Calculating a simply complicated return?
Since you have the balance at equal periods and the cash flows at the period ends, the best return calculation in this case is the true time-weighted return. See http://en.wikipedia.org/wiki/Time-weighted_return#Formulae  So, notwithstanding some ambiguity about your figures, here is a calculation using the first three periods from your second table. Giving a total return over the three periods of -23.88% If the periods are months, multiply by four to annualise.
"John Bensin's answer covers the math, but I like the plain-English examples of the theory from William Bernstein's fine book, The Intelligent Asset Allocator.  At the author's web site, you can find the complete chapter 1 and chapter 2, though not chapter 3, which is the one with the ""multiple coin toss"" portfolio example I want to highlight. I'll summarize Bernstein's multiple coin toss example here with some excerpts from the book.  (Another top user, @JoeTaxpayer, has also written about the coin flip on his blog, also mentioning Bernstein's book.) Bernstein begins Chapter 1 by describing an offer from a fictitious ""Uncle Fred"": Imagine that you work for your rich but eccentric Uncle Fred. [...] he   decides to let you in on the company pension plan. [...] you must pick   ahead of time one of two investment choices for the duration of your   employment: Certificates of deposit with a 3% annualized rate of return, or, A most peculiar option: At the end of each year Uncle Fred flips a coin. Heads you receive a 30% investment return for that year, tails a   minus 10% (loss) for the year. This will be hereafter referred to as   ""Uncle Fred’s coin toss,"" or simply, the ""coin toss."" In effect, choosing option 2 results in a higher expected return than option 1, but it is certainly riskier, having a high standard deviation and being especially prone to a series of bad tosses.  Chapters 1 and 2 continue to expand on the idea of risk, and take a look at various assets/markets over time.  Chapter 3 then begins by introducing the multiple coin toss example: Time passes. You have spent several more years in the employ of your   Uncle Fred, and have truly grown to dread the annual coin-toss   sessions. [...] He makes you another offer. At the end of each year,   he will divide your pension account into two equal parts and conduct a   separate coin toss for each half [...] there are four possible   outcomes [...]: [...]   Being handy with numbers, you calculate that your annualized return   for this two-coin-toss sequence is 9.08%, which is nearly a full   percentage point higher than your previous expected return of 8.17%   with only one coin toss. Even more amazingly, you realize that your   risk has been reduced — with the addition of two returns at the   mean of 10%, your calculated standard deviation is now only 14.14%, as   opposed to 20% for the single coin toss. [...] Dividing your portfolio between assets with uncorrelated results increases return while decreasing risk. [...]   If the second coin toss were perfectly inversely correlated   with the first and always gave the opposite result [hence, outcomes   1 and 4 above never occurring], then our return would always be 10%.   In this case, we would have a 10% annualized long-term return with zero risk! I hope that summarizes the example well.  Of course, in the real world, one of the tricks to building a good portfolio is finding assets that aren't well-correlated, and if you're interested in more on the subject I suggest you check out his books (including The Four Pillars of Investing) and read more about Modern Portfolio Theory (MPT)." I have asked myself this exact same question many times. The analysis would be simple if you invested all your money in a single day, but I did not and therefore I would need to convert your cash transactions into Index fund buys/sells. I got tired of trying to do this using Yahoo's data and excel so I built a website in my spare time. I humbly suggest you try my website out in the hopes that it helps you perform this computation: http://www.amibeatingthemarket.com/ I don't think it has to be either-or. You can profitably invest inside the SIMPLE. (Though I wouldn't put in any more than the 1% it takes to get the match.) Let's look at some scenarios. These assume salary of $50k/year so the numbers are easy. You can fill in your own numbers to see the outcome, but the percentages will be the same. Let it sit in cash in the SIMPLE. You put in 1%, your employer matches with 1%. Your account balance is $1,000 (at the end of the year), plus a small amount of interest. Cost to you is $500 from your gross pay. 100% return on your contributions, yay! Likely 0-1% real returns going forward; you'll be lucky to keep up with inflation over the long term. Short term not so bad. Buy shares of index ETFs in the SIMPLE; let's assume the fee works out to 10%. You put in 1%, employer matches 1%. Your contributions are $500, fees are $100, your balance is $900 in ETFs. 80% instant return, and possible 6-7% real long term returns going forward. Buy funds in the SIMPLE; assume the load is 5%, management fee is 1% and you can find something that behaves like an index fund (so it is theoretically comparable to above). 1% from you, 1% from employer. Your contributions are $500, load fees are $50, your balance is $950. 90% instant return, and possible 5-6% real long term returns going forward (assuming the 6-7% real returns of equities are reduced by the 1% management fee). (You didn't list out the fees, and they're probably different for the different fund choices, so fill in your own details and do the math.) Invest outside the SIMPLE in the same ETFs or equivalent no load index funds; let's assume you can do this with no fees. You put in the same 1% of your gross (ignoring any difference that might come from paying FICA) into a self directed traditional IRA. At the end of the year the balance is $500. So deciding whether or not to take the match is a no brainer: take it. Deciding whether you should hold cash, ETFs, or (one of two types of) funds in your SIMPLE is a little trickier. "First, to mention one thing - better analysis calls for analyzing a range of outcomes, not just one; assigning a probability on each, and comparing the expected values. Then moderating the choice based on risk tolerance. But now, just look at the outcome or scenario of 3% and time frame of 2 days. Let's assume your investable capital is exactly $1000 (multiply everything by 5 for $5,000, etc.). A. Buy stock: the value goes to 103; your investment goes to $1030; net return is $30, minus let's say $20 commission (you should compare these between brokers; I use one that charges 9.99 plus a trivial government fee). B. Buy an call option at 100 for $0.40 per share, with an expiration 30 days away (December 23). This is a more complicated. To evaluate this, you need to estimate the movement of the value of a 100 call, $0 in and out of the money, 30 days remaining, to the value of a 100 call, $3 in the money, 28 days remaining. That movement will vary based on the volatility of the underlying stock, an advanced topic; but there are techniques to estimate that, which become simple to use after you get the hang of it. At any rate, let's say that the expected movement of the option price in this scenario is from $0.40 to $3.20. Since you bought 2500 share options for $1000, the gain would be 2500 times 2.8 = 7000. C. Buy an call option at 102 for $0.125 per share, with an expiration 30 days away (December 23). To evaluate this, you need to estimate the movement of the value of a 102 call, $2 out of the money, 30 days remaining, to the value of a 102 call, $1 in the money, 28 days remaining. That movement will vary based on the volatility of the underlying stock, an advanced topic; but there are techniques to estimate that, which become simple to use after you get the hang of it. At any rate, let's say that the expected movement of the option price in this scenario is from $0.125 to $ 1.50. Since you bought 8000 share options for $1000, the gain would be 8000 times 1.375 = 11000. D. Same thing but starting with a 98 call. E. Same thing but starting with a 101 call expiring 60 days out. F., ... Etc. - other option choices. Again, getting the numbers right for the above is an advanced topic, one reason why brokerages warn you that options are risky (if you do your math wrong, you can lose. Even doing that math right, with a bad outcome, loses). Anyway you need to ""score"" as many options as needed to find the optimal point. But back to the first paragraph, you should then run the whole analysis on a 2% gain. Or 5%. Or 5% in 4 days instead of 2 days. Do as many as are fruitful. Assess likelihoods. Then pull the trigger and buy it. Try these techniques in simulation before diving in! Please! One last point, you don't HAVE to understand how to evaluate projected option price movements if you have software that  does that for you. I'll punt on that process, except to mention it. Get the general idea? Edit P.S. I forgot to mention that brokers need love for handling Options too. Check those commission rates in your analysis as well." "I got $3394.83 The first problem with this is that it is backwards.  The NPV (Net Present Value) of three future payments of $997 has to be less than the nominal value.  The nominal value is simple:  $2991.   First step, convert the 8% annual return from the stock market to a monthly return.  Everyone else assumed that the 8% is a monthly return, but that is clearly absurd.  The correct way to do this would be to solve for m in  But we often approximate this by dividing 8% by 12, which would be .67%.  Either way, you divide each payment by the number of months of compounding.   Sum those up using m equal to about .64% (I left the calculated value in memory and used that rather than the rounded value) and you get about $2952.92 which is smaller than $2991.   Obviously $2952.92 is much larger than $2495 and you should not do this.  If the three payments were $842.39 instead, then it would about break even.   Note that this neglects risk.  In a three month period, the stock market is as likely to fall short of an annualized 8% return as to beat it.  This would make more sense if your alternative was to pay off some of your mortgage immediately and take the payments or yp pay a lump sum now and increase future mortgage payments.  Then your return would be safer.   Someone noted in a comment that we would normally base the NPV on the interest rate of the payments.  That's for calculating the NPV to the one making the loan.  Here, we want to calculate the NPV for the borrower.  So the question is what the borrower would do with the money if making payments and not the lump sum.   The question assumes that the borrower would invest in the stock market, which is a risky option and not normally advisable.  I suggest a mortgage based alternative.  If the borrower is going to stuff the money under the mattress until needed, then the answer is simple.  The nominal value of $2991 is also the NPV, as mattresses don't pay interest.  Similarly, many banks don't pay interest on checking these days.  So for someone facing a real decision like this, I'd almost always recommend paying the lump sum and getting it over with.  Even if the payments are ""same as cash"" with no premium charged." "Wow, this analysis really surprised me. Very complete and useful, but i think my teacher request was easier. He just said: ""Try to build a diversified portfolio. Then try to add a commodity (like silver or gold) and understand how the risk vary introducing an asset like this."" So, i'm basically making a stocks portfolio and i'm calculating its expected return and risk. (for example 40%FB, 10%JNJ, 20%GS, 10%F and 20%MCD) then i'm adding GLD (so now i have something like 20%FB, 10%JNJ, 10%GS, 10%F and 20%MCD 30%GLD) and i'm actually making an excel spreadsheet where i calculate all the: -Expected returns -St Deviation -Covariance  At the end i compare the returns and the risks on the 2 different portfolios." Log-returns are very commonly used in financial maths, especially quantitative finance.  The important property is that they're symmetrical around 0 with respect to addition. This property makes it possible to talk about an average return. For instance, if a stock goes down 20% over a period of time, it has to gain 25% to be back where you started.  For the log-return on the other hand the numbers are 0.223 down over a period of time, and 0.223 up to get you back to square 1. In this sense, you can simply take an arithmetic average and it makes sense.  You can freely add up or subtract values on the log-return scale, like log-interest rates or log-inflation rates.  Whereas the arithmetic mean of (non-log) returns is simply meaningless: A stock with returns -3% and +3% would have 0% on average, when in fact the stock has declined in price? The correct approach on direct price-returns would be to take a different mean (e.g. geometric) to get a decent average.  And yet it will be hard to incorporate other information, like subtracting the risk-free rate or the inflation rate to get rate-adjusted average returns. In short: Log-returns are easier to handle computationally, esp. in bulk, but non-log-returns are easier to comprehend/imagine as a number of their own. Ah, pardon me, so it's not *either* 10x return or zero, but also includes points in between there?  Is it path dependent? Do you have any history on the asset?  The usual crutches for dealing with unknown probabilities are using risk neutrality or arbitrage pricing, but if the market is inefficient then that will be an estimation at best. "It can be pretty hard to compute the right number. What you need to know for your actual return is called the dollar-weighted return. This is the Internal Rate of Return (IRR) http://en.wikipedia.org/wiki/Internal_rate_of_return computed for your actual cash flows. So if you add $100 per month or whatever, that has to be factored in. If you have a separate account then hopefully your investment manager is computing this. If you just have mutual funds at a brokerage or fund company, computing it may be a bunch of manual labor, unless the brokerage does it for you. A site like Morningstar will show a couple of return numbers on say an S&P500 index fund. The first is ""time weighted"" and is just the raw return if you invested all money at time A and took it all out at time B. They also show ""investor return"" which is the average dollar-weighted return for everyone who invested in the fund; so if people sold the fund during a market crash, that would lower the investor return. This investor return shows actual returns for the average person, which makes it more relevant in one way (these were returns people actually received) but less relevant in another (the return is often lower because people are on average doing dumb stuff, such as selling at market bottoms). You could compare yourself to the time-weighted return to see how you did vs. if you'd bought and held with a big lump sum. And you can compare yourself to the investor return to see how you did vs. actual irrational people. .02, it isn't clear that either comparison matters so much; after all, the idea is to make adequate returns to meet your goals with minimum risk of not meeting your goals. You can't spend ""beating the market"" (or ""matching the market"" or anything else benchmarked to the market) in retirement, you can only spend cash. So beating a terrible market return won't make you feel better, and beating a great market return isn't necessary. I think it's bad that many investment books and advisors frame things in terms of a market benchmark. (Market benchmarks have their uses, such as exposing index-hugging active managers that aren't earning their fees, but to me it's easy to get mixed up and think the market benchmark is ""the point"" - I feel ""the point"" is to achieve your financial goals.)" R = I ^ P R = return (2 means double)  I = (Intrest rate / 100) + 1 [1.104 = 10.4%] P = number of periods (7 years)  2 = 1.104 ^ 7 (you double your money in seven years with a yearly Intrest rate of 10.4%) I = R^(1/P) 1.104 = 2^(1/7) P = log(R) / log(I)  7 = log(2) / log(1.104) Simple math. Take the sale proceeds (after trade expenses) and divide by cost. Subtract 1, and this is your return. For example, buy at 80, sell at 100, 100/80 = 1.25, your return is 25%. To annualize this return, multiply by 365 over the days you were in that stock. If the above stock were held for 3 months, you would have an annualized return of 100%.  There's an alternative way to annualize, in the same example above take the days invested and dive into 365, here you get 4. I suggested that 25% x 4 = 100%. Others will ask why I don't say 1.25^4 = 2.44 so the return is 144%/yr. (in other words, compound the return, 1.25x1.25x...) A single day trade, noon to noon the next day returning just 1%, would multiply to 365% over a year, ignoring the fact there are about 250 trading days. But 1.01^365 is 37.78 or a 3678% return. For long periods, the compounding makes sense of course, the 8%/yr I hope to see should double my money in 9 years, not 12, but taking the short term trades and compounding creates odd results of little value. "The author is using an approximation to what you have exactly, which is called a ""true"" time-weighted rate of return. You have expressed the total time-weighted return for the period in question.  In order to express this as an annual rate, you may annualize it by adding one, raising to the 1/y power, and subtracting one again, for a period of y years. The alternative to a time-weighted return is a money-weighted return, which is actually another name for the internal rate of return." At what point does my investment benefit from compounded interest? Monthly? Quarter? Yearly? Does it even benefit? I think you are mixing things. There is no concept of interest or compounding in Mutual Funds. When you buy a mutual fund, it either appreciates in value or depreciates in value; both can happen depending on the time period you compare. Now, let's assume at the end of the year I have a 5% return. My $10,000 is now $10,500. The way you need to look at this is Given you started with $10,000 and its now $10,500 the return is 5%. Now if you want to calculate simple return or compounded return, you would have to calculate accordingly. You may potentially want to find a compounded return for ease of comparison with say a Bank FD interest rate or some other reason.  So if $10,000 become $10,500 after one year and $11,000 after 2 year. The absolute return is 10%, the simple yearly return is 5%. Or the Simple rate of return for first year is 5% and for second year is 4.9%. Or the Average Year on Year return is 4.775%. You've got the right idea, except that the stated interest rate is normalized for a 1-year investment.  Hence if you buy a 4-week bill, you're getting something closer to 4/52 of what you've computed in your question.  More precisely, the Treasury uses a 360 day year for these calculations, so you multiply the stated rate by (number of days until maturity)/360 to get the actual rate of return. 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)