Buying a House Versus Renting. Floyd Vest

Transcription

1 Buying a House Versus Renting Floyd Vest People often face the financial decision of whether to buy or rent. The following is an analysis of this question for a family considering houses in the $80,000 range. Based on the author s location in December of 2011, such a price would buy a threebedroom house with 1½ baths, den, kitchen, attached garage, brick, air conditioning and heating, slab foundation, and copper plumbing. From examination of rental advertisements, this house would rent for about $1000 per month. (Readers are encouraged to do research according to their own interests.) After consulting with a real estate agent, a family learned that for an $80,000 house, they could make a $5000 down payment with closing costs of $1000. Because of their small down payment, mortgage insurance is required. Although billed separately, the combination would amount to about 5% mortgage interest. The monthly payments on a 30year mortgage for $75,000 would be $ per month, totaling $ for the year. (See the Side Bar Notes for the calculation.) With the decline in the price in housing in recent years, they thought that if they sold the house after five years they might get $80,000, the original price. They would get their down payment back but lose the $1000 closing cost plus the five years short term savings interest at 1%, amounting to 1000( ) 5 = $1051. There would be expenses to sell such as a 6% brokerage fee of 0.06(80,000) = $4800 and closing costs to sell estimated at $1500. They would lose the yearly interest on the $5000 down payment which is 5000( ) = $ Adding these costs, we have Closing costs to purchase and interest $ Realtor s fee Closing costs to sell Lost interest on the $5000 down payment $ The yearly cost of ownership could be estimated as: Payments for principal and interest $ Maintain and replacement = Taxes and insurance ($ $900) Adding in the above expenses at per year $10, Comparing this with $12,000 per year for rent and the various personal preferences, inconveniences, and risks, renting may look more favorable. But let us do a more careful analysis. Buying a House Versus Renting December,

2 A Buy versus Rent Spreadsheet. We will assume 0% inflation in expenses and 0% appreciation in the value of the property. In Table 1, for the Renter, Column 2 gives Maintenance and Replacement expenses of $50 per month or $600 per year. Column 1 gives the Rental cost of $12,000 per year, and Column 3 gives Total rental cost of $12,600 per year. (A prospective buyer should use a conservative estimate of rent on equivalent property based on actual in the field research.) For the buyer, Column 5 gives the yearly mortgage interest expense, and Column 6 gives the yearly Principal reduction on the mortgage. Column 7 gives the yearly Maintenance and Replacement expense of $1800 per year, Column 8 gives the annual property taxes of $1200, and Column 9 the Insurance of $900 per year. Column 10 is the Yearly cost of homeownership with (10) = (5) + (6) + (7) + (8) + (9), of $ per year. Column 10.1 would be Tax Savings, which we will not claim. (See the Side Bar Notes.) Column 10.2 would be After tax cost for ownership. Year Rent Renter's Maintenance Expense Table 1 Buy verses Rent Cash Flow Spreadsheet (1) (2) (3) (5) (6) Total Rental Buyer's Interest Cost Expense Principal Reduction 1 $12,000 $600 $12,600 $ $ $12,000 $600 $12,600 $ $ $12,000 $600 $12,600 $ $ $12,000 $600 $12,600 $ $ $12,000 $600 $12,600 $ $ (7) (8) (9) (10) = (5)+(6)+(7)+(8)+(9) Year Buyer's Maintenance Expense Taxes Insurance Buyer's Yearly Cost 1 $1800 $1200 $900 $ $1800 $1200 $900 $ $1800 $1200 $900 $ $1800 $1200 $900 $ $1800 $1200 $900 $ (11.5) (12) (13) (14) = (13) (12) Year Investment Buyer's Yearly Cost Total Renter's Cost Net Difference 1 $6000 $ $12,600 $ $ $12,600 $ $ $12,600 $ $ $12,600 $ $ $12,600 $ Buying a House Versus Renting December,

3 Columns 11.5, 12, 13, and 14 bring together the investment and comparison of owning and renting. Column 11.5 gives the $6000 investment for Year 0, Column 12 gives Total yearly cost of ownership, which is $ for Years 1 to 5. Column 13 gives Total rental cost of $12,600 per year. Column 14 gives the Net difference in yearly costs of $ per year in favor of ownership. We will now move to Column 15, which considers calculations resulting from the sale of the house at the end of Year 5. Selling price $80, Realtor fee at 6% of $80, Closing costs to sell Paying off the mortgage 68, (See the Side Bar Note for calculation.) $ as net proceeds from sale after expenses. For Year 5, this gives a net sum of $ plus $ from Column 14 making $ for net proceeds for end of Year 5 for Column 15, to be used in further calculations. Calculating the internal rate of return (irr) on the investment. The standard method of evaluating the rate of return of the $6000 investment resulting in the indicated cash flows in Columns 14 and 15 is to calculate the internal rate of return, irr = i. (These cash flows are not actual cash but the net difference between renting costs and homeowner costs, implied income.) The irr is the rate that discounts the cash flows to the Initial Equity = CF0 = $6000. (See the Side Bar Notes for the derivation.) The equation we solve is (1) 6000 = (1 + i) (1 + i) (1 + i) (1 + i) (1 + i) 5. To solve for i = irr, one would use financial functions such as on the TI83/84. We will use the function: irr(cf0,cflist, CFFreqList) with CF0 = CFList = { , } and CFListFreq = {4,1}. Code and commentary: On the Home page, Code 2 nd { , nd } Sto> Alpha B Enter 2 nd { 4, 1 2 nd } Sto> Alpha C Enter 2 nd Finance 8 You see irr( Write ()6000, Code: 2 nd List Select B, 2 nd List Select C ) Enter and you read 63.31%. The average rate of return on the $6000 investment generating the cash flows is 63.31% per year. Perhaps we should rethink the decision of Buy verses Rent. Buying a House Versus Renting December,

4 Graphing a polynomial. The equation for i = irr can be changed into a polynomial by multiplying through by (1+ i) 5 giving (2) Yo = 6000x x x x x , where x = 1 + i. We arranged the polynomial so it has a positive leading coefficient. By Descarte s Rule of Signs, there is only one positive real zero. (See the Side Bar Notes.) To graph, use the Window: xmin = 10, xmax = 3, xscl = 1, ymin = 100,000, ymax = 1000, yscl = 100. Then Graph. You see the zero of and a relative minimum between 1 and 2 and a relative maximum between 1 and 0. Buying a House Versus Renting December,

5 Exercises Show all your work. Label all inputs and outputs. Include units. Give both mathematical formulas and financial calculator code when appropriate. Name your calculator. Summarize. 1. (a) For a $100,000 house with a 20% down payment and a 30year 5% mortgage, what are the payments for principal and interest? (b) Write an amortization schedule for the first two months with beginning balance, payment, interest, principal, and ending balance. (c) Assume $1500 closing cost to purchase. Taxes and insurance are $2500 per year. Sell in five years at $100,000 with $3000 closing cost to sell and 6% brokerage. Maintenance and replacement costs are $2000 per year. With shortterm savings interest rates at 1%, do an estimate of the yearly cost of homeownership, as on page (a) Write the formula for calculating the house payment, given in a Side Bar Note, in general terms using P, R, n, and i. Give the meaning of each variable. (b) Solve for each variable except i. (c) Write the expanded sum for P in terms of 1+i. (d) Derive the formula for P. 3. How much would the homeowner owe after the 61 st payment? How much of the payment is principal, and how much is interest? 4. Use a scientific calculator to check the irr = i in Formula Given a $6000 cost of replacing the heating and air conditioning system at the end of the fifth year, calculate the irr. If there is a negative cash flow, the practice is to finance it at the rate of borrowing money or at a savings rate. 6. One point in closing costs to purchase is one percent of the initial mortgage. With two points, what is the effective mortgage interest rate in our example? 7. (a) If a homeowner in our example made a full payment every two weeks, 21 per year, how soon would the mortgage be paid off? How much interest would be saved? (b) If half a payment is made every two weeks, how much is saved? 1 1 (1 + i) 8. Given P= R, can we it solve for i? Discuss in both numerical and i general terms. 9. For a 25% marginal income tax rate, what are the annual savings on deductions on mortgage interest and property taxes in Table 1? 10. List at least thirty skills a resourceful homeowner would use. Why is it that $1.00 saved is worth $1.50 earned? Buying a House Versus Renting December,

6 11. Sketch the graph of the polynomial in Formula 2. Use Descarte s Rule of Signs to check for negative zeros. 12. Graph the rational function from Formula 1, y = (1+i) 1 + Use different Windows to investigate different parts of the graph. Report horizontal and vertical intercepts, inflection points, vertical and horizontal asymptotes, local maximums and minimums. Use Trace. Sketch the graph. 13. If the house is foreclosed at the end of year 5 for the mortgage balance, how would the homeowner come out? Buying a House Versus Renting December,

7 Answers to Exercises ( ) i! n " 1! 1+ i % 2. (a) P = R $ ', where P is the amount of the mortgage or present value. R is $ ' # & the payment. The interest rate per month or per compounding period is i, and n is the number of months or compounding periods. (b) Use logs to solve for n. (c) P = R(1+ i) 1 + R(1+ i) R(1+ i) n. 3. To calculate how much they owe after the 61 st payment, calculate the present value of 299 payments. Interest from the 61 st payment = 0.05 (present value of 300 payments, 12 $68,870.49) = $ Principal = = $ Two points is 2(0.01)75,000 = $1500. To calculate the effective interest rate, use financial functions to solve 360 i ,500 = to get i = = %. i 12 1! ! n ( " % + # $ 21 & ' 7. (a) For 21 payments per year, solve 75,000 = for n, to get 21 ), n = two week periods or years. The total paid is ( ) = $99, The interest paid is 99, ,000 = $24, With 360 payments, the interest paid is 360(402.61) 75,000 = $69, The interest saved is $45, Buying a House Versus Renting December,

8 Side Bar Notes Calculating the Monthly Payment. The formulas is 1! !360 ( " % + # $ 12 & ' 75,000 = PMT ), Mortgage pay off. To get the mortgage payoff at the end of year 5, calculate the present value of the remaining 300 payments to be 1! !300 ( " % + # $ 12 & ' P = = $68, ), Derivation of irr = i. Let CF0 = the initial equity and CF0 = P 1 + P P 5 so that P 1 (1 + i) = CF1, P 2 (1 + i) 2 = CF2, P 3 (1 + i) 3 = CF3, P 4 (1 + i) 4 = CF4, and P 5 (1 + i) 5 = CF5. Then CF0 = CF1(1 + i) 1 + CF2(1 + i) 2 + CF3(1 + i) 3 + CF4(1 + i) 4 + CF5(1 + i) 5. Solve for i = irr and you have internal rate of return i. Descarte s Rule of Signs. The number of positive zeros of a polynomial P(x) is equal to the number of sign reversals or less than that by an even natural number. The number of negative zeros is equal to the number of sign reversals in P( x) or less than that by an even natural number. Appreciation of home value. Longterm inflation has averaged about 3.2%, and as high as 5% for 20year periods. The above table could include appreciation of home value at these rates, and inflation of costs. Income tax deduction for property taxes and mortgage interest have been available. The tax deductions reduce expense for mortgage interest and property taxes by the person s marginal income tax rate. (See the tax rate schedule below.) For 2010, the Standard Deduction was $7,700 for single and $11,400 for married filing jointly. In the above table, the sum of mortgage interest and property taxes is less than the standard deduction. But if the family had enough deductions to itemize, these homeownership deductions could be financial advantages. Assume they are married and file jointly, have over $11,400 in other deductions, are in the 25% bracket, and take the tax deduction for property taxes and mortgage interest. This makes the Tax savings in Column 10.1 and After tax buyer s yearly cost in Column This changes Column 14 and Column 15. What would be the After tax internal rate of return on the $6000 investment? Buying a House Versus Renting December,

9 Closing costs to purchase include Legal fees, Title insurance, Loan origination fee, Survey cost, Recording fee, Appraisal fee, Photos, Credit report fee, and Points. Points are tax deductible. The other expenses are capitalized. Do a thorough inspection of the house and the property. Even experts overlook needed major repairs. Actual experience on property tax and maintenance costs. From 1977 to 1984, square feet, 3 bedrooms, 1 bath, attached garage, brick, modern plumbing expenses 2 for maintenance and replacement averaged 13.3% of annual rental income, property taxes 12%, and insurance 4%. The owner did all of the maintenance. The irr was 26.39% after taxes. The expenses for maintenance and replacement varied from as high as 53% in a year to as low as 1.5%, and did not include appliances other than a gas kitchen stove. The author doublechecked these figures. Based on other sources, for the purchase price, property taxes are one to two percent, maintenance and replacement is 2.5% to 14%, and insurance is less than one percent. Housing prices have declined (2011). Nationally, average housing prices are down 32.7% from Banks own 600,000 properties they will sell. There are one million homes in foreclosure, four million behind on mortgage payments. But real estate markets and prices are local. Mortgage interest rates are at a low. To research mortgage interest rates and savings rates see bankrate.com Tax Rate Schedule (taxable income). Tax rate Married filing jointly Single 10% Up to $17,000 Up to $8,500 15% $17,001 to $69,000 $8501 to $34,500 25% $69,001 to $139,350 $34,501 to $83,600 28% $139,351 to $212,300 $83,601 to $174,400 33% $212,301 to $379,150 $174,401 to 379,150 35% $379,151 and more $379,151 and more Buying a House Versus Renting December,

10 References Griffith, Reynolds, Floyd Vest, and Don Leander, Calculating the Return on Home Ownership, Personal Financial Planning, July/August, Vest, Floyd, Reynolds Griffith, The Mathematics of the Return from Home Ownership, School Science and Mathematics, Nov Vest, Floyd, The Mathematics of Amortization Schedules, (in this course). Vest, Floyd, Evaluating Investments, IRR, MIRR, NPV, AM, TR, and HM, (in this course). Buying a House Versus Renting December,