Devastating portfolio declines and what it takes to recover from them – the math isn’t in your favor9/1/2019 Everyone thinks about gaining money when they invest, but too often we neglect how important it is to not lose money. Not losing money is so important, in fact, that one of the greatest financial investors in history (and very likely the greatest one alive today) espouses the following as his Rule No 1: Don’t lose money. What’s Rule No 2? Remember Rule No 1. Rule No 2 is obviously meant to be a little humorous, but Buffet is a serious man when it comes to investing and his rules are meant to illustrate a fundamental truth about investing – that truth is that it’s very hard to recover from a loss and that it gets harder and harder the deeper the loss. This is all best illustrated with examples. Sometimes, a good set of examples can do more for contributing to understanding than pages and pages of text. So, let’s go over three examples, each with increasing levels of severity of initial losses. In each example, we’ll break things down into 3 time periods – Time 0, Time 1, and Time 2:
A 25% Loss – Somewhat severe, but recoverable With a 25%loss, your $1000 declines to $750 – this represents a onequarter decline in your portfolio and would obviously be an unwelcome occurrence. Now, let’s take a look at what sort of returns you’ll need to recover by Time 2; let’s see what sort of returns in the subsequent time period you’ll need to make you whole again. As you can see from the table, a 33.33% gain is required in order for you to recover and get back to the initial $1000. A 10% return, 20% return, or even a strong 30%return in one time period simply won’t do it. That means if each time period is 1 year, even a 30% return in the year subsequent to your 25% loss won’t be enough. 30% is a solid return. The fact that it’s not enough should be the first hint that getting back to whole is a lot harder than dropping, from a mathematical/percentage perspective. It’s only going to get worse. A 50% Loss – Very severe, but you can recover if you stay prudent over the long term With a 50% loss, it’s a lot harder to recover. Now, it takes a 100% gain (doubling your postloss portfolio value) to get back to whole again. If you halve your portfolio, you’ll need to double it to bring it back to its original value. So, if a time period is one year, you’ll need to double your postloss portfolio value to get back to your Time 0 initial value. That’s very hard. You’d be far better off having avoided such a decline because it’ll be an uphill climb getting back to baseline again. This is what Warren Buffet’s Rule No 1 points to. A drop of 50% in your portfolio value is very severe and detrimental to your long term investing goals. It will take a 100% increase  doubling your portfolio  to recover from a 50% loss. This is tough, but it's doable  it might not happen in a single time period but over time a prudent and disciplined investor stands a chance at recovery. A 75% Loss – A devastating blow to a portfolio that will take some time to recover from With a 75%, things get really bad. Now, in order to get back to whole, you’ll need a 300% gain. A 300% gain is the same as quadrupling your money (4x return). As any investor knows, a 300% return is very hard to get – it usually takes years to achieve such returns in a welldiversified portfolio. Let’s think about this some more. As we keep increasing out Time 1 losses by 25% increments, the return needed to get back to whole by Time 2 goes up by way more than 25%. This is based on the underlying mathematics of portfolio returns, but we don’t need to get deep into that here. The above examples should clearly show how each time the loss gets more severe, the needed gain to get back to baseline gets more and more astounding. If you lose 75% of your portfolio’s value in a single year due to a very severe recession or, far worse, due to investing blunders, you’re going to have to make some incredible returns (300%) to recover. What makes you think you’ll beable to do that? It’ll likely take a number of years and some serious investing discipline to be able to recover in this way. A 75% portfolio decline is devastating to any portfolio. It will take a 300% return (quadrupling your money; a 4x return) to get back to whole again. This is very hard to do in a single time period. It might take years of prudent and disciplined long term investing to recover. This demonstrates why large portfolio declines are so detrimental and should be avoided. A bit more mathematical, for the mathematically inclinedFor those that are more mathematically inclined, let’s dig a bit deeper into the portfolio maths.
Let’s assume an initial portfolio value of a – this is your Time 0 value For any portfolio change (decline or increase) d, where is greater than 1 but less than 1, the portfolio value in the immediately subsequent period (Time 1) will be a x (1 + d) To get back to the initial portfolio value by Time 2, we’ll need to do something to the Time 1 value to get it back to a (which we stated above was our initial value) We can simply divide the Time 1 value by (1 + d) to get back to a – that’s [a x (1 + d)]/(1 + d) Dividing by (1 + d) is the same as multiplying by 1/(1 + d) – that’s the amount, no matter what our initial a is and what the change d ends up being, that we have to multiply the Time 1 portfolio value by Now, notice that if d is less than 0, 1/(1 + d) will be larger than 1. So, if d is 0.25 (corresponding to a 25% decline in our first example above), then 1/(1 + d) is 1/(1 – 0.25) which is 1/0.75. What’s 1/0.75? It’s 1.3333. That means you’ll need 1.3333 times the Time 1 value – this exactly represents an aprox 33% increase. Let’s do a 75% decline as in the third example above – now 1/(1 + d) is 1/0.25. That’s equal to 4, which represents a 300% increase over the Time 1 value. We can see that as d approaches 1 (moving towards a total loss), 1/(1 + d) gets bigger, but by a disproportionate amount. Can we derive a simple way to see how our 1/(1 + d) factor changes with changes in d? Yes – it’s easy using Calculus: d/dx[1/(1 + d)] = d/dx[(1 + d)^1] = [(1 + d)^2] x d/dx(1 + d) = [(1 + d)^2] x (0 + 1) so, the derivative is 1/(1 + d)^2 Calculus can be applied to lots of situations to better understand how things change. F We can see that by squaring the (1 + d) term, we’re increasing the effects of both positive and negative portfolio changes. If d < 0, then squaring (1 + d), which will be less than 1, will only make the factor smaller. By making that factor smaller, the entire factor gets bigger because dividing 1 by smaller and smaller numbers makes the result bigger and bigger. This should be very discouraging – the numbers tell us that negative effects are magnified when we think about the returns needed to recover. 
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