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, October 10, 2020

Covid Statistics

 And now for something completely different. 

This slidecast covers some of the key statistics that are being widely used to measure and try to control the pandemic in 2020 caused by the covid-19 virus.

To access it, click the link below.


Wednesday, September 30, 2020

Annuity Terms

As the previous post indicated, henceforth some of my posts will provide links to documents in the .pdf format in lieu of including the covered material in a post. But some posts will provide instead links to slidecasts that I have created using the scratch programming language developed at M.I.T..

The term slidecast is an abbreviation for slide podcast. The goal is to combine both visual information in a form similar to that in a slide presentation with audio information in a form similar to that of a podcast.  

The programs that I have written for this purpose are compatible with many computer browsers. They can also be used on some other devices such as cell phones, tablets, etc.. With computer browsers, it is possible to choose to have the narration in English, French or Spanish. However, for devices without keyboards, only English is available.  

The first slidecast covers annuity terms.

The terms offered by insurance companies and others who provide annuities change from time to time. In recent years there has been a substantial decrease in the amount of future income that can be purchased for a given current payment. This slidecast shows some of the changes in annuity terms and the driving force behind them. To access it, click on this link:


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.