What exactly is the profit and loss of a portfolio?
When we 'delta-hedge', we make the value of a portfolio 0.  No - you make the risk relative to some underlying 0.  The portfolio does have a value, but if whatever underlying you're hedging against changes slightly the value of your portfolio should not change. But, what is the derivative of a portfolio?  It's the instantaneous rate of change of the portfolio) relative to some underlying phenomenon.  With a portfolio of many stocks, there's not one single factor that drives the value of your portfolio. You have sensitivity to each underlying stock (price and volatility), interest rates, the market as a whole, etc. For simplicity, we might imagine a portfolio that has holdings in .... a call .... a stock .... and a bank account (to borrow and lend money).  You will have a delta relative to the stock and a delta relative to the underlying instrument on the option, etc.  Those can only be aggregated for each factor (e.g. if the call is an option on the same stock) Theta is the only one you can calculate for the portfolio as a whole - it will be the aggregate theta of all of your positions (since change in time is constant across all investments).  All of the others are not aggregatable since they are measuring sensitivities to different phenomena.
Variance of a single asset is defined as follows: σ2 = Σi(Xi - μ)2 where Xi's represent all the possible final market values of your asset and μ represents the mean of all such market values.  The portfolio's variance is defined as σp2 = Σiwi2σi2 where, σp is the portfolio's variance, and wi stands for the weight of the ith asset.  Now, if you include the borrowing in your portfolio, that would classify as technically shorting at the borrowing rate. Thus, this weight would (by the virtue of being negative) increase all other weights. Moreover, the variance of this is likely to be zero (assuming fixed borrowing rates). Thus, weights of risky assets rise and the investor's portfolio's variance will go up. Also see, CML at wikipedia. The difference is downside risk. Your CD, assuming you are in the US and the CD is purchased from a deposit bank, will be FDIC insured, your $10,000 is definitely coming back to you.  Your stock portfolio has no such guarantee and can lose money.  Your potential upside is theoretically correlated to the risk that some or all of your money may not be returned to you. A long put - you have a small initial cost (the option premium) but profit as the stock goes down. You have no additional risk if the shock rises, even a lot. Short a stock - you gain if the stock drops, but have unlimited risk if it rises, the call mitigates this, by capping that rising stock risk. The profit/loss graph looks similar to the long put when you hold both the short position and the long call.  You might consider producing a graph or spreadsheet to compare positions. You can easily sketch put, call, long stock, short stock, and study how combinations of positions can synthetically look like other positions. Often, when a stock has no shares to short, the synthetic short can help you put your stock position in place. "Financing a portfolio with debt (on margin) leads to higher variance. That's the WHOLE POINT. Let's say it's 50-50.  On the downside, with 100% equity, you can never lose more than your whole equity. But if you have assets of 100, of which 50% is equity and 50% is debt, your losses can be greater than 50%, which is to say more than the value of your equity. The reverse is true. You can make money at TWICE the rate if the market goes up. But ""you pay your money and you take your chances"" (Punch, 1846)." The biggest benefit to having a larger portfolio is relatively reduced transaction costs. If you buy a $830 share of Google at a broker with a $10 commission, the commission is 1.2% of your buy price.  If you then sell it for $860, that's another 1.1% gone to commission.  Another way to look at it is, of your $30 ($860 - $830) gain you've given up $20 to transaction costs, or 66.67% of the proceeds of your trade went to transaction costs. Now assume you traded 10 shares of Google. Your buy was $8,300 and you sold for $8,600.  Your gain is $300 and you spent the same $20 to transact the buy and sell.  Now you've only given up 6% of your proceeds ($20 divided by your $300 gain). You could also scale this up to 100 shares or even 1,000 shares. Generally, dividend reinvestment are done with no transaction cost.  So you periodically get to bolster your position without losing more to transaction costs.  For retail investors transaction costs can be meaningful.  When you're wielding a $5,000,000 pot of money you can make your trades on a larger scale giving up relatively less to transaction costs. "You're talking about modern portfolio theory. The wiki article goes into the math. Here's the gist: Modern portfolio theory (MPT) is a theory of finance that attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets. At the most basic level, you either a) pick a level of risk (standard deviation of your whole portfolio) that you're ok with and find the maximum return you can achieve while not exceeding your risk level, or b) pick a level of expected return that you want and minimize risk (again, the standard deviation of your portfolio). You don't maximize both moments at once. The techniques behind actually solving them in all but the most trivial cases (portfolios of two or three assets are trivial cases) are basically quadratic programming because to be realistic, you might have a portfolio that a) doesn't allow short sales for all instruments, and/or b) has some securities that can't be held in fractional amounts (like ETF's or bonds). Then there isn't a closed form solution and you need computational techniques like mixed integer quadratic programming Plenty of firms and people use these techniques, even in their most basic form. Also your terms are a bit strange: It has correlation table p11, p12, ... pij, pnn for i and j running from 1 to n This is usually called the covariance matrix. I want to maximize 2 variables. Namely the expected return and the additive inverse of the standard deviation of the mixed investments. Like I said above you don't maximize two moments (return and inverse of risk). I realize that you're trying to minimize risk by maximizing ""negative risk"" so to speak but since risk and return are inherently a tradeoff you can't achieve the best of both worlds. Maybe I should point out that although the above sounds nice, and, theoretically, it's sound, as one of the comments points out, it's harder to apply in practice. For example it's easy to calculate a covariance matrix between the returns of two or more assets, but in the simplest case of modern portfolio theory, the assumption is that those covariances don't change over your time horizon. Also coming up with a realistic measure of your level of risk can be tricky. For example you may be ok with a standard deviation of 20% in the positive direction but only be ok with a standard deviation of 5% in the negative direction. Basically in your head, the distribution of returns you want probably has negative skewness:  because on the whole you want more positive returns than negative returns. Like I said this can get complicated because then you start minimizing other forms of risk like value at risk, for example, and then modern portfolio theory doesn't necessarily give you closed form solutions anymore. Any actively managed fund that applies this in practice (since obviously a completely passive fund will just replicate the index and not try to minimize risk or anything like that) will probably be using something like the above, or at least something that's more complicated than the basic undergrad portfolio optimization that I talked about above. We'll quickly get beyond what I know at this rate, so maybe I should stop there." "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." Very likely this refers to trading/speculating on leverage, not investing. Of course, as soon as you put leverage into the equation this perfectly makes sense.  2007-2009 for example, if one bought the $SPX at its highs in 2007 at ~$1560.00 - to the lows from 2009 at ~$683.00 - implicating that with only 2:1 leverage a $1560.00 account would have received a margin call. At least here in Europe I can trade index CFD's and other leveraged products. If i trade lets say >50:1 leverage it doesn’t take much to get a margin call and/or position closed by the broker. No doubt, depending on which investments you choose there’s always risk, but currency is a position too. TO answer the question, I find it very unlikely that >90% of investors (referring to stocks) lose money / purchasing power. Anyway, I would not deny that where speculators (not investors) use leverage or try to trade swings, news etc. have a very high risk of losing money (purchasing power). You can only lose your 7%. The idea that a certain security is more volatile than others in your portfolio does not mean that you can lose more than the value of the investment. The one exception is that a short position has unlimited downside, but i dont think there are any straight short mutual funds. "Month to date For the month to date (MTD), the price on Feb 28th is $4.58 and the price on March 16th is $4.61 so the return is which can be written more simply as The position is 1000 shares valued at $4580 on Feb 28th, so the profit on the month to date is Calendar year to date For the calendar year to date (YTD), the price on Dec 31st is $4.60 and the price on Feb 28th is $4.58 so the return to Feb 28th is The return from Feb 28th to March 16th is 0.655022 % so the year to date return is or more directly So the 2011 YTD profit on 1000 shares valued at $4600 on Dec 31st is Year to date starting Dec 10th For the year to date starting Dec 10th, the starting value is and the value on Dec 31st is 1000 * $4.60 = $4600 so the return is $4600 / $4510 - 1 = 0.0199557 = 1.99557 % The year to date profit is therefore Note - YTD is often understood to mean calendar year to date.  To cover all the bases state both, ie ""calendar YTD (2011)"" and ""YTD starting Dec 10th 2010"". Edit further to comment For the calendar year to date, with 200 shares sold on Jan 10th with the share price at $4.58, the return from Dec 31st to Jan 10th is The return from Jan 10th to Feb 28th is The return from Feb 28th to March 16th is The profit on 1000 shares from Dec 31st to Jan 10th is $4600 * -0.00434783 = -$20 The profit on 800 shares from Jan 10th to Feb 28th is zero. The profit on 800 shares from Feb 28th to March 16th is So the year to date profit is $4." The management expense ratio (MER) is the management fee, plus all of the other costs required to run the fund, excluding any trading costs. Here's a pretty good explanation. There is no difference.  When dealing with short positions, talking about percentages become very tricky since they no longer add up to 100%.  What does the 50% in your example mean?  Unless there's some base amount (like total amount of the portfolio, then the percentages are meaningless.  What matters when dealing with long and short positions is the net total - meaning if you are long 100 shares on one stock trade and short 50 shares on another, then you are net long 50 shares. You have got it wrong. The profit or loss for smaller investor or big investor is same in percentage terms. There are two ways that an asset can generate value.  One is that the asset generates some revenue (e.g. you buy a house for $100,000 and rent it out for $1,000 per month) and the second way is that the asset appreciates (e.g you buy a house for $100,000, you don't rent it out and 5 years later you sell it for $200,000).  Stocks are the same. Capital losses do mirror capital gains within their holding periods. An asset or investment this is certainly held for a year into the day or less, and sold at a loss, will create a short-term capital loss. A sale of any asset held for over a year to your day, and sold at a loss, will create a loss that is long-term. When capital gains and losses are reported from the tax return, the taxpayer must first categorize all gains and losses between long and short term, and then aggregate the sum total amounts for every single regarding the four categories. Then the gains that are long-term losses are netted against each other, therefore the same is done for short-term gains and losses. Then your net gain that is long-term loss is netted against the net short-term gain or loss. This final net number is then reported on Form 1040. Example Frank has the following gains and losses from his stock trading for the year: Short-term gains - $6,000 Long-term gains - $4,000 Short-term losses - $2,000 Long-term losses - $5,000 Net short-term gain/loss - $4,000 ST gain ($6,000 ST gain - $2,000 ST loss) Net long-term gain/loss - $1,000 LT loss ($4,000 LT gain - $5,000 LT loss) Final net gain/loss - $3,000 short-term gain ($4,000 ST gain - $1,000 LT loss)   Again, Frank can only deduct $3,000 of final net short- or long-term losses against other types of income for that year and must carry forward any remaining balance.