Impact of Gender on Annuity Payments

Impact of Gender on Annuity Payments

 Introduction Females have longer life expectancy than males in virtually all countries.   Gender related differences in life expectancy make it more likely that females will out-live their retirement resources than will males. Females, because of their longer life expectancy, might choose to purchase a longer-term annuity.

Question:   A 75-year old person has $100,000 to spend on an annuity., which makes monthly payments for a fixed period.  She or he wants to reduce the probability of outliving the annuity to below 10 percent.

What annuity term would accomplish this goal for a male and for a female?

How does the longer life expectancy of the female affect the size of the monthly annuity payment?

Data Source: This analysis is based on the United States Life Tables, 2008 published on September 24, 2012 by the National Center for Health Statistics of the Centers for Disease Control and Prevention

Table Two of the report has life statistics for males and Table Three of the report has life statistics for females.   Both tables can be downloaded directly into an EXCEL Spreadsheet.The data in the Table below is from the CDC life tables. 


Age Total number of females alive at age x Proportion of 75-year-old females who survive to age X Total number of Males Alive at age X Proportion of 75-year Males surviving until age X
75 73,974 61,980
76 71,973 97.3% 59,531 96.0%
77 69,831 94.4% 56,962 91.9%
78 67,539 91.3% 54,268 87.6%
79 65,080 88.0% 51,437 83.0%
80 62,448 84.4% 48,469 78.2%
81 59,647 80.6% 45,390 73.2%
82 56,688 76.6% 42,227 68.1%
83 53,563 72.4% 38,994 62.9%
84 50,253 67.9% 35,694 57.6%
85 46,782 63.2% 32,360 52.2%
86 43,166 58.4% 28,996 46.8%
87 39,414 53.3% 25,650 41.4%
88 35,567 48.1% 22,372 36.1%
89 31,677 42.8% 19,213 31.0%
90 27,805 37.6% 16,223 26.2%
91 24,017 32.5% 13,448 21.7%
92 20,380 27.6% 10,928 17.6%
93 16,962 22.9% 8,691 14.0%
94 13,821 18.7% 6,754 10.9%
95 11,006 14.9% 5,122 8.3%
96 8,549 11.6% 3,783 6.1%
97 6,467 8.7% 2,719 4.4%
98 4,754 6.4% 1,898 3.1%
99 3,392 4.6% 1,286 2.1%
100 2,345 3.2% 844 1.4%


 Calculation the Annuity Term:

Based on this cohort of 100,000 females, 73,974 females have survived to page 75.   Around 90% of these females are still alive until somewhere between age 95 and 96.

In a cohort of 100,000 men 61,980 are still alive at age 75 and the 90% survival mark for these 75-year olds is reached somewhere between age 94 and 95.

Let’s interpolate to get an exact number of months for our annuity formula.

For females we get 96-75 (21) years plus (11.6-10)/(11.6-8.7) x 12 or (7) months.  The 75-year old female must buy an annuity of 259 months  to reduce the probability that she will outlive the annuity to 10 percent.

For males we get 94-75 (19) years plus (10.9-10)/(10.9-8.3) x 12 or 5 months (I am rounding up to fulfill the contract.)   The 75-year old male must buy an annuity of 233 months to reduce the probability that he will outlive the annuity to 10 percent.

Calculating the Impact on Annity Payments:

So now we calculate the annuity payments for the female and the male with the PMT function.   The only input that differs is the duration of the contract – 259 months for females and 233 months for males.

The annuity payment calculations were obtained from the PMT function in Excel.


Female Male
Rate 0% 0 0
Rate 3% 0.03 0.03
Rate 6% 0.06 0.06
NPER 259 233
PV $100,000 $100,000
PMT rate=0 $386.10 $429.18 ($43.08) -11.2%
PMT rate =3% $524.96 $566.77 ($41.81) -8.0%
PMT rate=6% $689.45 $727.62 ($38.17) -5.5%


Conclusions:   Females must buy a longer-term annuity to obtain the same reduction in longevity risk as a male.   This reduces their monthly annuity payment.

This annuity calculator on the web confirms that females receive lower annuity payments and or pay higher prices for a comparable annuity.  I am not familiar with the specific formulas used by this calculator or the product that it pertains to.   My sole interest here is to provide some insight on how gender determines longevity risk.







Survivor Bias and Stock Market Risk


Survivor Bias and Stock Market Risk


Shortly after the large stock price decline of Facebook on July 26,2018, CNBC posted an article with a chart of other large market-cap declines of U.S. stocks.   Every single company in the chart still exists as an ongoing company.   It appears as the chart was compiled from a database that only contain companies that are still listed.

What no longer actively traded companies might have made a list of the largest declines in equity value for large-cap companies?

Why does the exclusion of these companies from the chart create a misleading picture of the risk of investing in equity?

The chart lists three companies with large cap declines in 2018?  Why is the occurrence of so many recent large declines in big-cap equity a concern?

The article:

The Chart:

Market Cap Losses in Big Cap Stock


Date Decrease In Equity ($B)


Jul 26 2018 $114.50


sep 22 2000



Apr 3 2000


Apple Jan 24 2016


Exxon Mobile

Oct 15 2008


General Electric

Apr 11 2008



Feb 2 2018


Bank of America

Oct 7 2008



Apr 2,2018


Wells Fargo

Feb 5 2018



Jul 23 2002


JP Morgan Chase Sep 29 2008




What no longer actively traded companies might have made a list of the largest declines in equity value for large-cap companies?

Companies no longer actively traded, which experienced large one-day drops in equity value include   — Enron, Worldcom, the old General Motors, Lehman Brothers, Bear Stearns, and Merill Lynch.   There are probably many more.

Why does the exclusion of these companies from the chart create a misleading picture of the risk of investing in equity?

 First, this list suggests large drops in equity values occur less frequently than a chart composed from all firms that ever existed.

Second, a list of large declines composed of stocks that survived understates the potential loss of wealth from buying and holding stocks after a large decline.   In fact, all the stocks on the CNBC list have recovered nicely.

Third, an anaysis of risk, which includes bankrupt firms would encourage investors to seek greater diversification in terms of number of equity holdings, sector, and asset classes.

The chart lists three companies with large cap declines in 2018?  Why is the occurrence of so many recent large declines in big-cap equity a concern?

Facebook, Alphabet, and Amazon all had their large declines in 2018.   These are three of the four FANG stocks.   The smallest FANG company also has had a recent large percentage decline.

The most popular trendy sector of the stock market is now realizing extremely large one-day changes.   This appears to be happening with greater frequency.

The disproportionate number of 2018 large-cap declines in this chart convinces me that people who have made a lot of money in FANG and in other high-tech stocks need to take some money off the table when the stock prices reach new highs.

Of course, in the case of Facebook  this advise was more valuable prior to July 26.

The volatility of FANG names and other Tech stocks does not mean this is a good time to buy value stocks because the pending  increase in interest rates might hurt value stocks more than growth stocks

I would be alarmed by the large number of 2018 events even if the chart contained firms that were no longer actively traded.

Holdings of Ten Emerging Market Funds

Holdings of Ten Emerging Market Funds

This post started with a list of emerging market funds found at the site below.

I examine and describe the holding and recent returns on the 10 largest of these funds.

I comment on the holdings of these funds, the risk of these funds and the risk of some of the holdings.

List of Funds, Discussion of Geographic Diversifications, and YTD returns:

Some Information on Ten Emerging Market Funds
Symbol Fund Geographic Concentration YTD Returns
VWO Vangarud FTSE Emerging Market 7 of top ten holdings are in China -7.32%
IEMG I Shares Core MSCI Emerging Market 7 of top ten holdings are in China -6.95%
EEM I Shares MSCI Fund 7 of top ten holdings are in China -7.44%
SCHE Schwab Emerging Markets Fund 6 of top ten holdings are in China -7.34%
FNDE Schwab Fundamental Large  Firm Emerging Market Index Largest holding is from Korea and second and third largest holding are Russian.  Top 10 holdings also include companies from China, Brazil and Taiwan -6.71%
DEM Wisdom Tree Emerging Markets Equity Income Fund Top two holdings are Russian.   Most other top ten holdings are form China or Taiwan. -4.54%
RSX Van  Eck Vectors Russia ETF All holdings appear to be in Russia. 4.36%
GEM Goldman Sachs ActiveBeta Emerging Markets Equity ETF 7 of top ten holdings are in China -6.75%
SPEM SPDR Portfolio Emerging Markets ETF 6 of top 10 holdings are in China -6.54%
DGS Wisdom Tree Emerging Markets Small Cap ETF Top 10 holdings are less than 9 percent of all holdings.   Highly diversified, smaller companies. -7.34%


Some Observations:

Many of the funds have a very large share of their funds in China.   In fact, some of the emerging market funds could accurately be called China plays.

Three of the funds have substantial issues in Russia.  Also, most but not all, of the Russian holdings are in the oil and gas sector.  Important to research holding If concerns about corruptions and sanctions would deter you from investing in Russia.   Energy funds that are not in Russia may be preferable to Russian energy plays.

Nine of the Ten funds have negative YTD returns.  The fund that exclusively invests in Russia has a +4.36% return. This fund performed really poorly in prior years.  The YTD return on U.S. large cap stocks is around 6.0%, largely because the market has been up for the past week or so.

Can investments in emerging market ETFs provide insurance against a major downturn in the U.S. market?

 Short answer is probably not:   The downturn in emerging markets in the 2007 to 2009 crisis was larger than the downturn in large-cap U.S. stocks.

Go here for discussion of emerging market funds during the financial crisis

Thoughts on specific holding of emerging market funds:

Emerging market funds have some well know reputable holding and some firms with dubious reputations, and some firms with little name recognition.

The most successful holdings of the funds in this sector include: (1) Alibaba, (2) Tencent and (3) Baidu Inc.

Investors who want a taste of advanced emerging markets or China may be better off directly investing a small fraction of their wealth in these companies.

Emerging Markets During the Financial Crisis

Emerging Markets During the Financial Crisis

True or False:   During the 2007 financial crisis emerging market funds outperformed U.S. large Cap funds?

Answer:  False:

Evidence:   The chart below compares peak price in 2007 prior to stock market collapse and trough price in 2009 prior to the recovery.   The peak price for VLACX was on October 1, 2007.   The peak price of VWO, the largest emerging market fund, occurred a bit later, on October 22, 2007.   The trough price for both funds was realized on March 2, 2009.

U.S. Large Cap versus Emerging Market During the Financial Crisis
Funds Peak Before Crash in October 2007 Trough at End of Crash in March 2009 % Decline
VLACX U.S. Large Cap 28.1 12.6 55.3%
VWO Emerging Markets 57.5 19.0 67.0%

The collapse in the Vanguard emerging market fund was larger than the collapse in the Vanguard large-cap U.S. equity market.

Perhaps some other emerging market fund did better but I doubt it.

Review of Measuring Portfolio Valuation

My blogging ground to a halt the last few weeks because I was completing a paper “Measuring Portfolio Valuation. “  Will put link to paper here shortly.

My new paper looks at two issues.   The first issue involves correct and incorrect ways to measure the PE ratio of a portfolio of stocks.   The second issue involves the correct way to conduct statistical tests on valuation measures for groups of stocks.

The paper starts with a discussion of the limitations of the PE ratio, the most commonly used valuation measure for common stocks.   The PE ratio is undefined when earnings are negative and unstable when earnings are small. By contrast, the ratio of the difference between market cap and earnings to market cap (denoted (MC-E)/MC) has a clear economic meaning when earnings are negative and is not an outlier when earnings are low.  In addition, there is a one-to-one relationship between this ratio and the PE ratio.

Many investment firms use a weighted average of firm PE ratios to measure the PE ratio of their ETFs or mutual funds.   The firms often discard observations from firms with negative earnings and cap the PE ratio of firms with high PE ratios.   These methods are arbitrary and often tend to understate the valuation of stock prices relative to earnings.

The ratio of the sum of market caps of firms in a portfolio to the sum of earnings of firms in the portfolio is the correct way to measure the PE of a portfolio.   This measure of PE can include all firms even firms with negative earnings.  Moreover, small changes in earnings for firm with high PE ratio do not have a large impact on the overall portfolio PE ratio.

A second way to measure the PE ratio of a portfolio, which relies on the weighted average of the statistic ((MC-E)/MC) is presented and shown to be equivalent to the ratio of the sum of market cap to sum of earnings.   This result is motivated in the following blog post.

Two Ways to Calculate a Portfolio PE Ratio:

The paper contains a formal proof demonstrating the two methods of constructing a portfolio PE are identical.

Often analysts conduct hypothesis tests on portfolio financial ratios.   Tests based on PE ratios often provide misleading results because of problems measuring the PE ratio when earnings are negative or small.   Firms with negative earnings are routinely omitted from the sample.  The standard deviation and skew of portfolio PE ratios are often large making it difficult to reject a null hypothesis.

By contrast, statistical tests based on (MC-E)/MC do not require the omission of firms with negative earnings.  Moreover, the distribution of (MC-E)/MC appears normally distributed with few outliers.    As a result, statistical tests using this ratio are more reliable than statistical tests using PE ratios.

Two Ways to Calculate a Portfolio PE Ratio

Two Ways to Calculate a Portfolio PE Ratio 

Question:  The table below contains data on the market cap and the earnings for four high-tech firms.

Market Cap and Earnings for Four Tech Firms
Market Cap

($ B)


($ B)

AAPL 892.16 46.65
MSFT 585.37 21.2
AMZN 475.37 1.92
TWTR 13.11 -0.44797


In this post, I am asking you to use two methods to calculate the PE ratio of this four-stock portfolio and to confirm that both methods provide the same answer.

Method One:

Calculate the PE ratio of this portfolio by taking the sum of the market cap numbers for the four stocks and dividing by the sum of the earnings of the four stocks.

Method Two:

Calculate the ratio of (market cap minus earnings) divided by market cap for the four stocks.

Calculate a weighted average of the values (MC-E)/MC for the four stocks with the ratio weighted by MC.  Give the name to this weighted average the letter f.

Calculate 1/(1-f).

Show that the PE ratio from method one is identical to 1/(1-f).


The straight forward way to calculate the PE ratio by taking the ratio of the sum of the market caps to the sum of the earnings is presented below.

Portfolio PE Ratio – Method One
Market Cap

($ B)


($ B)

AAPL 892.16 46.65
MSFT 585.37 21.2
AMZN 475.37 1.92
TWTR 13.11 -0.44797
Total 1966.0 69.3 28.4


This four-firm portfolio has a PE ratio of 28.4.

The PE ration calculation for method two  is presented below.


Portfolio PE Ratio — Method Two
Market Cap Earnings (MC-E)/MC Weight
AAPL 892.16 46.65 0.9477 0.4538
MSFT 585.37 21.2 0.9638 0.2977
AMZN 475.37 1.92 0.9960 0.2418
TWTR 13.11 -0.44797 1.0342 0.0067
1966.01 1.0000
f 0.9647
1/(1-f) 28.4


The second method for calculating a PE ratio gives the same result as a the first – 28.4.

Implications:   The PE ratio of a portfolio can be expressed as function of the weighted average of the ratio of the difference between market cap and earnings of the firm to market cap of the firm.    This is a very useful result.

PE ratios of firms are frequently not useful.

First, the PE ratio can become very large when earnings are very small. This means it is misleading to look at a weighted average of PE ratios because one firm can have a a very large impact. In our current example, the PE ratio of Amazon is 248 and the weighted average PE ratio for the four stocks is  77.

Second, PE ratios have no economic meaning when earnings are negative.

The PE ratio of a firm with negative earnings would reduce the weighted average of PE ratios in a portfolio.  By contrast, (MC-E)/MC will be larger than 1 if E is less than 0.

A firm with slightly negative earnings would have a negative PE ratio with a larger absolute value than a firm with very large losses.  This ranking of firms is incorrect because larger losses should be associated with lower relative valuations.   By contrast, (MC-E)/MC will always rise when E falls.

By contrast, the ratio of the difference between market cap and earnings over market cap is inversely related to the valuation of a firm.   When earnings are negative this ratio is greater than one.   When earnings are zero the ratio equals one.   When earnings are very small the ratio approaches one and is not an outlier.  The ratio of the difference between the market cap and earnings to market cap is intuitively defined for all earnings and not impacted by outliers.


How to correctly calculate portfolio PE ratios?


How to correctly calculate portfolio PE ratios?

Question:   Two analysts are given the task of calculating the PE ratio of the DOW using the data below.   (Note:  Data on one firm is missing because of a recent merger.)

The first analyst uses method one and takes the ratio of the weighted average of market caps for the 29 companies to the ratio of earnings for the 29 companies.

The second analyst uses method two and takes the weighted average of the PE ratios of the 29 stocks.

What is the correct way to calculate the PE ratio for this portfolio?

What are the ramifications of using the wrong method to calculate the PE ratio for this portfolio?

The Data:

Financial Information on Stocks In The Dow
Share of Dow Market Cap ($B) Trailing Earnings ($B) Trailing PE
MMM 0.0645 124.9 5.2 23.87
AXP 0.0278 79.8 4.3 18.5
AAPL 0.0474 793.6 45.5 17.44
BA 0.0781 149.7 6.7 22.18
CAT 0.0383 73.7 0.1 696.65
CVX 0.0361 222.6 5.8 38.1
CSCO 0.0103 166.5 9.4 17.7
KO 0.0138 192.0 4.0 47.5
DIS 0.0303 152.1 8.7 17.48
XOM 0.0252 347.4 11.7 29.6
GE 0.0074 209.4 7.1 29.45
GS 0.0729 92.1 7.4 12.44
HD 0.0503 192.8 8.2 23.5
IBM 0.0446 135.2 11.3 12
INTC 0.0117 178.9 12.3 14.5
JNJ 0.0400 348.9 15.9 22
JPM 0.0294 336.1 23.8 14.1
MCD 0.0482 126.9 4.9 25.7
MRK 0.0197 174.6 5.0 34.7
MSFT 0.0229 573.7 20.9 27.5
NKE 0.0159 85.1 4.1 20.7
PFE 0.0110 212.3 8.1 26.1
PG 0.0280 232.0 14.2 16.3
TRV 0.0377 33.8 2.8 12.2
UTX 0.0357 92.7 5.2 17.7
UNH 0.0602 189.4 8.1 23.5
VZ 0.0152 201.9 15.9 12.7
V 0.0323 240.7 6.2 39.1
WMT 0.0240 233.4 12.4 18.8

Methodological Note:  Assume the columns of your spreadsheet are – (1) Share of DOW in A, (2) Market Cap in B, Trailing Earnings in C, and Trailing PE in D.

Also assume there are 29 rows, 1 to 29 for each variable.

The formula for method one is =SUMPRODUCT(a1:a29,b1:b29)/SUMPRODUCT(a1:a29,c1:c29)

The formula for method two is


Analysis:   The DOW PE ratio for method one is 20.5, a pretty high number compared to the historic norm of PE ratios for this index.

The DOW PE ratio for method two is 46.7, a number that is implausible for the portfolio of DOW stocks

Market Cap Weighted Total 203.4
Earings Weighted Total 9.9
Dow PE Ratio Method One 20.5
DOW PE Ratio Method Two 46.7

The PE ratio of one company in the DOW, CAT is 696, an extreme outlier.  This outlier drives up the weighted average of the PE ratios by a lot.

It is inappropriate to take the average of PE ratios because often a PE ratio for a particular company is an outlier or is below zero.

PE ratios below zero are economically meaningless.    For a discussion of how to calculate the PE ratio of a portfolio when some stocks in the portfolio have negative earnings go to the following site.

PE Ratios When Some Firms Have Negative Earnings

Many analysts deal with the issues of negative or outlier PE ratios by dropping firms from their analysis.     There is no need to drop firms when you calculate a portfolio PE ratio if you are using an appropriate method.

Evaluating Fund Performance

Evaluating Fund Performance

Investment funds, both ETFs and mutual funds, are usually compared on the basis of returns of arbitrarily selected holding periods.   Typically, the fund manager reports year-to-date returns and return for one, three, five, and ten years.  The discussion of fund risk is usually based on a subjective assessment of the risk of the assets in the fund.

The conventional approach to presenting statistics on fund performance is inadequate.   Funds can be purchased at any time, not just a few arbitrarily selected dates.   This post measures the mean and standard deviation of return for two popular funds when there are multiple possible purchase and sale dates for each fund.

Statistical tests are used to evaluate whether the observed difference in return and risk outcomes for two funds are statistically significant.

Question:   This post considers two of Vanguards most successful funds.  VFIAX is a fund that mimics the S&P 500 and VWELX a fund that is around 70% equity and 30% fixed income.

The 48 potential purchase dates for both of the two funds are the first day of each month starting in January 2002 and ending in December of 2005.

The 48 potential sale dates for the two funds are the first day of each month starting in January 2012 and ending in December of 2015.

  • Assume that each combination of purchase and sale dates is equally likely.
  • What are the expected return and the standard deviation of return for both funds?
  • What are the minimum and maximum returns for each fund?
  • Can we reject the hypothesis of identical variances for the two funds?
  • Can we reject the hypothesis the mean returns are identical?


There are 2304 (48 x 48) possible (purchase-sale) outcomes.  For each of these outcomes I calculate ln(AP2/AP1) where AP2 is the adjusted price in the 2012 to 2015 time period and AP1 is adjusted price in the 2002 to 2005 time period.

The mean standard deviation, minimum, and maximum for the two funds are presented below.

Returns from Two Funds
Fund Description Mean Standard Deviation Minimum Maximum
VWELX Stocks and bonds 0.763 0.177 0.341 1.14
VFIAX Stocks 0.692 0.226 0.178 1.21

Sample size 2304 based on 48 possible purchase dates between 2002 and 2005 and 48 possible sale dates 2012 and 2015.


  • The mean return of the bond/stock fund is higher than the mean return of the stock-only fund by around 10 percent.
  • The standard deviation of returns for the bond/stock fund is lower than the standard deviation of returns for the stock-only fund by around 21 percent.
  • The maximum return is higher for the stock-only fund by around 92 percent.
  • The minimum return is lower for the stock-only fund by around 6 percent.


Comment One:  The finding that the combined stock/bond fund has a larger mean return compared to the stock-only fund is extremely unusual because over long periods stocks tend to have higher returns than bonds.    However, the stock portfolios of the two funds differ.  The stock portfolio in VWELX is broadly diversified but does not track a specific index.   The stock portfolio in VFIAX tracks the S&P 500.  VWELX was able to get higher returns than VFIAX because its stock portfolio outperformed the S&P 500 while the bond portfolio lowered risk.   It also did not hurt that interest rates fell and bond prices rose in this time period.

Comment Two:  The stock-only portfolio was much more risky than the combined bond-stock portfolio.   This is evidenced both by the lower standard deviation and the higher minimum return.  The minimum return statistic measures the worst-outcome return.  The worst-outcome return for the combined stock-bond portfolio is around 92 percent higher than the worst-outcome return for the stock-only portfolio.

Tests of equal variances for returns:

A test of the hypothesis that the variances of return for the two portfolios are equal was conducted.   The F-statistic comparing the ratio of the two standard deviations was 1.63, which is significantly different from 1.0.    The hypothesis that the two variances are identical is rejected.

Tests of equal mean returns:

A test of the hypothesis that the mean returns for the two portfolios are equal was conducted.  The t-statistic for this hypothesis test was 12.9.   The hypothesis of identical means is rejected.

Technical Note:  I used STATA to make the calculations in this note.  Period one and period two data were placed in separate data sets.   The N to N merge provides the 2304 outcomes.

Concluding Thought:  The practice of presenting return numbers on investment funds for a few arbitrarily chosen holding periods is, in my view, not very useful.    The holding periods are arbitrary and subject to manipulation.   There is no measure of risk.

The technique presented here relies on many possible outcomes defined by different purchase and sale dates.   The multiple outcome approach allows for the presentation of risk measures.

The note shows that the performance of the VWELX fund was exceptional in this period.

A Note on Use of Geometric and Arithmetic Averages of Daily Stock Returns

Question:  The table below has price data and daily return data for Vanguard fund VB.   Calculate the arithmetic and geometric averages of the daily return data.   Show that the geometric average accurately reflects the relationship between the initial and final stock price and the arithmetic average does not accurately explain this relationship.

Daily Price and Returns For Vanguard 

Fund VB

Date Adjusted Close Daily Return
7/1/16 115.480674
7/5/16 113.99773 0.987158509
7/6/16 114.744179 1.006547929
7/7/16 114.913373 1.001474532
7/8/16 117.202487 1.019920345
7/11/16 118.128084 1.007897418
7/12/16 119.451781 1.011205608
7/13/16 119.10344 0.997083836
7/14/16 119.262686 1.001337039
7/15/16 119.402023 1.00116832
7/18/16 119.63093 1.001917112
7/19/16 119.202965 0.996422622
7/20/16 119.959369 1.006345513
7/21/16 119.481646 0.996017627
7/22/16 120.297763 1.00683048
7/25/16 120.019083 0.997683415
7/26/16 120.616248 1.004975584
7/27/16 120.347522 0.997772058
7/28/16 120.536625 1.001571308
7/29/16 120.894921 1.002972507
8/1/16 120.735675 0.998682773
8/2/16 119.12335 0.986645828

Analysis:   The table below presents calculation of the two averages and the count of return days.  The product of the initial value of the ETF, the pertinent average and the count of return days is the estimate of the final value.   Estimates of final ETF value are calculated for both the arithmetic average and the geometric average and these estimates are compared to the actual value of the stock on the final day in the period.

Understanding The Difference Between Arithmetic Mean and Geometric Mean Returns
Statistic Value Note
Arithmetic Average of Daily Stock Change Ratio 1.001506208 Average function
Geometric Average of Daily Stock Change Ratio 1.001479966 Geomean function
Count of Return Days 21 Count Function
Estimate of final value based on arithmetic average 119.1889153 Initial Value x Arithmetic Return Average x Count Days
Estimate of final value based on geometric average 119.12335 Initial Value X Geometric Return Average x Count Days
Ending Value 119.12335 Copy from data table

There is another way to show that the daily return should be modeled with the geometric mean rather than arithmetic mean.  The average daily return of the stock is (FV/IV)(1/n) – 1 where FV is final value and IV is initial value and n is the number of market days in the period, which for this problem is 21.

Using this formula we find the daily average holding period return is 0.001479966.  Note that 1 minus the geometric mean of the daily stock price ratio is also 0.001479966.

The geometric mean gives us the correct holding period return.

Expected Profit and Risk with Random Transaction Dates

Profit and risk when there are four random purchase dates and four random sale dates

Question:   In 2013 a person buys QQQ the high tech ETF) on one of four randomly selected dates determined by when the broker arranges a meeting.   I

The person who bought the QQQ shares in 2014 got fired in 2015.   As soon as the person was fired he realized he needed cash so he called his broker and said “SELL QQQ” The firing is a random event independent of the market and out of control of the person, which occurred on one of four dates.

The four potential purchase and four potential sales dates for the QQQ transactions are presented below.

Information on Potential Purchases and Sales of QQQ
Potential Purchase Date Purchase Price QQQ Quantity purchased $25,000/Price Potential Sale Date Sale Price
20-May-14 88.0 284.1 5-Jan-15 101.4
7-Jul-14 95.1 262.9 8-Aug-15 110.5
7-Aug-14 94.2 265.4 24-Aug-15 98.5
10-Sep-14 100.1 249.8 5-Nov-15 114.7

The person spends $25,000 on the purchase of QQQ in 2014 and sells all shares in 2015.

Assume no dividends are paid.

What are all possible profit outcomes from the purchase and sale of the QQQ securities?

What is the expected profit?

What is the variance of profit?

Analysis:  The number of share purchased is $25,000 divided by the purchase price; hence the purchase price determines the number of shares purchased.

Tabulation of Number of Shares Purchased
Potential Purchase Date Purchase Price QQQ Number of shares purchased
20-May-14 88.0 284.1
7-Jul-14 95.1 262.9
7-Aug-14 94.2 265.4
10-Sep-14 100.1 249.8

Revenue received after the sale is price at time of sale times the number of shares owned.

Profit after the sale is revenue minus the $25,000 initial investment.

There are four possible purchase dates and four possible sale dates.   The purchase and sale dates are independent so there are a total of 16 possible equally likely combinations of sale and purchase dates.   The probability of each purchase/sale combination is 0.0625 (0.25*0.25).

The profit calculation for the 16 purchase-sale combinations is presented in the table below.

Potential Profit Calculation for Four Purchase Dates and Four Sale Dates
Comb # Probability Purchase Date Sale Date Number of Shares Owned Sale Price Profit
1 0.0625 20-May-14 5-Jan-15 284.1 101.4 $3,807
2 0.0625 20-May-14 8-Aug-15 284.1 100.5 $3,552
3 0.0625 20-May-14 24-Aug-15 284.1 98.5 $2,984
4 0.0625 20-May-14 5-Nov-15 284.1 114.7 $7,586
5 0.0625 7-Jul-14 5-Jan-15 262.9 101.4 $1,656
6 0.0625 7-Jul-14 8-Aug-15 262.9 100.5 $1,420
7 0.0625 7-Jul-14 24-Aug-15 262.9 98.5 $894
8 0.0625 7-Jul-14 5-Nov-15 262.9 114.7 $5,152
9 0.0625 7-Aug-14 5-Jan-15 265.4 101.4 $1,911
10 0.0625 7-Aug-14 8-Aug-15 265.4 100.5 $1,672
11 0.0625 7-Aug-14 24-Aug-15 265.4 98.5 $1,141
12 0.0625 7-Aug-14 5-Nov-15 265.4 114.7 $5,441
13 0.0625 10-Sep-14 5-Jan-15 249.8 101.4 $325
14 0.0625 10-Sep-14 8-Aug-15 249.8 100.5 $100
15 0.0625 10-Sep-14 24-Aug-15 249.8 98.5 -$400
16 0.0625 10-Sep-14 5-Nov-15 249.8 114.7 $3,646
Min -$400
Max $7,586
Range $7,986

The minimum profit is -$400.   The maximum profit is $7,985.

The expected profit is obtained by taking the dot product or the sumproduct of the probability vector with the profit vector.   The variance was obtained from the computational formula.

Var (Profit) = E(profit2) – E(Profit)2

For a discussion of these calculations see the previous post.

The expected value and variance or profit from the purchase of QQQ on one of four dates in 2014 and the sale of QQQ on one of four dates in 2015 are presented below.

Expected Profit and Variance of Profit Calculations
E(PROFIT) 2555.4
E(PROFIT2) 11036765.0
E(PROFIT2)-E(PROFIT)2 4506556.2
E(PROFIT-E(PROFIT))2 4506556.2

Financial Discussion:

The purchaser of QQQ or any stock that buys randomly and is forced to sell because of random events unrelated to the market bears substantial risk compared to an investor with enough liquid assets who will not need to sell in an emergency.   Investors would be wise to consider the level of the market and their ability to hold through downturns prior to selling.  The experts say that stock market returns beat returns on other securities over the long haul.  But this investor was only able to hold for a year.

Outcomes could have been worse.   The broker put the investor in QQQ a relatively diversified ETF that focuses on tech stocks.  Had the broker put his client in one particular stock (say IBM) and the investor was forced to sell he would have realized a large loss.