding the fact that you probably want to have a buffer of additional money in case you live to be more than 90 (unlike an annuity, the buffer is not ‘lost’ when you die – it goes to your descendants)

So will you have enough? Well that depends on the volatility of the equity markets. The results are striking and highlight some important points.

I set up the model to re-run itself 1,000 times with the equity returns generated by a random function. Basically I told the computer to play this game 1,000 times where the return on average was 5% per year, but that this return should vary with a standard deviation of 25%. (Chapter 6 on risk discussed how this is roughly what risk has been in the past, and explained what that meant.) In roughly 68% of the cases the return on equities will be within a range of −20% to +30% (one standard deviation from the mean; average of 5% return and then add or subtract 25% depending whether it is a negative or positive 1 standard deviation move).

The 1,000 iterations of the model reveal greatly varying results. If equity markets are good between now and your retirement you will have far more money than you planned to spend. However, if equity markets between now and retirement are bad, the savings are far from enough to cover the future outgoings and you potentially have a serious problem. The results from the iterations are as follows:

The numbers above represent the outcomes of the 1,000 iterations and are based on the inputs above. The only thing that makes the outcomes vary so much is the impact of having the equity returns vary in risk by a standard deviation of 25% per year (more or less in line with what world equity markets have experienced).

The point I am trying to make is that the uncertainty of equity returns can lead to very large fluctuations in outcomes over a long time horizon and that you as an investor need to be aware of that and plan for it in your financial planning. Depending on how strong or weak equity markets are between now and retirement you might have less than half the money you need, or almost 10 times as much as you need. Those are pretty wild swings, brought about by decades of taking risk in the equity markets. I’m obviously not saying that you should rigidly stick to the strategy above, but the point remains that long-term financial planning can lead to a wide range of outcomes once you introduce risk into the portfolio.

So, in returning to our model where we ran 1,000 iterations, the outcome we should expect if we are exactly in the middle is the median, not the average. If this was 1,000 outcomes, then we should expect the 500th, or median outcome.

Figure 10.1 shows the amount of money in our investment account at the age of 67, and the percentage of iterations that fall within various bands.

A couple of things are probably obvious about the graph:

• Most cases cluster around the median return, but there are also a few very positive outcomes where we end up millionaires.
• There are a high number of outcomes where we were unlucky and ended up with savings short of what we needed for retirement.

In my view, this second point is critical and goes to the heart of investment management and the risks we are willing to take with our financial lives.

As outlined above, we need to have accumulated approximately £434,000 by the time we are 67 in order to be able to withdraw £20,000 a year until we are 90. We also know that in the average (median) case we would have assets far in excess of that requirement. But now we also know that in a large number of cases our savings will fall short of that required:

Cases above £434K: 81.2%

Cases below £434K: 18.8%

So in nearly 19% of cases you will not have enough money for your retirement goals.

To most people a probability of approximately 20% that you will fail to meet your goals would cause great concern, particularly as you probably want some sort of additional reserve in case you live past 90. But what can you do about it?

As a rational investor, the worst thing you can do is to abandon your principles and pursue the promises of higher returns from various active managers or investment schemes. While you may get lucky, on average it would only make the situation worse.

In effect we have a few choices:

• We can contribute more annually.
• We can start with a higher amount.
• We can accept a lower annual amount in retirement.
• We can shift the mix between the minimal risk and equity to lower our risk, but at the expense of a lower expected asset base at the age of 67.

The super-cautious saver

If you are unwilling to take any risk that your savings fall short in retirement you could invest only in the minimal risk asset and your savings at age 67 would be £497,000, and far in excess of the £434,000 needed. By allocating some money to equities your expected savings would go up, and while that extra money would surely be nice to have it comes with a risk that there will not be enough. The greater proportion you allocated to equity, the more your median savings will be, but you will also increase the risk of falling short. The security of allocating entirely to the minimal risk asset comes at the expense of having significantly lower expected assets at 67 than if you had allocated some assets to equities – but that is a choice you have.³

If we had done the same exercise and found that our retirement savings fall short when investing only in the minimal risk asset we would have been faced with some tough choices. The way you respond to this shortfall relates to your personal attitudes towards risk:

• You could contribute more to your portfolio on an on-going basis if you are fortunate enough to be able to do so, or find a way to have a larger initial amount.
• If you are very risk averse you may not be willing to accept any risk with your retirement funds and simply accept a lower annual retirement pot.
• If you increase the equity allocation, the additional expected returns from equities will increase the chance that you have enough money for retirement. But while you increase the chance of achieving the target savings and on average will have more money in retirement, you do so at the expense of potentially incurring a greater shortfall in the cases where future equity markets are bad.

Risk/return

We can use the example above to get a sense of the price we pay for the surety of having enough money at retirement.

In the 50% minimal risk/50% equity case we would expect to have approximately £653,000, while the minimal risk only portfolio would give us £497,000. A difference of over £150,000 to use in our retirement! And the cost of that expected additional amount? Accepting that there is roughly an 18% risk that you will not have enough money to take out £20,000 a year in retirement.

The decision whether those additional expected assets is worth the risk of a shortfall in retirement often depends on personal circumstances. What will be the impact on your life of falling short of assets in retirement? Will this be catastrophic or just a mild annoyance? Hard to tell. Similarly, what will be the impact on your life of having significant excess funds? Will it make you happier? Does it mean that you can afford a better life for your children than you had yourself? Or perhaps you can leave them with a nest-egg that you could not otherwise afford.

Below are the results of running the iterations with various fractions of the assets invested in the minimal risk asset:

The bottom line of the table illustrates the amount of assets we are left with in the fifth per cent worst outcome, or every twentieth case. It is no surprise that the fifth percentile number gets worse as we increase the equity portion. With the higher expected returns from equities come the higher risk and the 5% worse cases therefore get progressively worse as we increase the equity allocation.

Generalising the examples

Like any financial model this is the case of rubbish in, rubbish out. Models are only as good as the assumptions you put into them. The model above is based on some pretty generic assumptions about risk and return, and only incorporates the minimal risk asset and equities. You might disagree with those assumptions and be more aggressive about your return expectations or risk. You may want to run the model assuming 0% real return from your minimal risk asset in line with the current market, or introduce some risk to the return of the minimal risk asset.⁴ Or you could move away from having a 50/50 allocation to minimal risk assets/equities in the portfolio, depending on your risk tolerances.

Instead of continuing to adapt a financial planning example that may not be entirely relevant to your financial planning needs, I would encourage you to do this kind of financial analysis yourself or to get someone to help you with it. You can do financial modelling in Excel, in Google Drive or use one of the many financial software packages available. In any case, make sure you spend enough time figuring out how everything fits together, and confirm that the model fits your specific needs.

It could be that you want to get an overview of what $100,000 invested in equities would get you in 10 years’ time with the same assumptions on risks and returns (see Figure 10.2). Or it could be that you want to figure out the probability that the money you have put aside for your children’s education will suffice. You might be an insurance company or pension fund that wants to find the probability of having enough funds to meet your future liabilities. The list is endless.

In any case, building a financial model that considers the range of potential outcomes will probably spark your thinking about the impact that the various outcomes will have. Take the model and amend it to roughly suit your situation. Then start playing with the inputs, assumptions and allocations:

• Have you been too conservative/taken too much risk?
• Could you bear bad equity markets in the decades ahead – what if they are riskier?
• How much is your situation likely to change in future? How will that impact on your risk profile?

• Is it possible that your financial goals are incompatible with the risk you are comfortable taking?
• What is the lowest 5% of outcomes? Lowest 2%? Could you handle it? Would you be willing to accept that you could fall short in one out of 20 cases? One out of 50? Never?
• Do you have a ‘drop dead’ level of assets that you simply are not willing to fall below? How does that dictate your allocations?
• As ‘touchy-feely’ as it sounds, try to feel what would happen in various cases. What does your gut tell you? How would it affect how you sleep at night? Your career? Your marriage?
• What would happen to your other assets, job, etc. at the same time as any fall in the markets? Would they reinforce what is happening to you or offer respite?

Investing time in understanding the risks of your portfolio or financial goals is a worthwhile undertaking whether through a model like this or any other means. Even in the scenario when you don’t change your thinking or portfolio as a result, the increased understanding will probably lower the emotional strain of adverse events. You have an idea of what might happen and what the consequences could be for you.

A few important points to remember when using a model like the one above:

• The model is very simple and built with some very basic assumptions, but be cautious about those that add lots of products with high returns and low correlations. This will enable people to come up with much better risk/return outcomes, but as rational investors we do not think it can be done (indeed rubbish in, rubbish out).
• The model does not incorporate corporate bonds and government bonds that could be a worthwhile addition to the simple rational portfolio. This is a simplification. Incorporating those two asset classes would involve estimating correlations between them and equities, which is a science in itself (and a constantly moving number). Besides, adding other government and corporate bonds is not a magic bullet for the portfolio; while those asset classes are good diversifying additions to the portfolio they are not without risk.
• The model is very sensitive to the inputs, particularly on risk, and does not account for some of the issues with using standard deviation to gauge risk that was discussed earlier (in Chapter 6). Changing the inputs would make a massive difference to the range of outputs. If you do change the inputs, make sure you understand why – don’t make ‘nicer’ assumptions just because you like the results better. The expected real return on equities will not be 10% a year just because you put it in a spreadsheet.
• Keep in mind that all the returns are real returns. So the model is in today’s money. Clearly the actual numbers decades from now will look very different because of inflation.

Keeping it real

If we have got to the point where we have built a financial model that reflects our situation and implemented our investment plan by purchasing the right products (see Chapter 14 on products and implementation) we are doing really well. Unfortunately that is not the end of it.

We have to keep an eye on our portfolio and financial model and adapt it to changing circumstances and, if nothing else, then the passage of time. Imagine a scenario where the market has moved up 50% in a year. As we look at our financial planning we would be remiss if we didn’t somehow take our new and improved financial situation into account in our forward planning. It could be that with our higher asset base we are able to reach long-term financial goals with a far lower risk.

When we periodically look at our portfolio in the context of our overall financial health it may also be that our personal circumstances have changed, which in turn could affect our financial plans – we were promoted, fired, received an inheritance, got divorced, the uninsured car was stolen, our tax circumstances changed, etc. It all matters. How often we do this kind of review is a personal decision, but at minimum you should aim for yearly or whenever there has been great turbulence in the financial markets or your personal life. Since you should rebalance the portfolio periodically anyway that would be a good time to review its composition. I’ll come back to rebalancing later.

Reacting to disaster

In the aftermath of the 2008–09 financial crisis many investors were understandably left aghast. Many had lost far more money in their portfolios than they thought possible, and often did so as their house and other assets also plummeted in value. The gut reaction for many investors was to sell their equity exposure at or near the bottom of the markets, only to miss out on the great rally that followed. ‘A familiar story of retail investors abandoning their plans’, some financial planners lamented.

I don’t think things are that simple. During crashes like 2008 there is a natural tendency for everyone to have a view on the markets. The markets will dominate the headlines and be a topic of conversation at work, the gym, meals out, in homes and everywhere else. How can you not have a view?

The point is that we still don’t have a view. While many people with the benefit of hindsight say they saw the rebound, ‘just’ because there is great market turbulence does not mean that an investor is better able to predict market movements. We don’t consider ourselves smarter than the average dollar invested in the market, and that average dollar put the S&P 500 at an index value of under 700 in March 2009. The fact that four years later we see that same index trading around its all-time highs does not mean that we could predict in March 2009 that this would be the case.

When you look back at the 2008–09 crises or any crises preceding it, many people have a sense that there is a bottom somewhere, and great profits to be made for investors who find the bottom. And clearly that has often been the case. If you had stayed the course or invested more at exactly the right point in March 2009 (or July 1932) you would have made a lot of money. But you obviously did not know that then. For all you knew at the time, March 2009 was just a precursor to a really bad decline in the market, and you were scared to death. There is no guarantee that markets bounce back after a decline. Just ask investors who bought Russian equities in 1917 …

But that does not mean that there is nothing you can do. First of all, after bad declines in the market it is likely that the future riskiness of the market has gone up a lot in the general turmoil. While that does not give market direction at least you can prepare yourself for the increased risk. Those willing to bear the extra risk will probably see commensurate higher expected returns, but they have to be willing to accept that a lot of money could also be lost.

We know that losses like those in 2008–09 do happen with some frequency and it is at times like these that you hope to benefit from having had a conservative allocation policy, instead of selling your holdings in desperation.

In any case there are no easy fixes and we are faced with a few unpleasant alternatives:

• We can find a way to put more money aside in savings.
• We can accept a lower amount in retirement or the same amount for a shorter time period.
• We can reallocate between the minimal risk asset and equities if the large decreases have affected the risk we are willing to take with our portfolio.

Though it sounds like annoying hindsight, investment allocations are about ensuring that you don’t find yourself in the position of making panic sales to start with. Have enough of a buffer so that you avoid selling equities at what might be the bottom of the markets. Over the long run, equity markets are likely to far exceed government bond returns, but they will also be far more volatile and periodically lose you a lot. Make sure your allocations allow for that.

A few ways to think about portfolio allocations

Here are some suggestions:

• Drop dead allocation If you need £100 for heart surgery, don’t buy equities with your last £105. Have enough money in the minimal risk asset so that you certainly have enough in the short term, and only then start adding equities.
• Pick a return point and see if you can handle the risk You may expect 0.5% per year from the minimal risk asset and a premium to that of 4–5% from a broad equity index, both after inflation. You can pick the return point you are after, and then figure out the risk required to achieve your goals (see Figure 10.3).
  A word of caution: if you borrow money to achieve higher returns than those from equities keep in mind that those loans tend to get recalled exactly at the worst time and lock in your losses.
• Pick a risk point and see what returns you can expect How much risk you can handle is subject to your personality and individual circumstances. You can work out your worst 1%, 2%, 5%, 10%, etc. scenarios to see what various portfolios look like, and get an idea of the risk you are willing to take. Put this in the context of the consequences of failing to reach your financial goal and guide your allocations that way.

• Think about the flexibility of your financial goals As you consider your portfolio allocations it is worth keeping in mind what is implied by the end portfolio value you are trying to achieve. Does your goal suggest the minimum of financial survival or meeting a firm liability with its nasty potential consequences of failure? Or does your target really suggest more of a ‘nice to have’ lifestyle pot of assets? These considerations should play into the risk you take with the portfolio.
• Think about avoiding temptation Set your portfolio up so you have some flexibility in case the markets move unexpectedly. Reallocations are costly and retail investors are notorious for selling their equity holdings at the worst time. Generally avoid trading in and out of things excessively – it’s costly and will significantly reduce long-term returns.
• Use market experience Think about how you react to losing money as the markets inevitably drop on occasion. If you can’t sleep at night for worry about the future or sell after the decline then your equity allocation is probably too high. On the other hand, if you think ‘that’s nothing’ and want to add more equities then you were probably too risk averse in your allocations. Sometimes it can be hard to know exactly how you will feel until the moment happens, and it can be hard to predict how other parts of your life would be affected by a drop in the markets.
• Impact of non-financial assets and liabilities As you think about the risk in your investment portfolio, consider how other assets and liabilities are affected by the same factors. Make sure that not too many bad things can happen to you at once, both from a value and liquidity perspective. Should the world equity markets experience a calamity, it is possible that the value of your house, job (an equity broker will be affected more than a civil servant, etc.), and other assets all decline at the same time. Those things are not as liquid as the rational portfolio. Be sure to avoid a distress sale of your only liquid asset (the rational portfolio) in bad markets by minimising illiquid assets that all drop in value at the same time. Meanwhile, your liabilities are often more fixed.

It is often a great idea to put your portfolio objectives and risk tolerances into words in a simple ‘portfolio mission statement’. Unless you are so inclined, the statement does not have to be accompanied by a picture of a soaring eagle, but be something for you to look at occasionally, and particularly as your circumstances change. It helps to think of why you are saving up or allocating between equity and bonds and the maturity profile of those bonds. It could also be helpful for you to put into words how you would react and feel about bad things happening in your portfolio. If nothing else, this forces you to think about how you would react to various scenarios and potentially restructure your portfolio as a consequence of something not sitting right with you.

Stages of life

Your age will be a big factor in how you allocate your portfolio.

Early savers

It is generally the case that younger savers allocate a greater portion of their portfolio to riskier assets. They are in the early stages of saving and the cumulative benefits of even a small expected outperformance from a slightly riskier asset can add up to a large amount over the coming decades. Also, should markets be bad early on, savers have decades before they need the money and more time either for the investments to make up the shortfall, or for them to adapt their lifestyles or savings rate. My advice to you: take risk with your savings and put lots in the equity markets, be ready to lose a lot of it, but also keep enough in the bank so that you can afford a crisis. It’s a good time to learn about the markets and how you deal with the risks. You should also familiarise yourself with all the tax benefits that might arise from pensions or other savings (such as ISAs in the UK). Getting into the habit of saving up and maintaining the discipline to stick with it is something that will serve you very well, particularly as you begin to see the cumulative gains from being a saver.

Mid-life savers

Once you enter the 30s and 40s you pass into the ranks of the mid-life savers. You might be at your prime in terms of earnings power and starting to get a good sense of how things will turn out careerwise. For many mid-life savers, tax considerations will play a major role in the execution of their portfolio, and many will want to allocate a greater fraction to the minimal risk asset, perhaps in longer-term bonds, than they did a decade or so earlier. Whilst these savers have accumulated some savings, the potential added return from allocating to equities is still important in reaching their financial goals in retirement. And should the equity markets be bad in future there are still some working years to address the losses from those investments, either by saving up more and reducing current spending, expecting to work longer or reducing expected retirement spending. At this point in your life savings cycle you might start to get a sense of what your expenses in retirement will look like, and perhaps how many income-earning years you have left before retiring.

In terms of practically shifting your investments from equities into lower-risk assets as you age I would encourage you to do this as you either put money into your savings or take money out anyhow, as that reduces trading costs and potentially taxes.

Retirees

At the other end of the spectrum is someone already in retirement. Particularly those without a great amount of savings to see them through their remaining years typically have a far lower risk tolerance as there are fewer options to make up a shortfall if markets turn against them. At the risk of over-simplifying, if you don’t share the upside of having more savings (with limited years left to enjoy them), but would experience the painful downside, then don’t take risk and stay with minimal risk bonds. Of course estate planning and passing on assets to the next generation will play a major role here in terms of the exact structuring of your portfolio. Also think about what non-investment income you can expect in the form of company pensions, social security, etc. and compare that to your expected outgoings. The difference between the two will need to come from investment income, or liquidating part of your portfolio. While many rules of thumb don’t apply universally, if you stick to only spending 4% of your portfolio a year, you will probably be fine (you can increase that percentage as you grow older).

For those in retirement I would encourage you to get ready for the day when you can no longer handle your savings yourself, or even plan for eventually passing them on. Keep things simple; have only a couple of accounts and not too many investments, and make clear to whoever is going to take over the management of your assets how you want them managed and why.

For those retirees with savings in excess of what they need, the risk profile of the portfolio may be different. These retirees are no longer only investing for their own needs, but also for the longer-term needs of their descendants or whoever the assets will be going to. Since the time horizon for those descendants can be much longer term, the portfolio could well include some equities and a generally riskier profile than if it was just for the retirees.

It really does depend on your circumstances

Like most things in investing, allocations are highly subject to individual circumstances and risk tolerance. Figure 10.4 shows how an investor’s allocations may change over his or her life, ignoring the complication of risky government and corporate bonds in the rational portfolio.

A few rules of thumb

In researching this book I came across several publications on the topic of investment and savings which had rules of thumb that were pushed as gospel truths. While I think rules of thumb fail to incorporate individual circumstance and attitudes here are a few to consider:

• Your age in bonds (60 year old = 60% in bonds).
• Don’t withdraw more than 4% of your portfolio a year in retirement.
• If you react badly to losing money in your portfolio, reduce the equity exposure by 10%. Keep doing that until you are OK with portfolio falls.
• Equity exposure = amount of years until retirement in percentage terms. So with 10 years to retirement your equity exposure should be 10%.
• Equity exposure = 120 – your age.
• Put aside 10% of income for retirement.
• The value of your house should be less than three times your annual income. (It was staggering to see how often and by how much this rule was broken particularly pre-crash.)
• Base your retirement needs on 100% of pre-retirement expenses plus 10%.
• Have at least 7–10 times your annual income in life insurance.
• Have at least 6–8 months of living expenses in cash in the bank.

Clearly these are exactly that, rules of thumb. Obviously these rules don’t apply to everyone, or indeed most people (and some are even slightly contradictory). Instead, in this chapter I’ve tried to give you a sense of how to think about your individual risk and some ways to understand the probabilities of encountering tough conditions when saving.

Adding government and corporate bonds

In earlier chapters, I suggested that there could be merit in adding risky government and corporate bonds to the portfolio, but we have not incorporated them in the examples above. This is not out of neglect, but to keep the discussion above simple and practical. If we attempt to add risky government and corporate bonds to the excel model outlined earlier we would significantly add to its complexity and you would need to have a view on complex things like asset correlations. Taking an average correlation between risky government bonds, corporate bonds and equities would be too much of a simplification. During times of great distress, correlations between asset classes tend to go up (just like they do between various equity markets), and as we run the multiple scenarios we would need different correlations for different states of the equity markets, resulting in something like this:⁵

Equity return	Sub-AA government bond correlation	Corporate bond correlation
15%+	0.35	0.40
5% to 15%	0.28	0.32
−5% to 5%	0.22	0.28
−15% to −5%	0.34	0.38
Below −15%	0.55	0.67

What this adjustment incorporates is that the addition of risky government and corporate bonds adds less diversification when you need it the most, namely when equity markets are poor. Since the bonds I suggest adding here are sub-AA rated return generators, in a crisis they may not be seen as a ‘flight to safety’ and go up in value. Your minimal risk asset will probably be seen as a safe haven and increase in price.

At the risk of oversimplification, other government and corporate bonds will add less risk to the portfolio than equities, but will not be risk free. Depending on the exact mix of sub-AA government bonds you add, you could perhaps see the other government and corporate bonds as adding half the risk of equities, although realise that this more a guesstimate than a precise calculation.

Adding great complexity to the simple model misses the point that all this is far from an exact science. The point of the excel model is to show that even the best-laid plans are subject to market risk, and to point out how we can incorporate this element of chance in thinking about our portfolio allocations and planning. By making the model excessively complex we run the risk of giving a false sense of precision.

1 On Youtube you can find a brief and instructive video called ‘Geometric vs. Arithmetic Average Returns’.

2 There are a surprisingly large number of books and articles that seem to neglect this issue. A typical example will be an illustration that talks about $1,000 invested making an average of 5% a year so that after 30 years you will have made $4,320, ignoring the fact that high volatility in the yearly returns could significantly reduce the final value.

3 If you wanted minimal risk you should buy inflation-adjusted government bonds with maturities similar to when you will start needing the money. In this case, that would involve some of the longest-dated bonds available, which typically carry a higher yield than the 0.5% used in this example.

4 As you increase the risk of the minimal risk asset the importance of introducing correlation with your equity investment increases. This is a complication I have left out.

5 This is still a significant simplification. The correlations in the various return scenarios would be an average expected correlation. Since each crisis is unique there is no guarantee that the actual correlations would be like those in the table.