How to find an optimum linear combination of various investments?
"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."
"There are probably 3-4 questions here.  Diversification - A good index, a low cost S&P fund or ETF can serve you very well. If you add an extended market index or just go with ""Total market"", that might be it for your stock allocation. I've seen people with 5 funds, and it didn't take much analysis to see the overlap was so significant, that the extra 4 funds added little, and 2 of the 5 would have been it. If you diversify by buying more ETFs or funds, be sure to see what they contain.  If you can go back in time, buy Apple, Google, Amazon, etc, and don't sell them. Individual stocks are fun to pick, but unless you put in your homework, are tough to succeed at. You need to be right at the buy side, and again to know if, and when, to sell. I bought Apple, for example, long ago, pre-last few splits. But, using responsible a approach, I sold a bit each time it doubled. Has I kept it all through the splits, I'd have $1M+ instead of the current $200K or so of stock. Can you tell which companies now have that kind of potential for the future?  The S&P has been just about double digit over 60 years. The average managed fund will lag the S&P over time, many will be combined with other funds or just close. Even with huge survivor bias, managed funds can't beat the index over time, on average. Aside from a small portion of stocks I've picked, I'm happy to get S&P less .02% in my 401(k). In aggregate, people actually do far worse due to horrific timing and some odd thing, called emotions." Dollar Cost Averaging would be the likely balanced approach that I'd take. Depending on the size of the sum, I'd likely consider a minimum of 3 and at most 12 points to invest the funds to get them all working. While the sum may be large relative to my net worth, depending on overall scale and risk tolerance I could see doing it in a few rounds of purchasing or I could see taking an entire year to deploy the funds in case of something happening. I'd likely do monthly investments myself though others may go for getting more precise on things. I  found that  an application already exists which does virtually everything I want to do with a reasonable interface.  Its called My Personal Index.  It has allowed me  to look at my asset allocation all in one place. I'll have to enter: The features which solve my problems above include:  Note - This is related to an earlier post I made regarding dollar cost averaging and determining rate of returns.   (I finally got off my duff and did something about it) "When you have multiple assets available and a risk-free asset (cash or borrowing) you will always end up blending them if you have a reasonable objective function.  However, you seem to have constrained yourself to 100% investment.  Combine that with the fact that you are considering only two assets and you can easily have a solution where only one asset is desired in the portfolio.  The fact that you describe the US fund as ""dominating"" the forign fund indicates that this may be the case for you.  Ordinarily diversification benefits the overall portfolio even if one asset ""dominates"" another but it may not in your special case. Notice that these funds are both already highly diversified, so all you are getting is cross-border diversification by getting more than one.  That may be why you are getting the solution you are.  I've seen a lot of suggested allocations that have weights similar to what you are using. Finding an optimal portfolio given a vector of expected returns and a covariance matrix is very easy, with some reliable results.  Fancy models get pretty much the same kinds of answers as simple ones.  However, getting a good covariance matrix is hard and getting a good expected return vector is all but impossible.  Unfortunately portfolio results are very sensitive to these inputs.  For that reason, most of us use portfolio theory to guide our intuition, but seldom do the math for our own portfolio.  In any model you use, your weak link is the expected return and covariance.  More sophisticated models don't usually help produce a more reasonable result. For that reason, your original strategy (80-20) sounds pretty good to me.  Not sure why you are not diversifying outside of equities, but I suppose you have your reasons." "It includes whatever you want to do with your investment.  At least initially, it's not so much a matter of calculating numbers as of introspective soul-searching.  Identifying your investment objectives means asking yourself, ""Why do I want to invest?""  Then you gradually ask yourself more and more specific questions to narrow down your goals.  (For instance, if your answer is something very general like ""To make money"", then you may start to ask yourself, ""How much money do I want?"", ""What will I want to use that money for?"", ""When will I want to use that money?"", etc.)  Of course, not all objectives are realistic, so identifying objectives can also involve whittling down plans that are too grandiose. One thing that can be helpful is to first identify your financial objectives: that is, money you want to be able to have, and things you want to do with that money.  Investment (in the sense of purchasing investment vehicles likes stocks or bonds) is only one way of achieving financial goals; other ways include working for a paycheck, starting your own business, etc.  Once you identify your financial goals, you have a number of options for how to get that money, and you should consider how well suited each strategy is for each goal. For instance, for a financial goal like paying relatively small short-term expenses (e.g., your electric bill), investing would probably not be the first choice for how to do that, because: a) there may be easier ways to achieve that goal (e.g., ask for a raise, eat out less); and b) the kinds of investment that could achieve that goal may not be the best use of your money (e.g., because they have lower returns)." Failing some answers to my comment, I am going to make some assumptions: Based upon a quick review of this article I'd probably be in the Russell 2000 Value Index Fund (IWN).  Quite simply it gives you broad market exposure so you can be diversified by purchasing one fund. One of the key success factors is starting, not if you pick the best fund at the onset.  I can recall, 20 years ago being amazed (and it was quite a feat) at someone who was able to invest $400 per month.  These days that won't get you to the ROTH maximum and smart 20 somethings are doing just that. "There are two umbrellas in investing: active management and passive management. Passive management is based on the idea ""you can't beat the market."" Passive investors believe in the efficient markets hypothesis: ""the market interprets all information about an asset, so price is equal to underlying value"". Another idea in this field is that there's a minimum risk associated with any given return. You can't increase your expected return without assuming more risk. To see it graphically:   As expected return goes up, so does risk. If we stat with a portfolio of 100 bonds, then remove 30 bonds and add 30 stocks, we'll have a portfolio that's 70% bonds/30% stocks. Turns out that this makes expected return increase and lower risk because of diversification. Markowitz showed that you could reduce the overall portfolio risk by adding a riskier, but uncorrelated, asset! Basically, if your entire portfolio is US stocks, then you'll lose money whenever US stocks fall. But, if you have half US stocks, quarter US bonds, and quarter European stocks, then even if the US market tanks, half your portfolio will be unaffected (theoretically). Adding different types of uncorrelated assets can reduce risk and increase returns.  Let's tie this all together. We should get a variety of stocks to reduce our risk, and we can't beat the market by security selection. Ideally, we ought to buy nearly every stock in the market so that  So what's our solution? Why, the exchange traded fund (ETF) of course! An ETF is basically a bunch of stocks that trade as a single ticker symbol. For example, consider the SPDR S&P 500 (SPY). You can purchase a unit of ""SPY"" and it will move up/down proportional to the S&P 500. This gives us diversification among stocks, to prevent any significant downside while limiting our upside. How do we diversify across asset classes? Luckily, we can purchase ETF's for almost anything: Gold ETF's (commodities), US bond ETF's (domestic bonds), International stock ETFs, Intl. bonds ETFs, etc. So, we can buy ETF's to give us exposure to various asset classes, thus diversifying among asset classes and within each asset class. Determining what % of our portfolio to put in any given asset class is known as asset allocation and some people say up to 90% of portfolio returns can be determined by asset allocation.  That pretty much sums up passive management. The idea is to buy ETFs across asset classes and just leave them. You can readjust your portfolio holdings periodically, but otherwise there is no rapid trading. Now the other umbrella is active management. The unifying idea is that you can generate superior returns by stock selection. Active investors reject the idea of efficient markets.  A classic and time proven strategy is value investing. After the collapse of 07/08, bank stocks greatly fell, but all the other stocks fell with them. Some stocks worth $100 were selling for $50. Value investors quickly snapped up these stocks because they had a margin of safety. Even if the stock didn't go back to 100, it could go up to $80 or $90 eventually, and investors profit. The main ideas in value investing are: have a big margin of safety, look at a company's fundamentals (earnings, book value, etc), and see if it promises adequate return. Coke has tremendous earnings and it's a great company, but it's so large that you're never going to make 20% profits on it annually, because it just can't grow that fast. Another field of active investing is technical analysis. As opposed to the ""fundamental analysis"" of value investing, technical analysis involves looking at charts for patterns, and looking at stock history to determine future paths. Things like resistance points and trend lines also play a role. Technical analysts believe that stocks are just ticker symbols and that you can use guidelines to predict where they're headed.  Another type of active investing is day trading. This basically involves buying and selling stocks every hour or every minute or just at a rapid pace. Day traders don't hold onto investments for very long, and are always trying to predict the market in the short term and take advantage of it. Many individual investors are also day traders.  The other question is, how do you choose a strategy? The short answer is: pick whatever works for you. The long answer is: Day trading and technical analysis is a lot of luck. If there are consistent systems for trading , then people are keeping them secret, because there is no book that you can read and become a consistent trader. High frequency trading (HFT) is an area where people basically mint money, but it s more technology and less actual investing, and would not be categorized as day trading.  Benjamin Graham once said: In the short run, the market is a voting machine but in the long run it is a weighing machine. Value investing will work because there's evidence for it throughout history, but you need a certain temperament for it and most people don't have that. Furthermore, it takes a lot of time to adequately study stocks, and people with day jobs can't devote that kind of time. So there you have it. This is my opinion and by no means definitive, but I hope you have a starting point to continue your study. I included the theory in the beginning because there are too many monkeys on CNBC and the news who just don't understand fundamental economics and finance, and there's no sense in applying a theory until you can understand why it works and when it doesn't." The article links to William Bernstein’s plan that he outlined for Business Insider, which says: Modelling this investment strategy Picking three funds from Google and running some numbers. The international stock index only goes back to April 29th 1996, so a run of 21 years was modelled.  Based on 15% of a salary of $550 per month with various annual raises: Broadly speaking, this investment doubles the value of the contributions over two decades. Note:  Rebalancing fees are not included in the simulation. Below is the code used to run the simulation.  If you have Mathematica you can try with different funds.    Notice above how the bond index (VBMFX) preserves value during the 2008 crash.  This illustrates the rationale for diversifying across different fund types. You are suggesting something called dollar cost averaging (or its cousin, dollar value averaging) - http://www.investopedia.com/articles/stocks/07/dcavsva.asp  This is certainly a valid investment strategy, although personally, I feel that for long term investment, it is not necessary unless you plan on being an active trader. I still strongly encourage you to research these two methods and see if they would work well for your personal investment strategy and goals. As far as what sorts of investments for a taxable account, I have three general recommendations: As far as which company to use for your brokerage, I personally have accounts at Voya, TRowe Price and Fidelity. I would strongly recommend Fidelity out of those three, mostly due to customer service and quality and ease of use of their website. Vanguard is a great brokerage, but you don't have to choose them just because you plan to mostly invest in Vanguard funds. I also recommend you research how capital gains and dividend taxing works (and things like lost harvesting), so that you can structure your investments with taxes in mind. Do this ahead of time, don't wait until April of 2016 because it will be too late to save on taxes by then. Quick and dirty paper but pretty interesting.. I'm not in Portfolio Management but I probably would have ended up at the modal number as well.   I don't know the subject deeply enough to answer my own question, but is the bias always toward underestimation of variance? Or is that a complex of the way the problem was set up?   Another question I have for those in investment management; Would this impact asset allocation? "Everything you are doing is fine. Here are a few practical notes in performing this analysis: Find all the primary filing information on EDGAR. For NYSE:MEI, you can use https://www.sec.gov/cgi-bin/browse-edgar?action=getcompany&CIK=0000065270&type=10-K&dateb=&owner=exclude&count=40 This is the original 10-K. To evaluate earnings growth you need per share earnings for the past three years and 10,11,12 years ago. You do NOT need diluted earnings (because in the long term share dilution comes out anyway, just like ""normalized"" earnings). The formula is avg(Y_-1+Y_-2+Y_-3) / is avg(Y_-10+Y_-11+Y_-12) Be careful with the pricing rules you are using, the asset one gets complicated. I recommend NOT using the pricing rules #6 and #7 to select the stock. Instead you can use them to set a maximum price for the stock and then you can compare the current price to your maximum price. I am also working to understand these rules and have cited Graham's rules into a checklist and worksheet to find all companies that meet his criteria. Basically my goal is to bottom feed the deals that Warren Buffett is not interested in. If you are interested to invest time into this project, please see https://docs.google.com/document/d/1vuFmoJDktMYtS64od2HUTV9I351AxvhyjAaC0N3TXrA" Okay, I think I managed to find the precise answer to this problem! It involves solving a non-linear exponential equation, but I also found a good approximate solution using the truncated Taylor series. See below for a spreadsheet you can use. Let's start by defining the growth factors per period, for money in the bank and money invested:   Now, let S be the amount ready to be invested after n+1 periods; so the first of that money has earned interest for n periods. That is,  The key step to solve the problem was to fix the total number of periods considered. So let's introduce a new variable: t = the total number of time periods elapsed So if money is ready to invest every n+1 periods, there will be t/(n+1) separate investments, and the future value of the investments will be:  This formula is exact in the case of integer t and n, and a good approximation when t and n are not integers. Substituting S, we get the version of the formula which explicitly depends on n:  Fortunately, only a couple of terms in FV depend on n, so we can find the derivative after some effort:  Equating the derivative to zero, we can remove the denominator, and assuming t is greater than zero, we can divide by the constant ( 1-G t ):  To simplify the equation, we can define some extra constants:    Then, we can define a function f(n) and write the equation as:  Note that α, β, γ, G, and R are all constant. From here there are two options: Use Newton's method or another numerical method for finding the positive root of f(n). This can be done in a number of software packages like MATLAB, Octave, etc, or by using a graphics calculator. Solve approximately using a truncated Taylor series polynomial. I will use this method here. The Taylor series of f(n), centred around n=0, is:  Truncating the series to the first three terms, we get a quadratic polynomial (with constant coefficients):  Using R, G, α, β and γ defined above, let c0, c1 and c2 be the coefficients of the truncated Taylor series for f(n):    Then,  n should be rounded to the nearest whole number. To be certain, check the values above and below n using the formula for FV. Using the example from the question: For example, I might put aside $100 every week to invest into a stock   with an expected growth of 9% p.a., but brokerage fees are $10/trade.   For how many weeks should I accumulate the $100 before investing, if I   can put it in my high-interest bank account at 4% p.a. until then? Using Newton's method to find roots of f(n) above, we get n = 14.004. Using the closed-form approximate solution, we get n = 14.082. Checking this against the FV with t = 1680 (evenly divisible by each n + 1 tested): Therefore, you should wait for n = 14 periods, keeping that money in the bank, investing it together with the money in the next period (so you will make an investment every 14 + 1 = 15 weeks.) Here's one way to implement the above solution with a spreadsheet. StackExchange doesn't allow tables in their syntax at this time, so I'll show a screenshot of the formulae and columns you can copy and paste: Formulae:  Copy and paste column A: Copy and paste column B: Results:  Remember, n is the number of periods to accumulate money in the bank. So you will want to invest every n+1 weeks; in this case, every 15 weeks. I learned most of this stuff from 3 textbooks in school probably totaling $900 between the 3. I imagine you don't want to spend the cash on that. I would suggest finding a source online. A lot of the surface level stuff you are looking for can be found online on websites like Investopedia. They are a great resource and are free usually. Determine an investment strategy and that will likely answer this for you. Different people have different approaches and you need to determine for yourself what buy and sell criteria you want to have. Again, depending on the strategy there can be a wide range here as some may trade index funds though this can backfire in some cases. In others, there can be a lot of buy and hold if one finds an index fund to hold forever which depending on the strategy is possible. Returns can vary widely as an index can be everything from buying gold stocks in Russia to investing in short-term Treasuries as there are many different indices as any given market can have an index which could be stocks, bonds, a combination of the two or something else in some cases so please consider asset allocation, types of accounts, risk tolerance and time horizon in making decisions or consider using a financial planner to assist in drawing up a plan with allocations and how frequently you want to rebalance as my suggestion here. "The standard low-risk/gain very-short-term parking spot these days tends to be a money market account.  However, you have only mentioned stock. For good balance, your portfolio should consider the bond market too. Consider adding a bond index fund to diversify the basic mix, taking up much of that 40%. This will also help stabilize your risk since bonds tend to move opposite stocks (prperhaps just because everyone else is also using them as the main alternative, though there are theoretical arguments why this should be so.) Eventually you may want to add a small amount of REIT fund to be mix, but that's back on the higher risk side. (By the way: Trying to guess when the next correction will occur is usually not a winning strategy; guesses tend to go wrong as often as they go right, even for pros. Rather than attempting to ""time the market"", pick a strategic mix of investments and rebalance periodically to maintain those ratios. There has been debate here about ""dollar-cost averaging"" -- see other answers -- but that idea may argue for investing and rebalancing in more small chunks rather than a few large ones. I generally actively rebalance once a year or so, and between those times let maintainng the balance suggest which fund(s) new money should go into -- minimal effort and it has worked quite well enough.,)"