Using contingency tables to measure systematic risk.

Contingency Tables Versus Beta

Question:  Below are two contingency tables describing the relationship between daily stock prices and daily changes in the S&P 500 for two companies.   One of the companies is a high-beta firm.   The other company a low-beta firm.

Movements in Stock Price Versus S&P Company One
Stock Close<Prior Low Stock Between Prior Low and High Stock Close>Prior High
S&P Close < Prior Low 12 18 4
S&P Between Prior Low & High 26 80 28
S&P > Prior High 5 31 48

 

Movements in Stock Price Versus S&P Company Two
Stock Close<Prior Low Stock Between Prior Low and High Stock Close>Prior High
S&P Close < Prior Low 10 21 3
S&P Between Prior Low & High 19 68 47
S&P > Prior High 14 48 22

The rows of the contingency table were formed by comparing the S&P close to the value of the low or high of the S&P on the previous market day.

The columns of the contingency table were formed by comparing the closing stock price to the low or high of the stock price on the previous day.

What table depicts the high beta stock?    What table depicts the low beta stock?  Defend your answer?

Use Stata to calculate Kendalls Tau B and Spearman’s rank correlation coefficient for the two firms.

How do these non-parametric statistics differ for these two firms?

Discuss implications of these results for future research.

Analysis:

 The count of observations on the diagonal of the contingency tables starting from the top right cell to the bottom left cell of the table represent the number of instances where stock prices move in tandem with the market.

The proportion of observations on this diagonal is 55.6% for company one compared to 39.7% for company 2.

Beta is the most common measure of the movement of stock prices with the market so it is relatively clear that company one has a higher beta than company two.

In fact, company one is Apple and company two is Duke Power.   According to yahoo finance, the beta for Apple is 1.40 and the beta for Duke Power is 0.01.   The values of the proportions of observations on the diagonal of the contingency table appear to correspond to yahoo finance beta estimates.   (Sample size of two is admittedly very small.)

Kendall’s Tau and Spearman’s rho are two statistics commonly used to measure the association between row and column variables of a contingency table.    The table below lists these two statistics for these two companies

Comparison of Non-Parametric Measures of Association Between Stock Price Movements and Market Movements
Apple Duke Ratio Apple to Duke
Kendall’s

Tau

0.3672 0.0247 14.9
Spearman’s

Rho

0.3980 0.0658 6.0

Concluding Thoughts:   Financial analysts use beta to measure the price of a stock.   The preliminary results presented here indicate that contingency tables and non-parametric statistics can be used to measure the association between movements company stock price and the overall market.   More research will be available on this topic shortly.

Authors Note:   I am planning the launch of a monthly financial and economics newsletter.  Please follow this blog to obtain advanced notice of this product.

 

Buy or Rent a House?

  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.

In my next post, I will show that statistical tests based on samples of the ratio of the difference between the market cap and earnings to market cap are more useful than statistical tests based on PE ratios.

 

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!   

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.

A note on the difference between geometric and arithmetic averages

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.

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.

 

 

 

 

 

 

 

 

 

 

Student debt and qualifying for a mortgage 

Student debt and qualifying for a mortgage 

Excel Topics:  PMT function and Spreadsheet design

Question:  A person graduates from college and graduate school with $100,000 in student debt.   The interest rate on a 10-year student loan is 5% per year.   The person wants to buy a house that costs $300,000 with a 90% LTV loan. The home mortgage interest rate is 4.5% on a 30 year FRM.

Assume that in order to qualify for the house the person must meet two conditions.

Constraint One:  The ratio of mortgage interest to income must be less than 0.28.

Constraint Two: The ratio of total interest (mortgage and non-mortgage) interest must be less than 0.38.

How much income does this person need to qualify for a loan on this house?

Why might student debt have a smaller impact on the purchase of a $700,000 home than the purchase of a $300,000 home.

Analysis:   The analysis for the $300,000 home is laid out in the table below.

Mortgage Qualification Example for Borrower with Student Debt
Note
Student loan Amount $100,000 $0 Assumption
Interest Rate 0.05 0.05 Assumption
Number of Payments 120 120 Assumption
Student Loan Payment $1,061 $0 From  PMT Function
House Amount $300,000 $300,000 Assumption
LTV 0.9 0.9 Assumption
Loan Amount $270,000 $270,000 LTV * House Amount
Intrerest Rate 0.045 0.045 Assumption
Number of Payments 360 360 Assumption
Mortgage Payment $1,368 $1,368 From PMT Function
Total Loan Payments $2,429 $1,368 Sum Payments
Monthly Income Constraint One $4,886 $4,886 Student Loan Payment divided by 0.28
Monthly Income Constraint Two $6,391 $3,600 Mortgate Payment Divided by 0.38
Required Monthly Income $6,391 $4,886 Max of income over both constraints
Required Annual Income $76,696 $58,631 12* Max Income

Observations Pertaining to the $300,000 home for a person with and without student loans

A person with no student debt could qualify for this mortgage with an annual income of $58,630.

The person with the student debt needs an annual income of $71,585.

The impact of student debt on purchases of a larger home:   The allowable mortgage is determined by two constraints one involving mortgage debt only and the other involving the sum of mortgage and consumer debt.   When the mortgage debt is very large, constraint one (the mortgage debt constraint) will be the binding constraint.

Download the student debt and mortgage qualification spreadsheet by clicking below:
Continue reading Student debt and qualifying for a mortgage 

Understanding the Four Percent Rule

Question:   Is the 4.0% rule an appropriate guideline for determining the amount of savings a retiree should spend each year?

Background on the 4.0% rule:  Under the four percent rule (as I understand it ) the retiree’s expenditure in her first year of retirement is four percent of wealth in certain accounts.  It is more difficult to apply the 4.0% rule to total household wealth because house equity, a major component of wealth is not liquid.  (The application of the 4.0% rule to total wealth including house equity would at some point require the sale of the home.)

Whether strict adherence to the 4.0% rule leads to a smooth, stable, and sustainable consumption pattern for the household does  not depend solely on average asset returns or average inflation.   The timing of returns and the timing of inflation are also important.

A sharp decrease in returns at  the beginning of retirement could lead to a relatively quick depletion of assets if not accompanied by a decrease in spending.  By contrast, a sharp increase in returns at the beginning of retirement could allow retirees to spend more than allowed or provide a bequest to heirs.

The four percent rule could prove inadequate for workers who are planning to retire in 2017 or 2018 for two reasons.   First, both nominal and real interest rates are very low.   Second, stock valuations are currently extremely high.  Portfolio returns will fall precipitously if interest rates rise and stock prices fall.

Illustrating the 4.0% rule:  The simulations of whether the four percent return will lead to a successful retirement rely on two assumptions — the rate of return on assets and the inflation rate. Our model assumes that the retiree starts spending four percent of assets at retirement and continues to spend at this level in real terms adjusted for inflation.   The output of the model is the number of years (if ever) before the retiree spends all retirement funds.

We consider four scenarios — (1) 2.00% returns and 3.0% inflation all years, (2) 4.0% returns and 3.0% inflation rate for all years, (3) a -20% return the first year followed by 2.0% returns and 3.0% inflation, and (4) -20% return the first year followed by 4.0% return and 3.0% inflation.

Note that scenario two is better than scenario one because returns are higher in the second scenario.

Scenario three is similar to scenario one except for the fact that the market collapses in the first year.

Similarly scenario four is similar to scenario two but for the first-year collapse in asset prices.

The table below presents the number of years it takes for the person to deplete all assets using the four percent rule.

Adequacy of resource for 4% rule under four scenarios
Shock Return Inflation Rate Year Balance goes to $0
None 2.00% 3.00% 23
None 4.00% 3.00% 30
-20% first year 2.00% 3.00% 19
-20% first year 4.00% 3.00% 23

One bad year of returns in the first year of retirement resulted in a 7 year reduction of years with assets in the high return situation (simulation two minus simulation four) and a reduction of four years of assets in the low-return scenario (simulation one minus simulation three.)

The number of years prior to total depletion of assets among the four scenarios ranges from 19 years (scenario three) to 30 years (scenario two.)

 

Concluding Thoughts:

I suspect that these calculations understate the financial risk associated with adherence to the 4.0% rule, in general and in the current economic and financial situation.

The bad scenarios presented here involve a first year of retirement where the portfolio falls by 20 percent.   The collapse could be larger or more last more than one year.

President Trump believes that the the situation in Korea is the calm before the storm.   The same could be said for our financial markets and our economy.

The market is peaking.   The financial advisors and experts want to keep you fully invested argue that the market can run upwards for at least a few more years.   The more reputable analysts acknowledge that the valuations are high.   Long term interest rates remain low but when they rise (and they eventually will rise) bond prices fall.  The simultaneous collapse of bond prices and the reduction in stock valuations could will eventually lead to large losses at the beginning of retirement for a cohort of workers.

Some analysts suggest that investors who use the 4.0% rule should maintain a larger portion of their portfolio in equites.  I disagree.  The worse case scenario for the 4.0% rule involving poor stock and bond returns in a period of inflation actually occurred during the 1970s when inflation rose and stocks fell.   Luckily retirees in the 1970s had traditional pension plans that were not tied to the market.

President Trump believes that the the situation in Korea is the calm before the storm.   The same could be said for our financial markets and our economy.   I don’t believe the current calm before the storm will last.

 

 

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.

Interestingly, most financial analysts advise their clients to add more to their 401(k) plan rather than pay off their mortgage.  More discussion of this problem can be found below.

Reduce Mortgage or Add to Your 401(k) Near Retirement?