I want to approximate the total United States stock market with part of my k. I have access to two low-cost index mutual funds: What relative weights should I use for these two funds to be equivalent to something like VTSMX? How would one determine this aside from just going with Does it make sense to use that ratio?

That ratio is a decent approximation of the market cap distribution in VTSMX, although it's not perfect because the two funds you have access to, VIIIX and VIEIX, overlap somewhat in their holdings. We can verify that this is close to the best allocation using either of the two methods below. Remember that because of expense ratios among other factors , the returns to your replicated portfolio may not match the returns to the target portfolio exactly.

According to Morningstar, VTSMX has the market cap distribution shown in the first column. Morningstar places VTSMX in the Large Blend category, so the category averages in the right-hand column are for that category.

The benchmark index is the Russell index. VIIIX and VIEIX have these distributions:. We want to find allocations for VIIIX and VIEIX, i.

There are several methods that would work for this; both return the same solution:. Using the goal-seek feature of Excel, I found that allocating The second data column is VTSMX, the fund you're trying to replicate. The percent error, which is calculated as the absolute value of the difference between the allocations divided by the allocation in VTSMX, increases as market capitalization decreases, but that shouldn't bother you. Since allocation decreases as market cap decreases, even though the error grows as market cap decreases, that error represents an increasingly small amount of the fund's actual value.

This isn't a perfect system, because I'm abusing the notion of "weighted percent errors" a little here, but hopefully the idea is clear. In this specific case, matrices provide a much more robust, and arguably logical, strategy. From a mathematical perspective, this is the problem we're trying to solve:. If you try using the value of w1 that came from the fourth equation in any of the other equations besides the fifth one, in which it doesn't matter you'll find it won't work.

The same goes for the w1 that came from the third equation, or the second, or the first. It seems we have a problem. All is not lost, however. If you've ever taken an algebra course, you can see that the system of equations above looks like this matrix equation:.

I labeled the numerical matrix on the left-hand side of the equation A to save space.

united states - How can I replicate Vanguard's Total Stock Market Index using their S&P and extended stock market funds? - Personal Finance & Money Stack Exchange

The linear algebra theory probably isn't exciting to everyone as it is to me, so I'll keep it short. Since A is a 5x2 matrix with rank 2, it therefore has a left inverse , which we can calculate like this:. With our left-inverse in hand, we can quickly find the solution to the matrix equation like this:. It's also important to understand that this is the "best" solution to the problem; in other words, we have found the weights of VIIIX and VIEIX that replicates the market cap distribution of VTSMX as closely as possible.

If you've ever taken a statistics or econometrics course, you might recognize the calculation and application of the left-inverse as the ordinary least squares OLS estimation of the weights; that means that in this case, "closest" implies that the weights we found minimize the sum of the squares of the differences "errors" or "residuals" between our replicated portfolio and VTSMX.

VTSMX Vanguard Total Stock Mkt Idx Inv Fund VTSMX Quote Price News

If you substitute the weights w1 and w2 you found, either in Excel or using matrices, into the left-hand side of the original system of equations or the matrix equation in statistical parlance, you're calculating the linear predicted values. I was curious how the replicated portfolio would perform against VTSMX, so I ran a quick simulation in MATLAB to compare the performance of each portfolio.

First, I made a few assumptions:.

As you can see, the replicated portfolio closely matches VTSMX. The replicated portfolio outperforms the VTSMX portfolio partly because the replicated portfolio is comprised of institutional shares, which have much lower expense ratios than the investor shares the VTSMX uses the market cap distribution also makes a difference. Even if you invested in the VTSAX , the Admiral Shares equivalent of VTSMX, which has a much lower expense ratio of 0.

The distance has narrowed, but it's not enough for the Vanguard fund to beat your portfolio. Although these average returns may not accurately represent the holdings in VTSMX or your replicated portfolio and the average returns statistics for each market cap may represent slightly different holdings than those of the Vanguard funds, these nuances don't pose a problem in this example because I'm using the same benchmark to estimate the returns on VTSMX and the replicated portfolio.

By posting your answer, you agree to the privacy policy and terms of service. Sign up or log in to customize your list. Stack Exchange Inbox Reputation and Badges.

Questions Tags Users Badges Unanswered. Join them; it only takes a minute: Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top. Craig W 3, 3 23 Short answer That ratio is a decent approximation of the market cap distribution in VTSMX, although it's not perfect because the two funds you have access to, VIIIX and VIEIX, overlap somewhat in their holdings.

Long answer According to Morningstar, VTSMX has the market cap distribution shown in the first column.

VIIIX and VIEIX have these distributions: There are several methods that would work for this; both return the same solution: Goal-seeking in Excel Using the goal-seek feature of Excel, I found that allocating Matrices In this specific case, matrices provide a much more robust, and arguably logical, strategy. From a mathematical perspective, this is the problem we're trying to solve: If you've ever taken an algebra course, you can see that the system of equations above looks like this matrix equation: Since A is a 5x2 matrix with rank 2, it therefore has a left inverse , which we can calculate like this: With our left-inverse in hand, we can quickly find the solution to the matrix equation like this: First, I made a few assumptions: Another option would be to find a set of weights that hit the average year return for VTSMX.

I calculated the weighted expense ratio for the replicated portfolio in the same way I calculated the predicted values. Using the expense ratios of VIIIX 0. I ignored inflation, since it should apply equally to both VTSMX and the replicated portfolio I ignored transaction costs; I think this is a safe assumption to make when purchasing mutual funds.

The results of the simulation: Caveats Although these average returns may not accurately represent the holdings in VTSMX or your replicated portfolio and the average returns statistics for each market cap may represent slightly different holdings than those of the Vanguard funds, these nuances don't pose a problem in this example because I'm using the same benchmark to estimate the returns on VTSMX and the replicated portfolio.

Code Here is the MATLAB code for the simulation; I made the chart in Excel.

Approximating the US total stock market with these funds? : personalfinance

CraigW Happy to help; are you at ease with my change to the question's title? Title looks good to me. Jeremy Unfortunately, this bit of math is almost, but not quite specific to this question. CraigW I added a simulation of the returns to VTSMX and the replicated portfolio in your K, and the results are pretty close.

The lower expense ratios of the institutional shares among other factors allows your portfolio to outperform VTSMX. Sign up or log in StackExchange. Sign up using Facebook. Sign up using Email and Password.

Post as a guest Name. That's what I call a comprehensive answer! MathOverflow Mathematics Cross Validated stats Theoretical Computer Science Physics Chemistry Biology Computer Science Philosophy more 3. Meta Stack Exchange Stack Apps Area 51 Stack Overflow Talent.