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 99381 (51 pages with 24 formulas, derivations, examples, exercises, a
model exam, and answers. See pages 12-37 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
" 1! 1.04!30 %
40,000 = PMT $
'
# 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
" 1! (1.04)(!30+ X ) %
X
Y2 = 50,000(1.05) (1 – 0.06) – 2313 $
'.
0.04
#
&
The second term is the present values of the remaining payments. Let
Y3 = Y2 ÷ 11,500, which is the total return on one dollar invested after the sale of the
property at the end of year X.
Let Y4 = Y3(1/ X ) ! 1 . (See the derivation in the exercises.)
Y4 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
Y2 = 50000((1.05)^X)(1 – .06) – 2313((1 – 1.04^(-30 + X))/.04)
Y3 = Y2/11500
Y4 = Y3^(1/X) – 1.
Then run a table for Y3 and Y4 from X = 1 to 30 years, to get Table 1.
Some Mathematics of Investing in Rental Property
Spring, 2011
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X
2
3
Y3
1.1544
1.4467
Y4
0.07444
0.13101
.•
.•
.•
.•
.•
.•
.•
.•
.•
5
6
7
2.074
2.4103
2.7626
0.15708
0.15792
0.15623
.•
.•
.•
.•
.•
.•
.•
.•
.•
10
3.9238
0.14649
.•
.•
.•
.•
.•
.•
.•
.•
.•
20
9.2126
0.11743
.•
.•
.•
.•
.•
.•
.•
.•
.•
30
17.664
0.10045
Table 1. Y3 = Return on one dollar. Y4 = 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 Y3 = $17.67.
• The property has earned Y4 = 10% per year on the investment of $11,500.
• From Y2, 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 Y4 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, 2011
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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 = 3000. These cash
flows make the following equation where i = IRR.
2000 = 1000(1 + i)–1 + –2500(1 + i)–2 + 0(1 + i)–3 + 5000(1 + i0–4 +
3000(1 + i)–5
Some Mathematics of Investing in Rental Property
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To solve for i using the irr( function on the TI83/84, consider the following code and
commentary: On the home screen,
2nd { 1000, (-)2500, 0, 5000, 3000 2nd } STO→ 2nd
L1 2nd Finance ∨ ∨ to irr( Enter You see irr( (-)2000,
2nd L1 ) Alpha Solve and read 27.88 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 = 2000 – 1000(1 + i)–1 + 2500(1 + i)–2 – 5000(1 + i)–4 – 3000(1 + i)–5.
The other is the fifth degree polynomial with IE as the positive leading coefficient
Y3 = 2000(1 + i)5 – 1000(1 + i)4 + 2500(1 + i)3 – 5000(1 + i) – 3000.
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 = 1 + 0.2788 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, 2011
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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 2313 ÷ 12 = $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 Y2 = 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 Y3, the total return on one dollar invested.
(d) Calculate Y4, the rate of return on the investment.
4. Give and solve general formulas used to calculate Y4, the percent per year earned on
the investment, in terms of Y3 and Y2 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 Y2, Y3, and Y4 for X years, with the 4.5%
mortgage, give both algebraic formulas and TI83/84 formulas for the following:
(a) Give formulas for Y2.
(b) Give formulas for Y3.
(c) Give formulas for Y4.
(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 725.16, 1049.23,
1388.96, 1745.00, 2118.10, 2509.06, 22857.15 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 1 + 0.381 in X and do the calculations on the home screen on the
TI83/84.
Some Mathematics of Investing in Rental Property
Spring, 2011
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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: –102.17, –311.77, 1037.49, 852.31, 1153.42,
1082.14, 1451.95, 1295.21, 1723.61,
(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 + 0.06(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, 2011
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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
⎡1 − (1 + i )− N ⎤
PV = PMT ⎢
⎥ , 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.
1
bath,
2
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.
Some statistics on a rental: From 1977 to 1984, 1100 square feet, 3 BR, 1
Investing in rental property (July, 2011). Nationally, average housing prices are down
32.7% from 2006. 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, 2011
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Answers to exercises:
1. (a) The monthly payment for principal and interest is $190.99 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 $2455.66.
(b) Monthly payments are $202.67.
(c) For annual payments, the pay-off requires $19,430.95. For monthly payments,
$19,555.49.
(d)
Beginning
Principal
Ending
Month
balance
Interest Paid
Payment
paid
balance
1
$40,000
150.00
202.67
52.67
39,947.33
2
39,947.33
149.80
202.67
52.87
39,894.46
3
39,894.46
149.60
202.67
53.07
39,841.39
" 1! 1.045)!10 %
3. (a) Algebraically, Y2 = 50,000(1.05) (1 – 0.06) – 2455.66 $
' =
# 0.045 &
$105,274.04.
In terms of TI83/84, Y2 = 50000((1.05)^20)(1 – .06) – 2455.16((1 – 1.045^(-)10)/.045).
(b) For the details in this calculation, selling price = $132,664.89, selling expense =
$7959.89. It took $19,430.95 to pay off the mortgage. Proceeds after 6% expense,
$124,705. Net proceeds after paying off mortgage = $105,274.04.
105, 274.04
(c) Y3 =
= $91.54 is the proceeds from one dollar invested.
1150
(d) (1 + Y4 )20 = 91.54. Y4 = 25.3% per year average return on the investment.
20
4. (Purchase expense) × (1 + Y4)X = Net proceeds from sale = Y2.
Also (1 dollar)(1 + Y4)X = Dollar return for each dollar invested = Y3.
1
1
!
$X
Y2
x ! 1,
Solving, we have Y4 = #
and
Y
=
Y
'
1
(
)
4
&
3
" Purchase Expense %
Y2
where Y3 =
.
Purchase Expense
" 1! (1+ 0.045)!(30! X ) %
5. (a) Y2 = 50000(1.05)X(1 – 0.06) – 2455.16 $
'.
0.045
#
&
Y2 = 50000((1.05)^X)(1 – .06) – 2455.16((1 – 1.045^((-)30+X))/.045)
(b) Y3 = Y2/1150
1
(c) Y4 = (Y3 ) x ! 1
Some Mathematics of Investing in Rental Property
Spring, 2011
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(e) After 30 years, the average rate of return on the investment is 10.044%. One dollar
invested accumulated to $17.66. The net proceeds from the sale were $203,131. To
optimize the percent return, sell around year 6.
6. (a) Let IRR = i, this gives 5250 = 725.16(1+i)–1 + 1049.23(1 + i)–2 + 1388.96(1 + i)–3 +
1745(1 + i)–4 + 2118.10(1 + i)–5 + 2509.06(1 + i)–6 + 22,857.15(1 + i)–7.
7. (a) Y1 = 5250 – 725.16(1 + i)–1 – 1049.23(1 + i)–2 – 1388.96(1 + i)–3 –1745(1 + iº–4 –
2118.10(1 + i–5 – 2509.06(1 + i)–6 – 22857.15(1 + i)–7
(b) Y2 = 5250(1 + i)7 – 725.16(1 + i)6 – 1049.23(1 + i)5 – 1388.96(1 + i)4 – 1724(1 + i)3 –
2118.10(1 + i)2 – 2509.06(1 + i)1 – 22857.15.
(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
1.381 giving i = 0.381 = 38.1%.
8. (b) IRR = 9.36%.
9. (a) The after tax accumulation of cash flows and earnings is $8998.59.
(b) $23,654 + 8998.59 = $32,652.59. 3950(1 + i)9 = 32652.59.
i = 26.5% after tax MIRR.
10. (a) Code and commentary for the TI83/84 TVM Solver: 2nd Finance Enter ∨
to I% 4.5 Enter For PV 40000 Enter For PMT (-)202.67 Enter
For FV 0 Enter For P/Y 26 Enter Select PT:END ∧ ∧ ∧ To highlight N
Alpha Solve and read 241.68 two week periods.
( " 0.045 % ! n +
* 1! $ 1+
26 '& #
*
(b) 40,000 = 202.67 *
- . Solving for the exponent n using logs, gives
0.045
*
26
*)
-,
(
" 0.045 % +
* 40,000 $
! 0.045 $
# 26 '& log *1!
= (–n)log # 1+
.
*
26 &%
202.67
"
*
)
,
So n = 241.68567 two week periods or 9.2954 years.
(c) The interest saved is $23,979.67.
Some Mathematics of Investing in Rental Property
Spring, 2011
9
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 12 – 37 in this course.
Some Mathematics of Investing in Rental Property
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