About Me

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I am STANCO 25 Professor of Finance, Emeritus at Stanford University and a recipient of the 1990 Nobel Prize in Economic Sciences. More than you will ever want to know about me can be found at my Stanford web site: www.stanford.edu/~wfsharpe

To see titles of prior posts, click the triangle by the month. To then see a post, click its title.

Sunday, April 19, 2020

World Bond and Stock Returns in 2020Q1

In my ebook on Retirement Income Analysis know, I suggest that retirees consider investment in two main types of vehicles: Treasury Inflation Protected Securities (TIPS) and a World Bond/Stock fund (WBS). This post shows the performance of a proxy for the latter in the turbulent quarter when the COVID-19 pandemic first took its toll on almost every aspect of peoples' lives, including asset values.

As in previous blogs in this series, I will use two Vanguard ETFs (exchange traded funds) to represent investment in world stocks and bonds: VT (their world stock ETF) and BNDW (their world bond ETF). Their cumulative returns with dividends reinvested over the quarter are shown in the diagram below (using data from Yahoo Finance). Also shown is WBS, a portfolio that includes shares of the two ETFs in proportions intended to represent the relative values of bonds and stocks in the world (details of the construction of the latter are given later in this post). Note: this portfolio is not to be confused with Webster Financial Services, whose stock has the ticker symbol WBS.

As is well known, there was a disastrous fall in the values of stocks around the world in this period, with a subsequent rise that gave back roughly half of the losses. On the other hand, bond prices fell only modestly, then ended the quarter slightly higher than they started. Not surprisingly, the world bond and stock fund experienced smaller changes in value than stocks but larger ones than bonds. Here are some key statistics:

Clearly, this period was characterized by a disastrous fall in the values of stocks around the world. At the low point, the world stock ETF (with dividends reinvested) had lost almost one-third of its value. Fortunately equity prices recovered some of this loss, ending with a value slightly over 16% lower than it had been at the beginning of the quarter. In contrast, the world bond ETF lost less than 4% during the quarter, and ended with a gain of almost 3%. Finally, our portfolio of world bonds and stocks suffered a smaller maximum loss (roughly 17%) and ended down slightly less than 7%.

Now to the construction of the world bond and stock fund (WBS) used for the calculations above. The formula was very simple: the fund consisted of one share of the world stock fund (VT) and 1.01 shares of the world bond fund (BNDW), with dividends paid by each fund during the quarter re-invested in that fund when received. All ETF values were taken from the Yahoo Finance site.

But why did I use a portfolio with 1.0 shares of VT and 1.01 shares of BNDW? Because that would have provided relative values of world bond and stocks consistent with the proportions for the end of November given on the FTSE Asset Allocation Policy Calculator site (for more, see the November 2019 post in this blog).

Here is the formula:

   x = [ Ps*(1–k) ] / [Pb*k]

   x = the number of shares of VT per share of BNDW
   Ps = the price of a share of VT at the end of                         November
   Pb = the price of a share of BNDW at the end of                  November
   k = the proportion of the value of world stocks in               the FTSE world bond and stock fund at the end
         of November

Prices for the two ETFs at the end of November could have been obtained shortly after the markets closed. But k (the proportion of the value of stocks in the world bond and stock fund) at the end of a month is not available at the FTSE site until some day in the first half of the following month. Therefore the value of x at the end of a month cannot be calculated until the first half of the following month. The calculations in this post assumed a buy-and-hold strategy throughout the first quarter of 2020 with 1.01 times as many shares in BNDW as in VT.

Here are calculations using values from the ends of the months prior to and during the first quarter of 2020.

As can be seen x, the proportions of numbers of shares in the two in a world bond/stock fund, did not change dramatically throughout the period. However, one could have revised the portfolio at the end of January to include, 1.027 shares of BNDW per share of VT, then again at the end of February to include 1.028 shares of BNDW per share of VT, etc.. If this quarter is representative, monthly changes of this sort would make relatively little difference.

The extent to which a fixed ratio of share holdings of the two funds could represent the overall market of world stocks and bonds is, of course due to relative amounts of redemptions, new issues and payouts of the two types of investments. Concerning the latter, according to Yahoo Finance on April 17, 2020, the yields of the two ETFs were quite similar, with the yield of BNDW equal to 3.01% per year and that of VT at 2.90% per year. But until there is a single mutual fund or ETF designed to represent world values of both bonds and stocks it makes sense to at least make calculations such as those in the table above and to adjust the relative number of shares of the ETFs used to represent world bonds and stocks when prudent.

Tuesday, October 29, 2019

BNDW: The Vanguard Total World Bond Exchange Traded Fund

In prior blogs, I suggested that a useful way to implement a World Bond Stock (WBS) fund would be to invest in two Vanguard exchange traded funds – VT (the Total World Stock ETF) and BNDW (The Total World Bond ETF), with the proportions reset periodically based on information from the FTSE Adaptive Asset Asset Allocation calculator website. I discussed the FTSE site in a September post and VT in an October post. This post deals with BNDW.

The Vanguard site provided the following information for BNDW at the end of September 2019:

But how could Vanguard have covered 15,029 bonds with only 172.9 million dollars? The answer is given in an accompanying table:

Aha! The fund actually holds shares in only two securities – each of which is itself a Vanguard ETF. They are: the Total International Bond ETF (BNDX) and the Total (U.S.) Bond Market ETF (BND). At the end of October 2019, each had significantly more assets than BNDW, as the table below shows.

Note that the total value of BNDW shares is by far the lowest (in the $Millions, not $Billions). This presents an investor interested in world bonds with a dilemma. The easiest course would be to invest in BNDW, letting Vanguard worry about covering both U.S. and non-U.S. Bonds in market proportions. But it would be cheaper to invest in BND and BNDX. Of course one would then need to periodically adjust the proportions invested to reflect relative market values of U.S. and non-U.S. Bonds. This could be done using the proportions of the two ETFs in the most recent BNDW monthly report with adjustments for changes in their values since that date. Such a strategy would also cost less, since roughly half the assets would incur an expense ratio of 0.09%, and the rest only 0.04% for a total of approximately 0.065%. Of course it is entirely possible that assets in BNDW will grow and that Vanguard will then lower its expense ratio. But at present, a home-made BNDW using shares of BNDX and BND would cost less than BNDW itself.

[ Update: On Oct. 30 (after the above was written and published) the Vanguard website showed that the expense ratio for BND had fallen to 0.035%, the others remained the same. ]

This is not all. At present, BND and BNDX shares are considerably more liquid than those of BNDW. The table below, from October 28th 2019, shows the average daily volume traded over the previous 45 trading days and the average spread during that period between the highest bid price and the lowest ask price.

A portfolio that includes BND and BNDX will incur lower expenses and lower likely trading costs than one that includes only BNDW. But BNDW simplifies the investor's life, since Vanguard takes care of any rebalancing required to keep the proportions of U.S. and non-U.S. Bonds similar to those in the indices representing the bond markets. You pay your money and take your choice.

Practical investors may well choose to hold a three-ETF (BND, BNDX and VT) proxy for the world bond/stock (WBS) portfolio. But for research purposes, it is easier to use the slightly more expensive two-ETF (BNDW and VT) proxy. Henceforth, I will generally do just this.

Now back to the analysis of BNDW. Here is Vanguard's breakdown of the bonds in its two component ETFs at the end of September, 2019:

And the top ten holdings by country:

For bond aficionados, here are some key characteristics of the bonds held:

With bonds as with stocks, United States securities constitute by far the largest proportion of world portfolios. For stocks, the U.S. percentage is somewhat over 50%; for bonds it is somewhat under 50%. But there are many companies, industries and securities outside the United States. To hold a piece of the world bond/stock portfolio, one needs U.S. and non-U.S. Securities. VT and BNDW can provide both.

Wednesday, October 2, 2019

VT: The Vanguard Total World Stock Exchange Traded Fund

In the prior post, I suggested using Vanguard's Total World Stock Exchange Traded Fund for the equity portion of a World Bond Stock (WBS) fund. Following industry practice, henceforth I'll refer to it using its ticker symbol: VT.

As described earlier, VT attempts to track the performance of the FTSE Global All Cap Index which “... covers both established and still-developing markets.” The home page of the Vanguard website provides a search box in the upper right corner. Typing VT in it will provide considerable information about the fund at the end of the prior month. For example, here is a regional breakdown of the holdings at the end of August 2019:

Some of the fund's characteristics at the time were:

And the ten largest holdings were:

Interestingly, 9 of the 10 largest holdings were in stocks of firms headquartered in the United States (but doing business in many parts of the world). Moreover, stocks of firms based in North America constituted almost 60% of the total value of the index. A subsequent breakdown by country showed that of these, at of the end of August 2019 stocks of U.S. companies constituted 55.8% of the total value.

The ETF's prospectus (available at the Vanguard web site) indicates that the FTSE Global All Cap index “..includes ... companies located in 47 countries, including both developed and emerging markets. The fund invests in a broadly diversified sampling of stocks in the index that approximates the index’s key risk factors and characteristics.”

The web site indicated that as of August 31, 2019 the FTSE index had 8,881 stocks while the fund had 8,192. It also showed that the Beta value for the fund relative to the index (rounded to two decimal places) was 1.00, indicating that on average, a given percentage move in the index was accompanied by a similar percentage move in the value of the fund. The rounded R-squared value was also 1.00, indicating that percentage changes in the values of the index were associated with similar percentage changes in the value of the fund.

As of August 31, 2019 the web site showed the following:

Fund total net assets $ 16.6 billion
Share class total net assets $ 11.7 billion

This is a key piece of information. It indicates that the ETF VT shares are only one class of claims on a fund of some sort. Thus there must be one or more other share classes. In this case there are two others. One can invest in “admiral class” shares of VTWAX – a mutual fund with the same asset holdings, or “institutional class” shares of VTWIX, another mutual fund which also has the same asset holdings.

(Note: I inferred the total value of VTWIX shares at the time from the reported values of the other two classes and the reported total net assets of $16.1 billion.)

But which of these three to choose? Most retirees choosing to invest in a total world stock fund will be have less than $5 million for the purpose, so VTWIX is not a possible alternative. However, if more than $3,000 is available to invest, either VT or VTWAX could be chosen. Both have low expense ratios, but VT is slightly cheaper. And, as the table shows, at the time it was clearly the more popular of the two share classes.

The Vanguard web site provides information useful for those choosing between investing in an ETF using the Vanguard brokerage platform and investing instead in a similar Vanguard mutual fund. It points out that the minimum investment in an ETF is the cost of one share, while Vanguard mutual funds generally require at least $3,000 in holdings. Moreover, an ETF can be purchased or sold at various prices during trading hours using market orders "or more sophisticated approaches” while for mutual fund shares “Regardless of what time of day you place your order, you'll get the same price as everyone else who bought and sold that day (and) that price isn't calculated until after the trading day is over.” However, there could be a downside: “if you want to repeat specific transactions automatically an ETF wouldn't be a suitable investment (since) You can't make make automatic investments or withdrawals into or out of ETFs.”

The Vanguard site doesn't cover issues associated with the liquidity of the market for ETFs. Information on this (and much else) is available on the etf.com website. While markets are open, you can find current bid and ask prices and associated volumes. Here, for example, is information from etf.com on October 1, 2019, showing real-time quotes for VT from the Chicago Board Options Exchange. 

As can be seen, at the time there was considerable liquidity for those wishing to buy or sell shares at the “market price”. Alternatively, of course, one can place a limit order that will execute only at a stated price or better.

The ETF.com site also provides historic information on premiums and discounts – the amounts by which “... the market price exceeded (premium) and fell below (discount) its fair value/net asset value (NAV)”. Here are some historic values for VT, with the associated daily volumes of shares traded

At least during this period, disparities between share prices and net asset values were relatively small. There is a reason why this should be the case. A key feature of the ETF form of an index fund is the ability of an “Authorized Participant” to exchange shares of underlying securities for one or more “creation units” of the fund's shares. The ETF.COM site defines such a unit as “The smallest block of ETF shares that an Authorized Participant can either create or redeem at Net Asset Value (NAV) with the issuer in exchange for the underlying shares of the fund”. For VT, the size of a creation unit is 200,000 of the fund's shares, while the fee is 0.07%. The presumption is that any significant deviation of an ETF's share price from the net asset value of its holdings will cause an Authorized Participant to spring into action, reducing or eliminating the disparity.

Some people argue that mutual funds are for investors and ETFs for gamblers. It is certainly the case that many of the thousands of ETFs appear to have been designed for those who wish to bet that certain types of securities are periodically overpriced or underpriced. And some may well use this ETF to try to try profit from possible market mistakes. But it would seem that either VT or VTWAX could provide an efficient and relatively low-cost stock component of a two-instrument low-turnover world bond-stock fund. To obtain a slightly lower expense ratio and to be au courant, I'll use VT in posts that follow. But VTWAX could serve the purpose as well.

Tuesday, September 24, 2019

A Two-fund Proxy for the World Bond Stock Fund

In the preceding post, I updated the material in my ebook that showed how to use four mutual funds to approximate a fund including all traded risky stocks and bonds in the world. I also suggested that it might be easier to approximate such a “world bond stock fund” (WBS) with two traded instruments. This post shows how this can be done.

Vanguard offers a number of exchange traded fund (ETFs) which invest in various types of bonds and stocks. The two of interest for our purposes are:

The Vanguard Total World Stock ETF (ticker symbol: VT)
The Vanguard Total World Bond ETF (ticker symbol: BNDW).

Following convention, I'll refer to them henceforth using ticker symbols.

I'll have more to say about ETFs in general and the particular approach utilized by BNDW. But first, let's consider the investments included in the funds and a way to determine the appropriate amounts to invest in each.

Here are the product summaries from the Vanguard website:


  • Invests in both foreign and U.S. stocks.
  • Seeks to track the performance of the FTSE Global All Cap Index, which covers both well-established and still-developing markets.
  • Has high potential for growth, but also high risk; share value may swing up and down more than U.S. or international stock funds.
  • Only appropriate for long-term goals.


  • Seeks to track the performance of the Bloomberg Barclays Global Aggregate Float Adjusted Composite Index.
  • Broad, diversified exposure to the global investment-grade bond market.
  • Unique ETF of ETFs structure.
  • Intermediate-duration portfolio, with exposure to short-, intermediate-, and long-term maturities.
  • Provides current income with high credit quality.

While these may not be perfect representatives for the overall world bond-stock market, they should provide a reasonable approximation. But how much to invest in each one? The market capitalization for the stocks covered by the FTSE Global All Cap Index can easily be found online. The fact sheet for the end of August 2019 showed a total value of $51,447,853 Million (roughly 51.5 Trillion US Dollars). But values for the bonds in Bloomberg Barclays indices are not available to the general public.

Not to worry. FTSE Russell graciously provides and maintains an Adaptive Asset Allocation Policy Calculator that allows investors or advisors to back-test and update strategies that adapt asset allocations to market proportions, using an approach that I presented in a paper published in the May/June 2010 issue of the Financial Analysts Journal.

For an investor wishing to hold market proportions of bonds and stocks, the approach simplifies to simply adjusting proportions of bond and stock holdings periodically as needed to approximate those of the market as a whole. Moreover, the FTSE AAAP site provides estimates of these proportions as of the last trading day of every month.

To access the site, go to: https://research.ftserussell.com/Analytics/AAAP/Home/Index. You will arrive at the home page for the calculator. The upper-left portion of the page looks like this:

Simply click the text “Calculator” to the right of the “Home” tab to obtain the page for the calculator. The top left-hand portion should look like this:

If the region shown is not Global , click the nearby arrows until it appears.

The text at the bottom of this section indicates the market capitalization proportions of global (world) equity and bonds at the end of the month shown as the current month. This information is generally updated from 7 to 10 days after the end of the each month.

To get a more precise indication of the bond and stock proportions, look at the graph to the right of this first bar. Initially, it will look like this:

The policy weights can be ignored – our interest is in the equity and bond market weights (the black and grey curves). As can be seen, these have varied substantially over the period covered (from December 2002 to the end of the prior month).

To obtain more detailed information, you can simply move your mouse to the diagram to obtain a dashed vertical grey line, then move it to any desired month . For our purposes, the final month (on the far right) is germane. Here is the diagram for the end of August, 2019:

The most recent proportions are shown at the upper right of the graph. In this case FTSE estimated that at the end of August 2019 equities consituted 47.699% of the market and bonds 53.301%.

FTSE makes these functions of the AAAP Calculator available without registration. Those who wish to analyze policies involving combinations of stocks and bonds other than those of the market as a whole need to register to obtain the full functionality of the adaptive asset allocation approach. But this is not needed for our purposes.

Once the proportions of world bonds and stocks at the end of the preceding month have been obtained it is straightforward to calculate the number of shares of one of the two ETFs that need to be sold and the number of shares of the other that need to be purchased in order to bring the portfolio proportions of bonds and stocks close to estimated current proportions.

The google spreadsheet below does the job. The values in green are required inputs, those in black are computed by the spreadsheet, and those in red indicate the number of shares of one ETF to purchase and the number of the other to sell as well as the associated estimated values. The information in column E indicates the sources of inputs (in green) and the computations utilized by the spreadsheet (in black). When constructing your own spreadsheet you will need to enter formulas in columns C and D for the rows in which there is a formula in black in column E.  Once you have done this you can just type in current values for the items in green for your portfolio and click “run” to obtain the appropriate values for the numbers in red.

Due to the need to trade in integer numbers of shares the proceeds proceeds generated from the sale of shares of one of the etf's is designed to exceed the cost of the shares of the other etf. However, if execution prices differ significantly from the prices input in line 4, this may not be the case. In any event, the total value of the holdings of the two assets is likely to change only slightly.

There is little reason to make frequent changes in holdings of the two funds. The majority of the changes in their relative values will result from changes in the prices of assets already held rather than new issues, redemptions, dividends, interest payments and the like. It should suffice to check the relative values of your holdings against those shown in the AAAP calculator every month or so, changing your portfolio holdings only when the differences are substantial.

Sunday, August 18, 2019

The Four-fund Proxy for the World Bond Stock Fund

Chapter 7 of my ebook on retirement income suggests the use of a portfolio of four Vanguard mutual funds to serve as a proxy for the “World Bond Stock Fund”, which can serve as the “market portfolio” in analyses using the RISMAT software. This post will update the information needed for this approach. However, times have changed. It is now possible to approximate the world bond stock fund with only two funds. The next post will discuss this. Feel free to wait for it.

Now to the four-fund approach. The ticker symbols and names of the funds are:

VBTLX  Vanguard Total Bond Market Index Fund (admiral shares)

VTABX Vanguard Total International Bond Index Fund (admiral shares)

VTSAX Vanguard Total Stock Market Index Fund (admiral shares)

VFWAX Vanguard FTSE All-World ex-US Index Fund (admiral shares)

In each case, the shares are of the "admiral" class which provide the same gross returns but with lower fees than the "investor class" shares of the same fund. At the time the chapter was written in 2015, the minimum investment for admiral shares in each of the funds was $10,000. However, this has fallen as the funds have attracted more assets. As of this post in 2019, one can obtain admiral shares for any one of these funds for an investment of $3,000 or more.

The first and third funds invest in securities of United States companies, while the second and fourth hold only securities of companies based in other countries. In the mutual fund industry the term “international” generally means “non-U.S.” but I prefer to use the latter term for clarity. Funds that hold securities of U.S. and non-U.S. Companies are generally called “Global” – a convention that I will follow.

A fund's expense ratio indicates the proportion of one's investment that is dedicated to cover the expenses of fund management. It is generally expressed as an annual percent of value. Over the course of potentially many years in retirement such expenses, if high, can significantly affect one's standard of living. Here is a table from Chapter 7 of the ebook showing the funds and their expense ratios in 2015:

VBTLX: 0.06% per year
VTABX: 0.14% per year
VTSAX: 0.05% per year
VFWAX: 0.13% per year

And here are the expense ratios today:

VBTLX: 0.05% per year
VTABX: 0.11% per year
VTSAX: 0.04% per year
VFWAX: 0.11% 
per year
Happily, all the expense ratios have decreased, presumably due to the increased assets under management allowing fixed costs to be spread over larger pools of money.

In the ebook I suggested that the world bond/stock fund be proxied by a combination of these four funds, using the relative market values of the underlying securities in each of the four component sectors. But where to fund such values? Ideally, for each fund we would find the market capitalization at some recent date of the index being tracked, then adjust the holdings of the four funds so that their relative values would have equaled those of the corresponding indices at that date.

Here are the values of indices tracked by the four funds in 2015:

 Barclays U.S. Aggregate Float Adjusted Index

Barclays Global Aggregate ex-USD Float Adjusted Regulated Investment Company Index Currency Hedged

The Center for Research in Security Prices U.S. Total Stock Market Index

FTSE All-World ex U.S. Index

In 2019 the funds tracked were similar, although the names of the indices tracked by VBTLX and VTABX had been changed to reflect their acquisition by Bloomberg. Moreover, small changes had been made in their composition. The current names are:

 Bloomberg Barclays U.S. Aggregate Float Adjusted Index

Bloomberg Barclays Global Aggregate ex-USD Float Adjusted RIC Capped Index (USD Hedged).

At the time the book was written, Barclays bond index values were not available to the public at large, so I proposed using capitalization values from relatively similar funds provided by Citibank:

Citibank U.S. Broad Investment Grade Bond Index (USBIG)

Citibank World Investment Grade Bond Index (worldBIG) minus the value of the USBIG index.

Fact sheets for the Bloomberg Barclays bond indices are still not publicly available. The Citibank indices are now provided by FTSE, which does make fact sheets with capitalization values available to the general public.

Here are ways to find the fact sheets for the four indices. The search texts given will generally work but you might need to choose from multiple provided links. In most cases the desired fact sheet will appear in your browser, but a downloadable pdf file may be provided instead. As always with information on the web, things may change and some extra effort may be needed.

U.S. Bonds
    Search for: FTSE USBIG fact sheet

Global Bonds
     Search for: FTSE WorldBIG fact sheet

U.S. Stocks
     Search for: CRSP U.S. market fact sheet

Non-U.S. Stocks
      Search for: FTSE all world ex US fact sheet

The FTSE fact sheets provide market capitalizations as of the end of the most recent month but CRSP capitalization values are provided only as of the end of the most recent quarter.

Once you have index market capitalizations for the end of the most recent quarter, you can update the desired values for the four funds using the approach given in the ebook for estimating current market values or, if you have access to the MATLAB language, the program given in chapter 7.

But instead you might want to use the two-fund approach. For this, see the next post.