## 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:

https://financememos.com/2017/10/11/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.

## Discussing Buy Versus Rent Calculators

Discussing Buy Versus Rent Calculators

Realtor groups have created a number of on-line calculators that attempt to provide an objective view of the advantages of buying a home versus renting a home.  The link to one such calculator is presented below.

http://www.realtor.com/mortgage/tools/rent-or-buy-calculator/

First, I describe the calculator.   Then I critique it.

Description of Calculator:

The simple version of the buy/rent calculator at www.realtor.com allows one to put in an address and get financial estimates for renting or buying.  The more advanced and interesting version allows a consumer to select assumptions on costs of buying and cost of renting.

The key assumptions on the cost of buying involve home price, down payment mortgage term, buying costs, selling costs, house appreciation, real estate taxes and miscellaneous homeowner fees.

The most important renting costs include the initial rent and the yearly appreciation in rent.

Other key assumptions include the exemption of  \$500,000 on capital gains in housing, an investment return, and an inflation rate.

Based on the inputted assumptions the model provides an estimate of the amount of time that it takes for buying a home to be cheaper than renting a home.

The model at www.realtor.com assumes that buying costs are 4.0 percent of the purchase price and selling closing costs are 6.0 percent of the final sales price.   Due to transaction costs associated with home purchases, renting will be less expensive than buying for people who stay in a house for a short period of time.   The output of the model is the number of years it takes for buying to be less

Comments on the calculator at www.realtor.com

Comment One:  Often realtors and bankers persuade young homebuyers to use available cash for a down payment rather than immediately retire consumer or student debt.  The model does not have an option to explicitly consider the impact of credit card debt or student debt on the buy versus rent outcome.   The model does require input on the assumption of investment returns.   One way to model the impact of keeping debt is to increase the investment return assumption so that it equals cost of credit cards and student loans.   It would be useful if the model allowed for separate assumptions on investment return and the cost of existing debt.

Comment Two:   I modified one example to consider the breakeven point of a transaction with a 15-year FRM at current interest rates.   I found that buying was preferable to renting after a 6-year period for the 15-year FRM compared to 8 years for the 30-year FRM.   Essay Four provides more information on mortgage choice and lifetime savings.

Comment Three:   The model cannot be easily modified to allow for interest rate uncertainty associated with adjustable rate mortgages.

Comment Four: The model requires an assumption of average annual growth in house appreciation over the entire period and does not consider issues related to the uncertainty of future house appreciation.  House prices do not appreciate in a steady or reliable fashion.   The realtor’s model would have severely overestimated the value of buying a home during the 2004 to 2009 time period and would have underestimate returns from purchasing in 2011 or 2012.  The argument that housing prices would continue to rise was made quite strenuously in 2007 and was used to motivate unrealistic price appreciation assumptions in the breakeven analysis.

The house price appreciation assumption is usually based on what the analyst expects will occur.   An alternative approach would involve basing this parameter on the certainty equivalent.   A certainty equivalent is the guaranteed return that someone would accept rather than take a risk on a higher but uncertain return.

Comment Five. Many people are forced to move because of a new job or divorce.   The rent versus buy calculator does not allow for economic costs associate with moving when house prices fall and house equity turns negative.   Nor does the buy versus rent calculator consider economic costs associated with negative equity that make it difficult for a home buyer to refinance should interest rates fall.

The more relevant question not answerable from this calculator is it better for a person to buy now or reduce debt and buy in a couple of years.

Comment Six:  Often realtors will expect home sellers to put additional investments into the property prior to selling the home.  (Most recently in many neighborhoods realtors are pushing home sellers to install granite kitchen tops.)   The model does not include an option to consider likely upgrade costs.  It may be able to correct for this problem by reducing the price appreciation assumption in the model.  However, the need for upgrades appears to differ widely across properties.

Comment Seven:: The buy-sell calculator can also be used to evaluate mortgage properties financed with FHA loans.   The FHA loan program is geared for relatively small mortgages.  The program has a loan limit that varies across counties and can change over time.   The FHA loan program allows for down payments as low as 3.5% FHA loan costs include mandatory mortgage insurance premiums, part of which is paid up front.  Due to the insurance premiums the cost of the FHA loan is often one percent point higher than the cost of conventional loans.  Most often, the number of years it takes for a home buyer to break even on an FHA loan program will be substantially higher than the number of years it takes to break even on a transaction financed with a conventional loan.   Not surprisingly, the use of real estate break- even calculators is usually illustrated with conventional loan examples rather than FHA loan examples.

Comment Eight: The assumption regarding the rate of appreciation of rents has a major impact on the buy versus rent decision.    A larger percent of people are choosing to rent rather than buy consequently more rents are continuing to rise often at a rate that exceeds the increase in the value of the home.   In some markets it may be legitimate to assume a higher increase in rents than home prices.  This alternative assumption might persuade more people to buy rather than rent.

Comment Nine:  Realtors often argue that a house purchase should occur now rather than later because macroeconomic conditions are about to change.   Over the last three or four years realtors have argued that people should buy because the FED is about to raise interest rates.  An increase in interest rates induced by Fed policy would increase the cost of interest on a home but might also lower house prices.

The Fed will eventually raise interest rates but even Nobel Prize winning economists are confused about when this will happen.  Potential homebuyers should not rely upon the interest rate forecasts of realtors when determining whether or not or buy or rent a home.

Concluding thoughts on the Limitations of Buy Versus Debt Calculators:  My comments suggest that for a wide variety of reasons buy versus debt calculators often overstate the case for buying rather than renting a home.    The approach relies on subjective assumptions on a wide variety of economic variables.   Assumptions on the most crucial variable – the future growth of housing prices have been grossly inaccurate in the past.

The one factor that favors buying over renting in the current environment is that stock prices are currently at historic highs and long term interest rates are at historic lows.   I suspect that based on the current market conditions returns on real estate will outpace returns on financial assets in the near future.  Hence an assumption of a low future return on financial assets might be justified at this time.

The buy versus rent calculator does not accurately measure the benefits of delaying a home purchase until consumer debt and student loans are substantially reduced or eliminated.   Nor does the model allow for active consideration of costs, which might be incurred if a young worker with little initial house equity is forced to sell a home in order to take advantage of a new job opportunity.  Usually younger households will be much better off by delaying the home purchase and using all available funds to retire student loans and consumer debt.

## 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) Earnings (\$ 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).

Analysis:

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) Earnings (\$ 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.

## A House Equity and Mortgage Payoff Spreadsheet

A House Equity and Mortgage Payoff Spreadsheet:

Question:   A person buys a house and plans to either sell and move or pay off the mortgage in twelve years.

The person is considering taking out a 15-year or a 30-year fixed rate mortgage.

The assumptions on the home purchase, house equity growth, the cost of selling and moving, and the cost of funds for the payoff of the mortgage are presented in the table below.

Table One: Assumptions for 30-year vs 15-year FRM Comparison:

 Label 30-year FRM 15-year FRM Purchase Price of House \$500,000 \$500,000 Down payment percentage 0.9 0.9 Initial Loan Balance \$450,000 \$450,000 Mortgage Term 30 15 House appreciation rate 3.0% 3.0% Mortgage Interest Rate 4.0% 3.3% Years person owns house 12.00 12.00 Cost of selling and moving to a new home as % of house value 9.0% 9.0% Tax Rate on Disbursements from 401(K) Plan 30.0% 30.0%

Create a spreadsheet that provides estimates of house equity after the sale and move or mortgage payoff amounts after twelve years when the house buyer uses a 30-year FRM and when the house buyer uses a 15-year FRM

Base your mortgage payoff calculation on the assumption that the source of funds for the mortgage payoff are fully taxed funds from a 401(k) plan.

Spreadsheet:

http://wp.me/a2WYXD-4i

Results:

The results for the comparison of the 15-year and 30-year FRM for the assumptions presented in table one are presented in Table 2.

Table Two: Results for the 30-year vs 15-year FRM Comparison:

 30-year FRM 15-year FRM House Equity after Selling and Moving Costs \$318,303 \$540,109 Forecasted Mortgage Payoff Amount -\$472,025 -\$155,160

Observations on the 30-year vs 15-year FRM comparison:

The person taking out the 15-year FRM mortgage has around \$222,000 more in house equity at the end of the 12-year holding period.

The mortgage payoff calculation when funds are disbursed from a 401(k) plan includes tax on the disbursements.   Inclusive of the tax bill, the mortgage payoff amount is \$317,000 higher for the buyer who uses the 30-year FRM than for the buyer who uses the 15-year FRM.

Other Applications for the House Equity or Mortgage Payoff Spreadsheet:

Modify the mortgage payoff calculation to allow for a situation where funds for the mortgage payoff are obtained from three sources – (1) a savings account, (2) sales of common stock, and (3) disbursements from a 401(k) plan.   Treat tax rates as an endogenous variable in the new model.

Compare results for both mortgage types under the 90% LTV assumption to results under an 80% LTV assumption.

Run the model on 15-year and 30-year FRMs for holding periods ranging from 1 to 15 years.   How does the advantage of the 15-year FRM vary with holding period?

Authors Note:   This problem was discussed further in the post below.

Essay Nine: Retire Mortgage Debt or Accumulate in Your 401(k) Plan:

https://financememos.com/2015/10/09/essay-nine-retire-mortgage-debt-or-accumulate-in-your-401k-plan/

Essay nine points out that many financial advisors stress accumulation of wealth in 401(k) plans rather than mortgage balance reductions even when their clients are nearing retirement.  The major banks employing the same financial advisors issue mortgages and sponsor 401(k) plans.   As a result, the interests of the financial advisors and the interests of their clients are not automatically aligned.

This approach can backfire when stock markets underperform nearing retirement.

During working years. the tax code favors people with large mortgages and people who are contributing to their 401(k) plan.  However, after retirement the person who must disburse funds from a 401(k) plan often has a hefty tax bill.

## 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

=SUMPRODUCT(a1:a29,d1:d29)

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

http://www.dailymathproblem.com/2017/05/price-earning-ratios-for-portfolios.html

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?

Analysis:

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.

Observations:

• 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.

Comments:

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.

http://dailymathproblem.blogspot.com/2015/11/expected-value-and-variance-of-share.html

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.

## Measuring Returns for Different Investment-Consumption Patterns

Measuring Returns for Different Investment-Consumption Patterns

Question:   An investment advisor tells his client to invest \$1,000 per month in VFIAX (Vanguard S&P fund) for five years.   The person will then live off the proceeds in this fund for 36 consecutive months.

Calculate the return on assets from this investment/consumption plan for two different start dates – January 1, 2002 and January 1, 2003.

What is the NPV of investment returns from this investment strategy/ consumption plan on the same start dates?

What should investors who are planning to save for five years and spend for three years learn from this example?

Mutual funds and ETFs tend to advertise holding period returns based on specific purchase dates and specific sale dates.   These returns are based on the price of securities on two dates only.   What does the example presented here tell you about the usefulness of two-period return statistics reported by mutual funds?

Methodological Note:  The shares purchased each month are \$1,000/PVFIAX where PVFIAX is the price of the ETF.   I sum over 60 months to get the total shares purchased, which I will denote TSHARES. The formula for cash inflow for the 36 months are (1/36)*TSHARE*PVFIAX.

The cash inflow/outflow column and the date column are inputted into the XIRR function in Excel to give the IRR of the inflows/outflows on these particular dates. The XNPV function gives net present value of the cash flows.

Analysis:

The value of VFIAX reached its pre financial crisis high in 10/2007 and reached its crisis trough in 02/2009.   Hindsight is 20/20 but it appears as though diversification prior to the downturn would have been beneficial.

What follows are return calculations for the two scenarios.

Results are in the table below.

 Returns for Two Investment/Consumption Scenarios Invest Period Consumption Period IRR NPV 2002/2006 2007/2009 12.04 \$15,766 2003/2007 2008/2010 2.98*e-9 \$801

Observations:

• The person who stopped saving in December 2006 did fairly well despite the financial crisis.The IRR for this investor was 12.04 %.   The NPV of the investments was \$15,766.   (NPV calculation assumes a5 percent cost of capital.)
• The person who stopped investing in December 2007 realized a return only slightly higher than 0 percent.The NPV of this person’s investment was around \$800.

Discussion of Investment Strategy:

In my view, a 100 percent VFIAX strategy is unwise for an investor with this type of investment and consumption period.

How to fix this problem is a more difficult question.  It is important to note that the strategy of putting 100 percent of funds in VFIA for an investor with a start date of January 1 2009 or January 1, 2010 did quite well.

529 plans offer life-cycle funds that drift towards a more conservative investment as the person nears the date where he must spend money.   Lifecycle funds would have done reasonably well for both of the scenarios considered here.  However, the life-cycle approach creates miserable results when the market does poorly in the first few years of the investment period and then rebounds.

My view on how to solve this problem is evolving.  A 60/40 (stock/bond) portfolio would have done well in these time periods but I don’t believe that it will work in the next crash.  Interest rates are now very low and I expect in the next crisis bonds and stocks will crash together.   Perhaps allocating some resources into an inflation-indexed bond fund would help balance returns during the next crisis.

The trend in investment is toward investment in passively managed funds like the ones offered by Vanguard.    This is at best a partial solution.   Investors need help in allocating money across several passively managed funds.  This includes advice on initial allocations and reallocation over time.

I believe there is a need for an actively managed fund that invests exclusively in passively managed funds and reallocated assets across funds as market conditions change.

Note on traditional holding period statistics:  The value of VFIAX in January 2002 was 17.9.  In December of 2010 the value of VFIAX was 39.5.   The return for this 7.9 year  holding period was at 10.5%.

 Holding Period Calculation Jan-03 17.9 Dec-10 39.5 Holding Period in Years 7.92 ROR 10.5%

However a person who started investing in January 2003 and started spending in January 2008 earned squat!

The mutual funds can legally and honestly report great eight-year or ten-year holding return but their clients aren’t doing particularly well.

Such a surprise!

## How much house can a student borrower qualify for?

How much house can a student borrower qualify for?

This answer depends on the maturity of the student loan.

Situation:

Consider a person with a \$100,000 student debt.

• The person can either pay the debt back over a 10-year period or a 20-year period.
• The student loan is this person’s only consumer debt.
•  The person earns \$80,000 per year.
• The student loan interest rate is 7.0 percent.
• The mortgage interest rate is 4.0 percent.
• The mortgage term is 30 years.

Questions:

• How much mortgage can the person qualify for if the person keeps the student loan at 10 years?
• How much mortgage can the person qualify for if the person changes the student loan term to 20 years?
• What is the increased cost of the student loan payments involved by switching from a 10-year to 20-year student loan?

Answer:   I developed a spreadsheet that calculates the maximum allowable mortgage this person can qualify for.

In order to qualify for a mortgage two conditions must hold.

• Monthly mortgage payments must be less than 28% of income.
• Monthly mortgage and consumer loan payments must be less than 38% of income.

The procedure used to calculate the allowable mortgage is as follows:

• First, I calculate the maximum allowable mortgage payment based on zero consumer debt.   This value is 28 percent of monthly income.
• Second, I calculate the maximum allowable mortgage payment consistent with mortgage payments and consumer debt payments equal to 38 percent of income.   This is done by backing out the student loan and allocating the rest to mortgage debt.
• Third, I insert mortgage interest rate, term and payment info into the PV functions to get the mortgage amount
• Fourth, The allowable mortgage is the minimum of the mortgage totals consistent with the two constraints.

The calculations for the two situations presented in this problem are presented in the table below

 Mortgage Qualification Example for Borrower with Student Debt row # Student Loan Information Note 1 Student loan Amount \$100,000 \$100,000 Assumption 2 Interest Rate 0.07 0.07 Assumption 3 Number of Payments 120 240 Assumption 4 Student Loan Payment \$1,161 \$775 From  PMT Function Mortgage Information 5 Rate 0.035 0.035 6 Term 360 360 Income Assumption 7 Income \$80,000 \$80,000 Assumption 8 Constraint One:  Maximum monthly mortgage payment consistent with this income assumption \$1,867 \$1,867 28% of monthly income 9 Constraint Two:  Maximum monthly consumer and mortgage payments consistent with income \$2,533 \$2,533 38% of monthly income 10 Maximum mortgage consistent with constraint one. \$415,697 \$415,697 pv of mortgage rate number of periods, and pmt where mortgage rate and payments are assumptions baed on the market and product chosen and payment is max allowable given   income 11 Allowable mortgage payment consistent with constraint two given required student debt \$1,372 \$1,758 Row 9 minus Row 7 12 Max mortgage consistent with borrowing contraint two. \$305,593 \$391,505 Use PV function with rate and term set by market and product and payment the amount of mortgage payment after required consumer payments 13 Allowable mortgage debt \$305,593 \$391,505 Minimum of Row 10 and Row 12

An increase in the term of the student loan from 10 to 20 years increases the size of a mortgage a household can qualify for from \$305,000 to \$391,000.

Getting the extra mortgage is not cheap.  The increased student loan term causes total student loan payments to go from \$139.000 to \$186,000.

Concluding thoughts:  Most people who have \$100,000 in student debt will have to refinance the student loan if they are going to buy a house.