How Math Fattens Your Wallet
July 10, 2005
I spend my weeks taking financial topics and trying to explain them in plain English. But sometimes, words just aren't enough.
Indeed, to get a decent handle on investing, you really need to grasp some basic mathematical concepts. Sound scary? Don't worry, you aren't about to get dragged into some mind-numbing numerical quagmire. Instead, what follows is a gentle stroll though a few key investment ideas and the math that backs them up.
Bearing Down
To clock healthy investment returns, you need to take risk. But if you take too much risk, you could suffer a financial setback that will take years to recover from.
At issue is the brutal math of investment losses. Suppose you pick some bum investments and your $100,000 portfolio tumbles to $75,000. To recoup this 25% loss, a 25% bounce back won't cut it. That would get your $75,000 to just $93,750. Instead, to make back your full 25% loss, you need a 33% gain.
The steeper the loss, the more daunting the climb back. Suffer a 50% drubbing and you will need a 100% gain to return to even. Lose 75% and you will require a 300% recovery.
True, a 75% loss might seem unlikely. Yet the Nasdaq Composite Index fell 78% from its March 2000 peak to the October 2002 market low. Anybody whose portfolio had mirrored the Nasdaq was looking at a steep 355% climb to get back to even.
Digging Out
How long would it take to earn that 355%, assuming 8% annual performance? The temptation is to divide the 355% by 8%, putting the recovery time at 44 years.
But in fact, it isn't quite that grim. Instead, assuming an 8% return, it will take Nasdaq investors a little under 20 years to recover their bear-market losses.
To understand why, let's assume your $100,000 portfolio earns 8% this year and 8% next year. The first year's 8% gain will boost your portfolio's value to $108,000. That means you start the second year with more money and so get more benefit from the second year's 8% return. Result: Your wealth climbs to $116,640 by the end of the second year, giving you a cumulative two-year gain of 16.64%.
Taking Time
As the years roll by, this investment compounding can generate eye-popping performance. At a steady 8% annual return, you would earn a cumulative 47% after five years, 116% after 10 years, 585% after 25 years and 4,590% after 50 years. Impressed? It's amazing what you can amass with a little money and a lot of time.
Indeed, every so often, newspapers will carry stories about folks who die in their nineties and, to the shock of friends and neighbors, leave behind estates worth millions of dollars. The stories always have the same basic elements: These millionaires never earned a lot of money, they lived modestly, they drove used cars and they didn't have grand homes.
Often, the newspapers will speculate that these folks were brilliant investors. But the explanation is usually more prosaic.
First, these people weren't big spenders, so they had plenty of money to save. Second, they lived more than a decade longer than most other folks, so they enjoyed years of extra investment compounding. Adopt those two strategies and you could also be rich.
No, I can't tell you how to live 10 or 15 years beyond your life expectancy. But you can get those extra years of investment compounding at the other end, by starting to save as soon as you get out of school, rather than waiting until your 30s or 40s. If you do that, you will have a 10-year or 20-year head start on your neighbors. And there's a good chance you will retire with a seven-figure portfolio.
Averaging Up
All this, of course, assumes you earn decent investment gains and avoid big losses. That is easy enough to do. You simply spread your money across a broad array of stock and bond mutual funds and avoid rolling the dice on a few stocks or a single market sector.
But many folks can't resist such high-stakes investing. What explains this penchant for risk? I blame it partly on a mixture of financial illusion and bad math.
We hear about stocks and sectors that are up 200% or 300% and we imagine that we, too, could own the next hot investment. Yet, ironically, the existence of these big winners means we are actually less likely to pick market-beating stocks.
How come? In most years, the investment world's big winners score such huge gains that they skew the stock-market averages higher, so that a majority of stocks end up trailing behind the market averages.
Consider an extreme example. Assume you have a market of 10 stocks. Nine notch 10%, while one soars 100%. The average return would be 19%, with nine out of the 10 stocks trailing that average. The implication: If all you own are one or two stocks, you could make out like a bandit. But the odds are, you will end up with market-lagging results.
Losing Less
You can reduce the risk of badly trailing the market by favoring stock funds over individual stocks. But, unfortunately, when folks opt for stock funds, they almost always plunk for actively managed funds. To be sure, some actively managed funds succeed in beating the market. But most fail, typically by around 1.5 percentage points a year. That is why I am a big advocate of low-cost index funds, which will give you the market's performance minus annual expenses of maybe 0.2%.
Let's say the stock market clocks 8% a year over the next 60 years, taxes take 20% of your annual return and inflation steals another three percentage points. If you buy index funds, the 8% raw return will become 7.8% after expenses. Taxes will trim that to 6.24%. Knock off three points for inflation, and you will have a "real" return of 3.24%.
Meanwhile, if you opt for an actively managed fund and lag behind the market by 1.5 percentage points, the market's 8% will become 6.5%, taxes will reduce that to 5.2% and inflation will bring your return down to 2.2%.
Garnering 2.2% rather than 3.24% may not sound like a big deal. Suppose, however, that you start investing as soon as you get out of school and suppose you live another 60 years. Over those 60 years, a 2.2% annual gain will turn $10,000 into $36,902. But if you score 3.24% a year, you will amass $67,745 -- an impressive 84% more.
运用数学知识让你的钱包鼓起来
我花了数周的时间研究了一下金融问题，并试图用深入浅出的方式对它们作出解释，然而我发现，有时候仅仅使用文字还是不够的。
实际上，为了能得心应手地进行投资，你真的需要掌握一些基本的数学概念。听上去很吓人？别担心，这并不是要你绞尽脑汁地计算一些数字难题，相反，接下来我将带你轻松地了解一些关键的投资理念、以及它们背后的数学知识。
遭遇亏损
为了获得丰厚的投资回报，你需要承担风险。不过如果你冒太大的风险，你又可能会遭遇挫折，需要几年才能把财务状况恢复过来。
这里就涉及一个残酷的投资亏损数学问题。假设你投资不利，10万美元的投资组合缩水至只有75,000美元，为了挽回这25%的亏损，投资组合需要反弹的幅度并不是25%，因为这只能让你的投资组合从75,000美元增加到93,750美元。要完全回到原来的水平，你需要让投资组合增值33%。
亏损的幅度越大，收回投资的难度也就越大。如果你的投资组合缩水50%，要恢复到盈亏相抵的水平，需要反弹100%；如果缩水75%，则需要反弹300%。
当然，亏损75%的可能性并不大。不过，那斯达克综合指数曾经从2000年3月的高点一路跌至2002年10月的低点，跌幅达到了78%。因此对于同期追踪那斯达克综合指数的投资者而言，要想弥补所有的亏损，需要让自己的投资组合增值355%。
摆脱困境
假设每年的收益率是8%，那么要赚到这355%需要多长时间呢？人们可能会很自然地用355%除以8%，答案就是44年。
不过，事实上情况并没有这么糟糕。相反，如果年投资回报率是8%，追踪斯达克综合指数的投资者只需要不到20年的时间就能把亏损弥补回来。
为了便于理解，让我们假设你拥有100,000美元的投资组合，并且今年和明年的投资回报率都是8%。第一年8%的投资回报率会让你的投资组合增值到108,000美元，这意味著你第二年投资组合的基数变大了，因此8%的投资回报率就能带来更多的收益。结果是：到第二年年底，你的投资组合增值到了116,640美元，这样算下来，你两年累计的投资回报率就是16.64%。
提早行动
随著时间流逝，上述投资的表现会让你瞠目结舌。如果年投资回报率稳定在8%，5年后累计的投资回报率将达到47%，10年后达到116%，25年后为585%，50年后更是高达4,590%。很惊人吧？很少的一笔钱在经历若干年后居然能产生如此大的收益。
实际上，报纸常常会刊登这样的文章，讲的是某人在90多岁时去世了，让其朋友和邻居惊讶的是，他们留下了价值数百万美元的不动产。这些故事往往总是有相同的一些基本内容：这些百万富翁们从来没有赚过大钱，他们生活简朴，开著二手车，住房也不宽敞。
这些报纸常常会猜测这些人可能是出色的投资者，但对此做出的解释却更为实际。
首先，这些人都不是挥金如土的人，因此他们攒了很多钱；其次，他们比一般人多活了10年以上，因此能享受到额外的投资收益。具备了上述两个条件，你也可以变得很有钱。
我无法告诉你如何多活10年或15年，但你可以从一出校门就开始攒钱，而不是等到三、四十岁，这样从另一个角度看也是延长了投资的年限。如果你这样做了，你可能就比你的邻居提早行动了10年或20年，就很有可能在退休时口袋里装著一份7位数的投资组合。
分散投资
当然，所有这些都是假设你能获得丰厚的投资回报，并且没有出现巨额的亏损。要做到这一点很简单。你只要将资金分散投资在多种不同的股票和债券上，避免将赌注都押在少量股票或是一类股票上。
但许多人都抵御不了此类高风险的投资。为什么人们会对风险趋之若鹜呢？我认为这部分是市场假象和错误计算混合作用的结果。
我们都听说过一些股票和类股上涨200%或300%，我们认为自己也许能挑中下一只热门股。然而，具有讽刺意味的是，这些大赢家的存在意味著我们在实际中选中跑赢大市的股票的可能性减小了。
为什么呢？在多数年份，投资世界中的大赢家表现是如此地出色，以致于股市大盘也受其推动而有更好的表现，如此一来，绝大多数股票只能被甩在了大盘的后面。
看看以下这个极端的例子。假设你持有10只股票，其中9只上涨了10%，剩下一只则飙升了100%。这10只股票的平均回报率为19%，因此你的10只股票中，有9只股票的表现逊于平均水平。由此可见，如果你只持有一只或两只股票，你有可能会大赚一笔，但也有可能你的投资会落了个弱于大盘的结局。
提高收益
你可以通过投资股票基金而非单只股票降低上述风险。然而不幸的是，在选择股票基金时，人们总是倾向于积极管理型基金。有一点是肯定的，部分积极管理型基金的确成功地跑赢了大市，但多数此类基金还是无法做到这一点，其每年的收益率一般较大盘低1.5个百分点左右。这也正是我积极推荐低成本指数基金的原因，指数基金每年的投资回报率与大盘一致，只是要减去每年大约0.2%的费用。
假设股市在未来60年里的年投资回报率为8%，你获得的收益的应税税率为20%，另外通货膨胀率为3%，如果你购买了指数基金，在剔除0.2%的费用后，投资回报率为7.8%；在计算入纳税和通货膨胀因素后，实际的投资回报率为3.24%。
如果你购买的是积极管理型基金，它的回报率一般较大盘低1.5个百分点，如果大盘是8%，它就是6.5%，在扣除纳税和通货膨胀因素后，其投资回报率变成了2.2%。
好像2.2%与3.24%相比差距并不大，但设想一下，如果你从一出校门就开始投资，持续60年的时间。在这60年当中，如果年投资回报率是2.2%，那么你的10,000美元就能变成36,902美元；但如果年投资回报率为3.24%，你将能获得67,745美元，比前者多了84%。
