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.

Saturday, August 8, 2020


Please click this link for the pdf version:  Post 8

To make it easier for me to create posts for this blog, and to provide each one in a format that can, if desired, easily be read and saved on your local computer, I have chosen to use the pdf format beginning with this post. You should be able to view this post and those to follow in your browser, and save any such pdf file if you choose to do so.  

Thursday, July 16, 2020

Treasury Inflation-Protected Securities (TIPS) in the first half of 2020

In the book on Retirement Income Analysis, I argue that for an investor who purchases goods and services with U.S. Dollars, the only riskless securities are Treasury Inflation Protected Securities (TIPS) issued by the U.S. Government. These bonds make coupon payments semi-annually that are adjusted to take into account the most recent Consumer Price Index (CPI) for goods and services in the United States. At such a bond's maturity date, an amount is paid that is also adjusted to take into account the CPI at the time. There is one additional feature: if at the maturity of a TIPS issue the CPI is lower than it was at the time the bond was issued, the original par value is paid rather than a smaller amount. With this one highly unlikely exception, the coupon payments and principal payment from a TIPS bond can be considered riskless in real terms.

At any given time there are TIPS outstanding with different times to maturity. For each one, a real yield to maturity can be calculated based on its current market price and future guaranteed coupon and principal payments. At its web site for Daily Treasury Real Yield Curve Rates, the U.S. Treasury provides yields to maturity for five different maturities (5, 7, 10, 20 and 30, years). Here is a link:


The following graph shows the rates for each market date in 2020 through July 10, 2020.

Throughout this period, longer maturity bonds provided higher yields to maturity than those with shorter maturities. In bond jargon: the real yield curve was upward-sloping. That said, there were days when the differences among yields with different maturities was very small indeed, but at other times they differed significantly. In most months there was a downward trend in yields at each maturity. But there was one major exception: in the middle of March, yields changed from negative values to positive values, reflecting significant declines in the prices of the underlying bonds. The reason is not hard to guess. The gravity of the Covid-19 pandemic was becoming apparent at the time and prices of securities of many types fell dramatically. TIPS were no exception. However, the panic did not last very long. By the early part of April, TIPS prices were near their values a month earlier and thus yields to maturity were back to previous levels. Thereafter, as the second quarter progressed, yields continued to fall, reflecting rising prices for the securities.

But there is more drama here. At the beginning of the year, TIPS with shorter maturities provided almost no real yield, but those with longer maturities provided at least some increases in purchasing power. Later, yields fell and turned to negative values. In early July the yield to maturity on 5-year TIPS was close to -1% per year and those of every maturity were priced to provide negative real yields to maturity. Thus for any time period up 30 years, an investor in TIPS could plan on receiving less purchasing power than he or she invested.

Forward yields

While the yields to maturity tell the story of long-term returns from TIPS, it is useful to do some additional calculations to obtain estimates of “forward yields”. Consider two TIPS – one maturing in five years, the other in seven years. Think of the latter as composed of two investments: the first requires investment today and pays off in five years. The second requires a “forward” commitment today to invest a dollar in five years and will then pay off two years later in year seven. Assuming no coupon payments and using subscripts to denote the initial year and the final year for an investment we can write the relationship among two standard yields to maturity and a forward yield to maturity as:

(1+y0-5)5 (1+y5-7)2 = (1+y0-7)7

The first and last expressions use the types of yields-to-maturity that are provided on the Treasury website and plotted in our diagram – these are sometimes called spot rates. The second term includes a forward rate for a commitment made today to invest in year 5 to obtain a payment in year 7. Obviously it is a simple matter to compute the forward rate, given the two spot rates. The following diagram shows the results of such calculations for each of the combinations of adjacent spot rates in the previous diagram. For completeness, the spot rates for years 0 through 5 are also shown.

If each TIPS paid only at maturity, it would be possible to actually achieve the forward rate for an interval. For example, one could invest in a 7-year TIPS with the proceeds from selling short a 5-year TIPS. This would require a known future payment in 5 years for the short position and provide a known future receipt in 7 years for the long position. The net result would earn the the forward rate for years 5 to 7. Unfortunately, the actual world is not that simple since actual TIPS provide coupon payments every six months. It is true that from time to time an institutional investor or broker will purchase a TIPS and sell one claim for its coupon payments and another for its principal payment at maturity (a “stripped TIPS”) which will be equivalent to a zero-coupon TIPS. But such exotica are generally not available for individual investors. The bottom line is that the forward TIPS yields can best be interpreted as indicators rather than explicitly achievable investments.

To see how derived forward rates can be useful, consider the 7-year TIPS on July 10, the last year in our example. The yield to maturity was -0.87% per year. It would be natural to think of the security as providing -0.87% per year for seven years. But wait. The yield on a 5-year TIPS at the time was -0.95% and the forward rate for years 5 through 7 was -0.67%. It might be more useful to think of the 7-year TIPS as providing -0.95% on average for five years, then -0.67% on average for another two years. Of course, if we had used data for TIPS with maturities in years 1, 2, 3, 4, 5, 6 and 7 we could have broken the overall yield into even more pieces. Those interested in making such detailed calculations can find current TIPS quotes at https://www.barrons.com/market-data/bonds/tips.

But why calculate forward rates at all? One answer is that they might be useful predictors of future spot rates. But not perfectly so since the forward rate for a future interval of time may reflect both a forecast of spot rates at that time and a premium for taking the risk of making a commitment to invest at a future date. Thus a forward rate might be greater than predictions of the associated future spot rates. If so, these data suggest that on July 10 2020, TIPS yields suggested that investors expected negative relatively riskless real rewards for deferring consumption for decades into the future.

To provide some perspective, here is a longer record of yields to maturity on 10-year TIPS, from the Federal Reserve Bank of St. Louis FRED site (https://fred.stlouisfed.org/series/WFII10):

As can be seen, 10-year TIPS yields have been negative before (mainly in 2012) but turned positive thereafter until 2020. We can only hope that this might happen again.

In any event given current TIPS rates, retirees need more savings than before to achieve any given future standard of living with certainty. Some will respond to this fact by saving more and/or spending less today. Others may decide that saving is less attractive and spend more than previously intended. Moreover, the latter tendency may be encouraged as well by the pandemic (“eat, drink and be merry, for tomorrow you die”). One might argue that relatively few investors are following this advice since if more did, TIPS prices would be lower and thus yields to maturity higher. But here, as always, it is important to remember that prices in security markets are influenced disproportionately by the rich.

One thing is certain: current TIPS yields are far lower than in most earlier times. All the examples in my ebook on Retirement Income Analysis used computations that assumed a riskless real rate of return of 1% for all maturities with a return premium for bearing the risk of a world bond/stock portfolio. Of course the accompanying RISMAT software makes it easy for the user to change either or both of these numbers since they are simply properties of a market object. 

But in a world of low or negative riskless real rates of return, some retirees may conclude that if they survive COVID, they might well run out of money thereafter.

Other than that, have a nice day!

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.