Negative properties of continuously compounded returns
What you're missing is the continuous compounding computation doesn't work that way.  If you compound over n periods of time and a rate of return of r, the formula is e^(r*n), as you have to multiply the returns together with a mulitplicative base of 1.  Otherwise consider what 0 does to your formula.  If I get a zero return, I have a zero result which doesn't make sense.  However, in my formula I'd still get the 1 which is what I'm starting and thus the no effect is the intended result. Continuous compounding would give e^(-.20*12) = e^(-2.4) = .0907 which is a -91% return so for each $100 invested, the person ends up with $9.07 left at the end.  It may help to picture that the function e^(-x) does asymptotically approach zero as x tends to infinity, but that is as bad as it can get, so one doesn't cross into the negative unless one wants to do returns in a Complex number system with imaginary numbers in here somehow. For those wanting the usual compounding, here would be that computation which is more brutal actually: For your case it would be (1-.20)^12=(0.8)^12=0.068719476736 which is to say that someone ends up with 6.87% in the end.  For each $100 had in the beginning they would end with $6.87 in the end. Consider someone starting with $100 and take 20% off time and time again you'd see this as it would go down to $80 after the first month and then down to $64 the second month as the amount gets lower the amount taken off gets lower too.  This can be continued for all 12 terms.  Note that the second case isn't another $20 loss but only $16 though it is the same percentage overall. Some retail stores may do discounts on discounts so this can happen in reality.  Take 50% off of something already marked down 50% and it isn't free, it is down 75% in total.  Just to give a real world example where while you think a half and a half is a whole, taking half and then half of a half is only three fourths, sorry to say.  You could do this with an apple or a pizza if you want a food example to consider. Alternatively, consider the classic up and down case where an investment goes up 10% and down 10%.  On the surface, these should cancel and negate each other, right?  No, in fact the total return is down 1% as the computation would be (1.1)(.9)=.99 which is slightly less than 1. Continuous compounding may be a bit exotic from a Mathematical concept but the idea of handling geometric means and how compounding returns comes together is something that is rather practical for people to consider. You cannot use continuous compounding for returns less than or equal to 100% because a natural logarithm can only be taken for a positive amount.  This answer includes the accurate way to ascertain r, for which many people use an approximation. For example, using -20% monthly return for 12 months:- -0.2 -0.223144 0.0687195 Checking: 0.0687195 True Now trying -100% monthly return:- -1. Indeterminate Why?  Because a natural logarithm can only be taken for a positive amount.  So the latter calculation can not be done using (logarithmic) continuous compounding. Of course, the calculation can still be done using regular compounding.  For -100% the results go to zero in the first month, but -150% produces a more interesting result: -1.5 -11920.9 Well, one can easily have rates below -100%.  Suppose I start with $100, and end up with $9 after a year. What was my rate of return? It could be -91%, -181%, -218%, or -241%, or something else, depending on the compounding method.  We always have that the final amount equals the initial amount times a growth factor G, and we can express this using a rate r and a day count fraction T. In this case, we have T = 1, and B(T) = B(0) * 0.09, so: So, depending on how we compound, we have a rate of return of -91%, -181%, -218%, or -241%. This nicely illustrates that:
The literal answer to your question is that a number of different types of mutual funds did not have significant downturns in 2008. Money Market Funds are intended to always preserve capital. VMMXX made 2.77% in 2008. It was a major scandal broke the buck, that its holders took a 3% loss. Inverse funds, which go up when the market goes down, obviously did well that year (RYARX), but if you have a low risk tolerance, that's obviously not what you're looking for. (and they have other problems as well when held long-term) But you're a 24-year-old talking about your retirement funds, you should have a much longer time horizon, at least 30 years. Over a period that long, stocks have never had negative real (inflation-adjusted) returns, dating back at least to the civil war. If you look at the charts here or here, you can see that despite the risk in any individual year, as the period grows longer, the average return for the period gets tighter and tighter. If you look at the second graph here, you see that 2011 was the first time since the civil war that the trailing 30-year return on t-bills exceeded that for stocks, and 1981-2011 was period that saw bond yields drop almost continuously, leading to steady rise in bond prices. Although past performance is no guarantee of future results, everything we've seen historically suggests that the risk of a broad stock-market portfolio held for 30 years is not that large, and it should make up the bulk of your holdings. For example, Vanguard's Target retirement 2055 fund is 90% in stocks (US + international), and only 10% in bonds. The number you are trying to calculate is called the Internal Rate of Return (IRR).  Google Spreadsheets (and excel) both have an XIRR function that can do this for you fairly simply. Setup a spreadsheet with 1 column for dates, 1 column for investment. Mark your investments as negative numbers (payment to invest).  All investments will be negative. Mark your last row with today's date and today's valuation (positive). All withdrawals will be positive, so you are pretending to withdrawal your entire account for the purpose of calculation. Do not record dividends or other interim returns unless you are actually withdrawing money. The XIRR function will calculate your internal rate of return with irregularly timed investments. Links: Article explaining XIRR function (sample spreadsheet in google docs to modify) The short answer is that you would want to use the net inflow or net outflow, aka profit or loss.   In my experience, you've got a couple different uses for IRR and that may be driving the confusion.  Pretty much the same formula, but just coming at it from different angles.   Thinking about a stock or mutual fund investment, you could project a scenario with an up-front investment (net outflow) in the first period and then positive returns (dividends, then final sale proceeds, each a net inflow) in subsequent periods.  This is a model that more closely follows some of the logic you laid out. Thinking about a business project or investment, you tend to see more complicated and less smooth cashflows.  For example, you may have a large up-front capital expenditure in the first period, then have net profit (revenue less ongoing maintenance expense), then another large capital outlay, and so on.   In both cases you would want to base your analysis on the net inflow or net outflow in each period.  It just depends on the complexity of the cashflows trend as to whether you see a straightforward example (initial payment, then ongoing net inflows), or a less straightforward example with both inflows and outflows.   One other thing to note - you would only want to include those costs that are applicable to the project.  So you would not want to include the cost of overhead that would exist even if you did not undertake the project. "An answer can be found in my book, ""A Modern Approach to Graham and Dodd Investing,"" p. 89 http://www.amazon.com/Modern-Approach-Graham-Investing-Finance/dp/0471584150/ref=sr_1_1?s=books&ie=UTF8&qid=1321628992&sr=1-1 ""If a company has no sustained cash flow over time, it has no value...If a company has positive cash flow but economic earnings are zero or less, it has a value less than book value and is a wasting asset. There is enough cash to pay interim dividends, bu the net present value of the dividend stream is less than book value."" A company with a stock trading below book value is believed to be ""impaired,"" perhaps because assets are overstated. Depending on the situation, it may or may not be a bankruptcy candidate." "The other two folks here are right with the math and such, so I'll just throw some intuition out there for you.  The basis for this valuation model is really just tacking the Gordon growth model (which is really just a form of valuing a perpetuity) onto a couple of finite discounted cash flows. So that ending part is the Gordon growth model *at the future point* discounted back to the present.  The Gordon growth model uses a ""next period"" dividend for the very simple reason that it's the next one you'd get if you bought the stock.  Is that explanation clear enough, or were some of these points not adequately explained in your class? I'll help a bit more, if I can." As user14469 mentions you would have to decide what type of properties you would like to invest in. Are you after negatively geared properties that may have higher long term growth potential (usually within 15 to 20km from major cities), or after positive cash-flow properties which may have a lower long term growth potential (usually located more than 20km from major cities).  With negative geared properties your rent from the property will not cover the mortgage and other costs, so you will have to supplement it through your income. The theory is that you can claim a tax deduction on your employment income from the negative gearing (benefits mainly those on higher tax brackets), and the potential long term growth of the property will make up for the negative gearing over the long term.  If you are after these type of properties Michael Yardney has some books on the subject. On the other hand, positive cash-flow properties provide enough rental income to cover the mortgage and other costs. They put cash into your pockets each week. They don't have as much growth potential as more inner city properties, but if you stick to the outer regions of major cities, instead of rural towns, you will still achieve decent long term growth. If you are after these type of properties Margaret Lomas has some books on the subject. My preference is for cash-flow positive properties, and some of the areas user14469 has mentioned. I am personally invested in the Penrith and surrounding areas.  With negatively geared properties you generally have to supplement the property with your own income and generally have to wait for the property price to increase so you build up equity in the property. This then allows you to refinance the additional equity so you can use it as deposits to buy other properties or to supplement your income. The problem is if you go through a period of low, stagnate or negative growth, you may have to wait quite a few years for your equity to increase substantially. With positively geared properties, you are getting a net income from the property every week so using none of your other income to supplement the property. You can thus afford to buy more properties sooner. And even if the properties go through a period of low, stagnate or negative growth you are still getting extra income each week. Over the long term these properties will also go up and you will have the benefit of both passive income and capital gains. I also agree with user14469 regarding doing at least 6 months of research in the area/s you are looking to buy. Visit open homes, attend auctions, talk to real estate agents and get to know the area. This kind of research will beat any information you get from websites, books and magazines. You will find that when a property comes onto the market you will know what it is worth and how much you can offer below asking price. Another thing to consider is when to buy. Most people are buying now in Australia because of the record low interest rates (below 5%). This is causing higher demand in the property markets and prices to rise steadily. Many people who buy during this period will be able to afford the property when interest rates are at 5%, but as the housing market and the economy heat up and interest rates start rising, they find it hard to afford the property when interest rate rise to 7%, 8% or higher. I personally prefer to buy when interest rates are on the rise and when they are near their highs. During this time no one wants to touch property with a six foot pole, but all the owners who bought when interest rates where much lower are finding it hard to keep making repayments so they put their properties on the market. There ends up being low demand and increased supply, causing prices to fall. It is very easy to find bargains and negotiate lower prices during this period. Because interest rates will be near or at their highs, the economy will be starting to slow down, so it will not be long before interest rates start dropping again. If you can afford to buy a property at 8% you will definitely be able to afford it at 6% or lower. Plus you would have bought at or near the lows of the price cycle, just before prices once again start increasing as interest rates drop. Read and learn as much as possible from others, but in the end make up your own mind on the type of properties and areas you prefer. Another disadvantage is the inability to value commodities in an accounting sense. In contrast with stocks, bonds and real estate, commodities don't generate cash flows and so any valuation methodology is by definition speculative. But as rhaskett notes, there are diversification advantages. The returns for gold, for instance, tend to exhibit low/negative correlation with the performance of stocks. The question is whether the diversification advantage, which is the primary reason to hold commodities in a multi-asset class portfolio through time, overcomes the disadvantages? The answer... maybe. "He's calculating portfolio variance. The general formula for the variance of a portfolio composed of two securities looks like this:  where w_a and w_b are the weights of each stock in the portfolio and the sigmas represent the standard deviation/risk of each asset or portfolio.  In the case of perfect positive or negative correlation, applying some algebra to the formula relating covariance to the correlation coefficient (rho, the Greek letter that looks like ""p""):  tells us that the covariance we need in the original formula is simply the product of the standard deviations and the correlation coefficient (-1 in this case).  Combining that result with our original formula yields this calculation:  Technically we've calculated the portfolio's variance and not it's standard deviation/risk, but since the square root of 0 is still 0, that doesn't matter. The Wikipedia article on Modern Portfolio Theory has a section that describes the mathematical methods I used above. The entire article is worth a read, however." You miss the step where the return being doubled is daily. Consider you invested $100 today, went up 10%, and tomorrow you went down 10%. Third day market went up 1.01% and without leverage - got even. Here's the calculation for you: day - start - end 1      $100    $120  -  +10% doubled 2      $120    $96   -  -10% doubled 3      $96     $97.94 - +1.01% doubled So in fact you're in $2.06 loss, while without leveraging you would break even. That means that if the trend is generally positive, but volatile - you'll end up barely breaking even while the non-leveraged investment would make profits. That's what the quote means. edit to summarize the long and fruitless discussion in the comments: The reason that the leveraged ETF's are very good for day-trading is exactly the same reason why they are bad for continuous investment. You should buy them when there's a reasonable expectation for the market to immediately go in the direction you expect. If for whatever reason you believe the markets will plunge, or soar, tomorrow - you should buy a leveraged ETF, ride the plunge, and sell it in the end of the day. But you asked the question about volatile markets, not markets going in one direction. There - you lose. Ok, so disregarding passivity, could you help me through a simplified example?   Say I only had two assets, SPY and TLT, with a respective weight of 35 and 65% and I want want to leverage this to 4x.  Additionally, say daily return covar is:  * B/B .004% * B/S -.004% * S/S .02%   Now, if I read correctly, I should buy ATM calls xxx days in the future.  Which may look like: Ticker, S, K, Option Price, Delta, Lambda  * TLT  $126.04  $126.00  $4.35  0.50  14.5 * SPY  $134.91  $134.00  $6.26  0.55  11.8  ^ This example is pretty close but some assets are far off.   I feel like I'm on the wrong track so I'll stop here.    I just want to lever up my risk-parity. Margin rates are too high and I'm docked by Reg-T. Keep in mind there are a couple of points to ponder here: Rates are really low. With rates being so low, unless there is deflation, it is pretty easy to see even moderate inflation of 1-2% being enough to eat the yield completely which would be why the returns are negative. Inflation is still relatively contained. With inflation low, there is no reason for the central banks to raise rates which would give new bonds a better rate. Thus, this changes in CPI are still in the range where central banks want to be stimulative with their policy which means rates are low which if lower than inflation rates would give a negative real return which would be seen as a way to trigger more spending since putting the money into treasury debt will lose money to inflation in terms of purchasing power. A good question to ponder is has this happened before in the history of the world and what could we learn from that point in time. The idea for investors would be to find alternative holdings for their cash and bonds if they want to beat inflation though there are some inflation-indexed bonds that aren't likely appearing in the chart that could also be something to add to the picture here. "Compounding is just the notion that the current period's growth (or loss) becomes the next period's principal. So, applied to stocks, your beginning value, plus growth (or loss) in value, plus any dividends, becomes the beginning value for the next period. Your value is compounded as you measure the performance of the investment over time. Dividends do not participate in the compounding unless you reinvest them.  Compound interest is just the principle of compounding applied to an amount owed, either by you, or to you. You have a balance with which a certain percentage is calculated each period and is added to the balance. The new balance is used to calculate the next period's interest, which again adds to the balance, etc. Obviously, it's better to be on the receiving end of a compound interest calculation than on the paying end. Interest bearing investments, like bonds, pay simple interest. Like stock dividends, you would have to invest the interest in something else in order to get a compounding effect. When using a basic calculator tool for stocks, you would include the expected average annual growth rate plus the expected annual dividend rate as your ""interest"" rate. For bonds you would use the coupon rate plus the expected rate of return on whatever you put the interest into as the ""interest"" rate. Factoring in risk, you would just have to pick a different rate for a simple calculator, or use a more complex tool that allows for more variables over time. Believe it or not, this is where you would start seeing all that calculus homework pay off!" You question is very hard to answer as it is tough to put a value on how much bad your added investment in evil companies would cause and also how much value the charities add.  However, there has been a bunch of really good work on socially responsible investing in general.  This paper might be too technical for some but the conclusion section is very readable and clear. The big worry about socially responsible investing from a financial standpoint is that it will lower returns in the long run.  The paper above and others show fairly clearly that as long as you only exclude a few classes of stocks and still have a fairly broad base that the expected returns are similar.  The main issue though is some socially responsible funds have much higher fees. So the usual advice applies, do your research to make sure your investments are well diversified and have low fees.   As long as the index is fairly broad you can consider the difference between the fees on the socially responsible index and investing in a more common index as the long run cost.  Then you can balance that cost and having more money for charity against the benefits of not investing in evil companies. Supply and demand for a particular bond may be such that the market price exceeds the par value for the bond at maturity. This is when you get a negative yield. Especially when volatility is high, people will actually pay money to park it in treasuries for an amount of time. But when compared to a > 25% vol in the equities market over that same period, taking a 5% or less hit doesn't sound nearly as bad! I've been a retail trader for close to 7 years and while I have a specialized futures account, I use Interactive Brokers for my other trading. They charge per share or contract rather than per trade (good for smaller accounts  or if you want to piece into and out of positions). You can also trade just about anything. Futures, options, options on futures, individual stocks, ETFs, Bonds (futures), currencies. The interface is pretty good as well. I have seperate charts (eSignal) so I'm not sure how good their charting is