# Some Mathematics of Investing in Rental Property. Floyd Vest

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1 Some Mathematics of Investing in Rental Property Floyd Vest Example 1. In our example, we will use some of the assumptions from Luttman, Frederick W. (1983) Selected Applications of Mathematics of Finance and Investment, UMAP Module (51 pages with 24 formulas, derivations, examples, exercises, a model exam, and answers. See pages for this topic.) In the following example of purchase and investment in a rental property, the investor paid \$50,000 for the property. He made a 20% down payment of \$10,000. The closing costs to purchase was 0.03(50,000) = \$1500. The buyer had invested \$11,500. For the yearly appreciation of the property we will use 5%. We will assume the closing costs to sell are 6% of the selling price. Also, we assume that the mortgage interest rate is 4%. (See the following real world example for such numbers.) As an approximation to the monthly payments for principal and interest over 30 years, we calculate PMT where " 40,000 = PMT 1!1.04!30 % \$ ' # 0.04 & and solve with PMT = \$2313 per year, about \$193 per month. (See the Side Bar Notes for financial formulas.) We assume that rent collected pays principal and interest on the mortgage, insurance, property taxes, vacancy, and maintenance. (See the following real world example.) Consider the possibility of the sale of the property at the end of X years. The funds left from the sale after appreciation, paying closing costs to sell, and paying off the mortgage are Y 2 = 50,000(1.05) X X ) " 1! (1.04)(!30+ % (1 0.06) 2313 \$ '. # 0.04 & The second term is the present values of the remaining payments. Let Y 3 = Y 2 11,500, which is the total return on one dollar invested after the sale of the property at the end of year X. Let Y 4 = Y 3 (1/ X )!1. (See the derivation in the exercises.) Y 4 is the percent per year earned on the investment when sold at the end of year X. Enter these formulas in a TI83 (or other calculator) as Y 2 = 50000((1.05)^X)(1.06) 2313((1 1.04^(-30 + X))/.04) Y 3 = Y 2 /11500 Y 4 = Y 3^(1/X) 1. Then run a table for Y 3 and Y 4 from X = 1 to 30 years, to get Table 1. Some Mathematics of Investing in Rental Property Spring,

2 X Y 3 Y Table 1. Y 3 = Return on one dollar. Y 4 = Rate of return on the purchase expense. X is in years. Examining the table, we observe that at the end of 30 years: If the property is sold, each dollar invested has returned Y 3 = \$ The property has earned Y 4 = 10% per year on the investment of \$11,500. From Y 2, the recovery from the sale is \$203,131. To maximize the rate of return from the sale, the optimal time to sell is around year six. (See the Y 4 column.) Example 2. Are these figures realistic? Answer: They have been in the past. Consider an actual example of a single family rental purchased for \$19,000 in 1977 and sold for \$52,000 in 1998, a period of approximately 21 years. At the time of purchase there were VA and FHA assumable mortgages for about 4%. Over the 21 years, you can calculate the appreciation rate to be 4.9%. (Do the calculation.) The closing costs to purchase, and the closing costs to sell were negligible since no realtor was used. The investment to purchase by the buyer was \$2200. From after two years, the rent paid more than the cost of principal and interest, property taxes, insurance, maintenance, and vacancy. (There was a positive cash flow.) This continued for 21 years with the exception of two additional years. The owner did all of the management and almost all of the maintenance and replacement. Since 2200(1 + r) 21 = 52,000, the before tax rate of return, on the initial investment, from the sale was r = 16.25% per year. This happened during a period of years when inflation had averaged near 5%. Appreciation is directly proportional to inflation, other factors not interfering. These calculations do not add in the positive cash flows for each of the many years. Some Mathematics of Investing in Rental Property Spring,

3 How a rental makes money. If things work out, rental property makes the investor a profit by: 1) Positive cash flows. 2) Appreciation of the property. 3) The use (interest on) of tax savings from the depreciation deduction that is taken at the marginal income tax rate, and which must be paid back at the time of sale. 4) The profit (tax savings) on depreciation resulting from recapture (paid back at the time of sale) at the capital gains rate which is usually less than the marginal income tax rate. 5) You could also include the interest earned from investing the monthly cash flows. During this period of time, if one had acquired twenty rent houses, the person would be close to a millionaire. The acquisition of millions of dollars is important because millions of dollars are required for funding retirement and family security. (For calculations of the amount of money required to fund retirement, see Vest, Floyd, Taking the Long View of Life, HiMap Pull-Out, Consortium 83, 2002, in this course.) Often, a person s wage from their job pays about the cost of living. To get ahead and meet their long-term responsibility, requires of a person a second source of income to be invested in long-term investments. To do a more precise analysis of the above 1977 to 1998 property, one would run out a spreadsheet with annual calculations of rent collected, maintenance expenses, mortgage expenses, depreciation taken as a deduction on income tax, income taxes paid on the positive cash flow, etc. (A total of about 24 columns with rows representing years.) Finally the proceeds from the sale is entered. Then one would calculate the internal rate of return (IRR) (or the MIRR, whichever you prefer) on the investment. The IRR is the annual rate that discounts the cash flows to the initial equity. To calculate IRR, one usually uses an iterative routine on a computer or calculator. If one uses the MIRR function on a spreadsheet, it will request a reinvestment rate for positive cash flows, a cost of money rate for negative cash flows, and accumulate the cash flows and their earnings to be added to the net proceeds from the sale, then calculate the MIRR. The MIRR can be calculated after income taxes. (See Exercise 9.) Example 3. Consider the following end of year cash flows, CFs, and initial equity, IE, in dollars. (Taken from Chapter 14 of the TI83/84 manual.): IE = 2000, CF1 = 1000, CF2 = 2500, CF3 = 0, CF4 = 5000, CF5 = These cash flows make the following equation where i = IRR = 1000(1 + i) (1 + i) 2 + 0(1 + i) (1 + i (1 + i) 5 Some Mathematics of Investing in Rental Property Spring,

4 To solve for i using the irr( function on the TI83/84, consider the following code and commentary: On the home screen, 2 nd { 1000, (-)2500, 0, 5000, nd } STO 2 nd L1 2 nd Finance to irr( Enter You see irr( (-)2000, 2 nd L1 ) Alpha Solve and read The IRR = i is in percent form. To illustrate some the mathematics involved, we write the cash flow formula two ways. One is with Initial Equity as the leading positive constant: Y1 = (1 + i) (1 + i) (1 + i) (1 + i) 5. The other is the fifth degree polynomial with IE as the positive leading coefficient Y3 = 2000(1 + i) (1 + i) (1 + i) (1 + i) Substitute X for 1 + i and graph both curves for X from 2 to 3, and Y from 20,000 to 20,000. You get two interesting graphs where both cross the X-axis at approximately 1.3 since X = 1 + i = is a sufficiently close estimate for the zero for both equations. Notice that X = 0 is a vertical asymptote and by extending the X range, you can see that Y = 2000 is a horizontal asymptote. Some Mathematics of Investing in Rental Property Spring,

5 Exercises Show all your work. Put on units. Summarize. Identify variables. 1. In the above discussion, to simplify, mortgage calculations were done in years, not months. (a) Using 360 months for a 30 year 4% mortgage of \$40,000, calculate the monthly payment. (b) Why are the monthly payments less than the annual payment of = \$193? 2. Consider the example discussed at the beginning of this article except that the mortgage interest rate is 4.5%. (a) How much are annual payments? (b) How much are monthly payments? (c) How much money is required to pay off the mortgage at the end of twenty years? (d) Build an amortization schedule for the first three months of the mortgage. 3. Following the above example with mortgage interest rate at 4.5%, and after 20 years, give both algebraic and TI83/84 formulas and with both calculate the following: (a) Give formulas for Y 2 = the amount remaining from the selling price after 5% appreciation for twenty years and after selling expenses and calculate. (b) Explain all the details and values in the parts of the formula. (c) Calculate Y 3, the total return on one dollar invested. (d) Calculate Y 4, the rate of return on the investment. 4. Give and solve general formulas used to calculate Y 4, the percent per year earned on the investment, in terms of Y 3 and Y 2 and X years. Give one formula based on the purchase expense and one based on one dollar invested. 5. For the purpose of building a table for Y 2, Y 3, and Y 4 for X years, with the 4.5% mortgage, give both algebraic formulas and TI83/84 formulas for the following: (a) Give formulas for Y 2. (b) Give formulas for Y 3. (c) Give formulas for Y 4. (d) Run the table from 2 years to 30 years. (e) Discuss the table. 6. The following are initial equity and yearly cash flows from an actual spreadsheet: IE = 5250, End of year cash flows beginning with year one are , , , , , , in dollars. (a) Write the IRR equation. (b) Verify that IRR = 38.1%. You can use STO and RCL on an ordinary scientific calculator, or put in X and do the calculations on the home screen on the TI83/84. Some Mathematics of Investing in Rental Property Spring,

6 7. For the IRR equation in Exercise 6: (a) As in Example 3, write the associated rational function formula with IE as positive leading term. (b) Write the associated seventh degree polynomial with IE in the positive leading term as in Example 3. (c) In what interval of the X-axis would you expect to find a root for X = 1 + i? (d) What interval on the X-axis would make an interesting graph? (e) Graph the rational function. Graph the polynomial. What do they have in common? Keep adjusting the Range until you can report the asymptotes. 8. Given IE = 20,534.97, and the following end of year cash flows beginning with year one, 7200, 6000, 2400, 1200, 6000, 3600, 600 in dollars. (a) Write the cash flow formula where i = IRR is the rate which discounts the cash flows to the initial equity. (b) Calculate IRR. 9. Consider an investment with the following actual end of year after tax cash flows, in dollars, for the years 1977 to 1985: , , , , , , , , , (a) Calculate the after tax accumulations at the end of 1985 at 6% per year with a marginal income tax rate of 34% (yearly average at the time for this investor). You may want to put (1 0.34) in STO and use RCL on an ordinary scientific calculator for repeated calculations, or on the TI83/84, put it in X and calculate on the home screen. (b) The after tax proceeds from the sale was \$23,654 (at a then 28% capital gains income tax rate). Add the after tax accumulation of cash flows and earnings to this figure and use the purchase expense of \$3950 to calculate the Modified Internal Rate of Return (MIRR). 10. On the 4.5% thirty year mortgage for \$40,000, with monthly payments of \$202.67, if you made payments every two weeks, 26 payments per year, how many years would it take to pay off the mortgage? (a) Solve with the TVM Solver on the TI83/84. (b) Solve algebraically. (c) How much interest was saved? Some Mathematics of Investing in Rental Property Spring,

7 Side Bar Notes Using TI83/84 symbols: Compound Interest Formula FV = PV(1 + i) N, where i = the interest rate per compounding period, N = the number of compounding periods, FV = future value, and PV = present value. Formula for the Present Value of an Ordinary Annuity used in mortgage calculations PV = PMT 1 (1 ) + i N, where i = the interest rate per compounding period, N = the i number of compounding periods, PMT = payment, PV = present value. In the TI83/84, I% = annual interest rate in percent form. Some statistics on a rental: From 1977 to 1984, 1100 square feet, 3 BR, bath, attached garage, brick, modern plumbing: vacancy rate averaged 2% of the months per year, expense for maintenance and replacement 13.3% of annual rental income, property taxes 12% of annual rental income, insurance 4% of annual rental income. These figures came from a cash flow spreadsheet giving annual after tax cash flows and after tax proceeds from the sale, with the IRR of 36.65% before taxes, and 26.39% after taxes. Investing in rental property (July, 2011). Nationally, average housing prices are down 32.7% from Banks own 600,000 properties they will sell. There are one million houses in foreclosure, and four million behind on mortgage payments. New home supply is at a 50-year low. Mortgage interest rates are low. Inflation is expected to increase. Are you a good investor? If you are capable of doing financial mathematics, you are capable of becoming a good investor. Not everybody is a good investor. From Money magazine, August, 2011: 57% of sales materials, from free lunch seminars advertising financial investments, include misleading or unwarranted claims. 20% of seniors have been sold bad investments, paid too high fees, or been taken by fraud. Seniors lost \$3 billion last year to financial predators. Presenters talk about complicated costly products variable equity indexed annuities, reverse mortgages, private real estate investment trusts. It s is not a level playing field, presenters know their products, seniors know little. Be alert to protect your aging seniors. Some Mathematics of Investing in Rental Property Spring,

8 Answers to exercises: 1. (a) The monthly payment for principal and interest is \$ per month. (b) These monthly payments are less than the \$193 because money is applied earlier to principal and thus reducing interest. 2. (a) At 4.5%, annual payments are \$ (b) Monthly payments are \$ (c) For annual payments, the pay-off requires \$19, For monthly payments, \$19, (d) Month Beginning balance Interest Paid Payment Principal paid Ending balance 1 \$40, , , , , , (a) Algebraically, Y 2 = 50,000(1.05) 20 " (1 0.06) !1.045)!10 % \$ ' = # & \$105, In terms of TI83/84, Y 2 = 50000((1.05)^20)(1.06) (( ^(-)10)/.045). (b) For the details in this calculation, selling price = \$132,664.89, selling expense = \$ It took \$19, to pay off the mortgage. Proceeds after 6% expense, \$124,705. Net proceeds after paying off mortgage = \$105, (c) Y 3 = 105, = \$91.54 is the proceeds from one dollar invested (d) (1 + Y 4 ) 20 = Y 4 = 25.3% per year average return on the investment. 4. (Purchase expense) (1 + Y 4 ) X = Net proceeds from sale = Y 2. Also (1 dollar)(1 + Y 4 ) X = Dollar return for each dollar invested = Y 3. Solving, we have Y 4 = where Y 3 = Y 2 Purchase Expense.! Y 2 \$ # & " Purchase Expense % 1 X '1 and Y4 = ( Y 3 ) 1 x!1, 5. (a) Y 2 = 50000(1.05) X X ) " 1! ( )!(30! % (1 0.06) \$ '. # & Y 2 = 50000((1.05)^X)(1.06) (( ^((-)30+X))/.045) (b) Y 3 = Y 2 /1150 (c) Y 4 = ( Y 3 ) 1 x!1 Some Mathematics of Investing in Rental Property Spring,

9 (e) After 30 years, the average rate of return on the investment is %. One dollar invested accumulated to \$ The net proceeds from the sale were \$203,131. To optimize the percent return, sell around year (a) Let IRR = i, this gives 5250 = (1+i) (1 + i) (1 + i) (1 + i) (1 + i) (1 + i) ,857.15(1 + i) (a) Y1 = (1 + i) (1 + i) (1 + i) (1 + iº (1 + i (1 + i) (1 + i) 7 (b) Y2 = 5250(1 + i) (1 + i) (1 + i) (1 + i) (1 + i) (1 + i) (1 + i) (c) Let X = 1 + i for the graph. A person might do a graph from 1 to 2 expecting a zero for X between 1 and 2. (e) For Y1 you get vertical and horizontal asymptotes. Both curves cross the X-axis at giving i = = 38.1%. 8. (b) IRR = 9.36%. 9. (a) The after tax accumulation of cash flows and earnings is \$ (b) \$23, = \$32, (1 + i) 9 = i = 26.5% after tax MIRR. 10. (a) Code and commentary for the TI83/84 TVM Solver: 2 nd Finance Enter to I% 4.5 Enter For PV Enter For PMT (-) Enter For FV 0 Enter For P/Y 26 Enter Select PT:END To highlight N Alpha Solve and read two week periods. 1! ! n ( " % + * # \$ 26 & ' - (b) 40,000 = * - * Solving for the exponent n using logs, gives * - 26 )*,- ( " 40, % + * # \$ 26 & ' - log * 1! -! = ( n)log \$ * " # 26 % &. * - ), So n = two week periods or years. (c) The interest saved is \$23, Some Mathematics of Investing in Rental Property Spring,

10 Teachers Notes For the basics of mathematics of finance, see Luttman or Kasting in this course. For exercises on graphing and using iterative programs for solving financial functions, see Vest, Floyd, What Can a Financial Calculator Do for a Mathematics Teacher or Student, (1989) and Computerized Business Calculus, (1991) and Beezer, Robert A., Closing in on the Internal Rate of Return, (1996) in this course. For an example of a cash flow spreadsheet with calculation of IRR, see Vest, Floyd, and Reynolds Griffith, The Mathematics of the Return from Home Ownership (1991) in this course. A lifetime file on finance. Have your students set up a lifetime file on finance and financial mathematics including lessons in this course. Include this article because they may at sometime want to invest in real estate. In later years, some of your students will thank you. Take it from personal experience. Code for using the TI83/84 TVM Solver is found in the answer to Exercise 10. Code for using the irr( function is in Example 3. A derivation of the IRR = YTM concept can be found in Vest, Floyd, CD s Bought at a Bank verses CD s Bought from a Brokerage (2011) in this course. A problem with IRR, as used on investments, is discussed. For interesting problems on the financial mathematics of real estate investments, see Luttman, pages in this course. Some Mathematics of Investing in Rental Property Spring,

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