The University of British Columbia
Diploma Program in Urban Land Economics
Sample Final Examination
BUSI 121 — FOUNDATIONS OF REAL ESTATE MATHEMATICS
Time:
3 Hours
Date:
Sample Final Exam
Instructions
This examination consists of two sections:
Part One:
Part Two:
Written Answers
Multiple Choice Questions
70%
30%
100%
This examination consists of twenty-one (21) pages. Please check to ensure that you have a complete
examination.
No outside materials are allowed other than a hand-held, cordless, silent calculator that is NOT alphanumeric
and programmable.
Part One (written answers) MUST be answered in the appropriate booklet(s) provided. Where applicable, show
all calculations so that partial credit can be awarded if your solutions merit it.
Part Two (multiple choice questions) MUST be answered, in pencil, on the multiple choice answer sheet
provided. Instructions on this procedure are contained within the examination. Answers recorded in any other
manner will not be considered.
Please note that formulae pages for your use are attached to the examination.
Answer ALL Questions.
IMPORTANT
This examination is the property of the UBC Real Estate Division
Educational Testing Service.
No part of this examination is to be removed from this examination room.
If any portion of the examination paper is missing from the exam envelope,
the examination will not be marked.
The University of British Columbia
BUSI 121 – Sample Final Examination
2
PART ONE – Written Answers
Record your answers in the booklet(s) provided. Enter your student number in the box marked Student
Number. Where appropriate, show ALL calculations required and clearly LABEL ALL GRAPHS drawn.
Marks
1.
George Parrot, the owner of Friends Development Co., is in need of an interest accruing
construction loan with progress advances. George anticipates that he will need to make five
draws on the loan as follows:
End of Month
1
2
5
8
10
4
Amount
$110,000
$ 75,000
$ 21,000
$ 7,000
$ 3,000
The Fraser Bank of B.C. has agreed to George’s schedule and requires a return of j2 = 12%
on its construction loans. How much will the Friends Development Co. owe at the end of the
tenth month?
2.
The following represents the number of bedrooms in a population of five houses:
2, 1, 4, 3, 10
For this distribution:
1
1
1
2
2
2
3
(a)
(b)
(c)
(d)
(e)
(f)
(g)
calculate the mean
calculate the median
calculate the mode
calculate the standard deviation
calculate the variance
calculate the coefficient of variation
explain which measures of central tendency and dispersion are best and why?
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
3
Marks
3.
For a population of single family dwelling units, the standard deviation of the sales prices is
$100,000.
2
(a)
Assuming a sample size of 25 are taken from this population, what is the standard
error of the mean?
2
(b)
Calculate the sample size of the mean required if you want to achieve a standard error
of the mean of no more than $10,000.
4.
The following are the ages of all the sales personnel in a large real estate sales office:
20
21
21
21
22
22
22
22
23
24
26
26
26
27
29
30
32
33
33
33
33
33
33
33
34
35
36
36
36
36
37
37
39
40
42
43
45
47
47
47
47
47
48
49
49
50
50
52
52
53
55
56
57
59
62
65
65
66
67
69
2
(a)
Place this data into 5 groups with appropriate ranges. (Ex. Age 20-29)
2
(b)
Construct a frequency distribution for this grouped data, showing both the absolute
group frequency and the relative group frequency.
2
(c)
Construct a histogram of the grouped data.
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
4
Marks
5.
Consider the following hypothetical data regarding the average price of homes in Canada over
an eleven year period:
Year
1
2
3
4
5
6
7
8
9
10
11
Average Price
$150,000
$160,000
$171,000
$167,000
$162,000
$210,000
$245,000
$255,000
$295,000
$305,000
$309,000
4
(a)
Calculate a three year moving average on the sales price data.
2
(b)
Why would a moving average be preferable to using the raw data to examine long term
trends?
6.
Jill has the opportunity to invest in Whitewater, a recently developed waterslide park in the
sunny Okanagan. The forecasted net cash flows for the waterslide park are as follows:
Year
1
2
3
4
5
6
7
8
9
10
2
(a)
Net Cash Flow
$ 400,000
$ 400,000
$ 180,000
$ 180,000
$ 450,000
$ 950,000
$ 950,000
$ 950,000
$ 950,000
$ 950,000
If Jill requires a return of 9% per annum, compounded annually, is this investment
attractive at the asking price of $2,000,000? Show calculations.
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
5
Marks
(b)
Assume Jill purchases the waterslide park for $1,750,000 and she wishes to earn
j1 = 9% per annum on her investment. For this investment:
2
(i)
Determine the profitability index.
2
(ii)
Determine the present value ratio.
2
(iii)
Assuming that she purchases the waterslide park for $1,750,000, calculate the
internal rate of return on Jill’s investment.
7.
Theo wants to purchase Pavel’s house. Three years ago, when Pavel purchased the property,
he arranged a mortgage of $850,000. The mortgage is written with a 5 year term and a 25 year
amortization period. The loan has payments of $6,487.32 per month. At the end of the 5 year
term Pavel will owe $783,153.54. Pavel has just paid the 36th payment of the loan. Theo offers
Pavel $1,000,000 for the property, consisting of the assumption of the $813,069.55 book value
of the mortgage, plus a downpayment of $186,930.45. The current rate on comparable 2 year
mortgages is 6.5% per annum, compounded semi-annually.
4
(a)
What is the market value of Theo’s offer?
3
(b)
All other things being equal, would the market value of Theo’s offer be higher or
lower if the original loan had been fully amortized. Explain. (Note: no recalculation
is required)
8.
Matt Stamin is considering investing in his old friend Charles Smith’s new restaurant, the U.K.
Grill. In return for a $35,000 investment today, Charles has promised to pay back Matt the
following cash flows at the end of each of the next five years:
Year
1
2
3
4
5
Cash Flows
$5,000
$8,000
$7,500
$9,000
$9,750
Charles insists this investment offers an excellent return. Matt, having received bad advice
from Charles in the past, has done his homework this time, and has found out that the market
discount rate for similar investments is 12% per annum, compounded annually. Calculate the
following return measures for Matt’s investment, assuming Matt does receive the cash flows
described above:
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
Marks
2
2
2
2
(a)
(b)
(c)
(d)
2
(e)
9.
2
2
2
1
Net Present Value (NPV)
Internal Rate of Return (IRR)
Present Value Ratio
Internal Rate of Return (IRR) if Matt is able to reinvest the yearly cash flows at 8%
per annum, compounded annually.
Based on these return measures, briefly outline your opinion whether Matt should
invest or not?
Stellar Financing has arranged a loan in the amount of $250,000, written at an interest rate of
8.75% per annum, compounded semi- annually. The loan calls for monthly payments, a five
year term, and an amortization period of twenty-five years. You have been asked by the
borrower to help explain the interest consequences of this financing arrangement. Specifically,
you must calculate:
(a)
(b)
(c)
(d)
10.
6
The principal and interest portions of the 12th payment.
How much interest and principal will be paid during the term of the loan.
How much interest will be paid during the 5th year.
The outstanding balance at the end of the term of the loan.
Mr. Brown has approached Ms. Jane, a mortgage broker, about a mortgage loan for the
purchase of a new house. Ms. Jane has agreed to arrange a loan with a face value of $130,000,
at an interest rate of 9.5% per annum, compounded semi-annually, with a 25 year amortization
and a 5 year term. The loan will be repaid with quarterly payments rounded up to the next
higher dollar. Included in the face value of the loan are a $4,000 commission for Ms. Jane and
survey and legal fees totalling $2,000.
4
(a)
2
(b)
What is the cost of the funds advanced to Mr. Brown, expressed as an effective annual
rate?
If a mortgage investor came along and purchased Mr. Brown’s mortgage for an
amount greater than the outstanding balance, would this be a discounted mortgage or
a bonused mortgage? Explain the difference between the two.
END OF PART ONE
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
7
MULTIPLE CHOICE ANSWER SHEET INSTRUCTIONS
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
8
PART TWO – Multiple Choice Questions – 1 mark each
Record your answers on the multiple choice answer sheet provided. Answers recorded in any other manner
will NOT be considered. Each question has only ONE correct answer.
The next THREE (3) QUESTIONS are based on the following information:
Frank Nomad recently inherited a small fortune. He has several investment options and is analyzing them to
determine which has the highest return. His options are as follows:
1.
Term
(years)
Amount received
(at term end)
A
B
C
$500,000
$480,000
$490,000
1
2
3
$600,000
$542,000
$550,000
20%
20.36%
20.6022049%
8.3%
The semi-annual periodic rate (isa) earned on Investment B is:
(1)
(2)
(3)
(4)
3.
Amount to invest
(today)
The effective annual rate (j1) earned on Investment A is:
(1)
(2)
(3)
(4)
2.
Investment
12.5245083906%
6.16716925379%
3.0835846269%
6.2622541953%
The weekly periodic rate (iw) earned on Investment C is:
(1)
(2)
(3)
(4)
12.24489%
3.85185548108%
0.0740741439%
0.074907552%
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
4.
9
A house purchase closes on October 12 th. Funds will be advanced on that date such that, with interest
accrued, $223,500 will be owing on November 1 st. The interest rate to be charged on the loan will be
8.75% per annum, compounded monthly. What amount will be advanced to the borrower on October
12th? It is not a leap year.
(1)
(2)
(3)
(4)
$ 222,453.66
$ 222,488.00
$ 222,381.74
$ 222,434.86
The next TWO (2) QUESTIONS relate to the following information:
Francine and Balfour Shufflebottom have found a home that they simply must own. However, in order to
purchase the home, they must obtain financing.
5.
The Shufflebottoms can only afford to pay $1,500 per month. If the loan has an amortization period and
term of 15 years and a market rate of j1 = 12.55%, what is the maximum they can borrow?
(1)
(2)
(3)
(4)
6.
$ 125,782.61
$ 143,615.70
$ 123,430.55
$ 121,381.05
The Shufflebottoms have found a lower interest rate at a different financial institution. If the rate is j 1
= 12.45%, and payments are still $1,500 per month over a 15 year amortization period, what is the
maximum loan the Shufflebottoms can obtain?
(1)
(2)
(3)
(4)
$126,392.34
$124,057.66
$122,023.64
$144,530.49
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
10
The next THREE (3) QUESTIONS are based on the following information:
The following table outlines financial information with respect to commercial leases. In each instance, lease
payments are made in advance.
Lease
Present Value
A
$200,000.00
B
C
7.
annually
25 years
$16,000.00
j4 = 12%
monthly
25 years
$900.00
j12 = 18%
quarterly
25 years
6.14305453319%
6.96960443041%
6.23739733068%
6.85222206243%
$ 85,451.90
$ 86,164.61
$ 86,306.42
$ 87,017.78
The required payment for Lease C is:
(1)
(2)
(3)
(4)
10.
Lease Term
The present value of Lease B is:
(1)
(2)
(3)
(4)
9.
Payment
Frequency
Nominal Rate
The nominal rate per annum, with semi-annual compounding, for Lease A is:
(1)
(2)
(3)
(4)
8.
$150,000.00
Periodic
Payment
(in advance)
$ 6,628.60
$ 6,931.38
$ 6,833.76
$ 6,539.49
You have just won a lottery that will pay you $10,000 at the end of each year, forever. If interest rates
are j1 = 9%, what is the present value of your winnings?
(1)
(2)
(3)
(4)
$ 111,111.11
$1,000,000.00
$ 109,000.00
impossible to calculate, due to insufficient data
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
11
The next THREE (3) QUESTIONS are based on the following four graphs:
11.
(A)
(B)
(C)
(D)
The equation for the line x - 2y = -4 corresponds to which graph?
(1)
(2)
(3)
(4)
12.
A
B
C
D
The equation for the line 5x - 2y = -10 corresponds to which graph?
(1)
(2)
(3)
(4)
A
B
C
D
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
13.
12
The equation for the line x - y = 1 corresponds to which graph?
(1)
(2)
(3)
(4)
A
B
C
D
The next TWO (2) QUESTIONS are based on the following metes and bounds description of a property:
•
•
•
•
•
14.
move 22 feet 45E to the east (N45EE) to the southwest corner of the property;
move 80 feet due east (N90EE) to the southeast corner of the property;
move 92.4 feet 30E to the northeast (N30EE) to the northeast corner of the property;
move 126.2 feet due west (S90EW) to the northwest corner of the property;
move 80 feet due south to the southwest corner of the property, to close or "bound" the property.
Which of the following shapes best approximates the shape of this property?
(1)
(2)
(3)
(4)
A.
B.
C.
D.
A
B
C
D
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
15.
13
What is the area of this property?
(1)
(2)
(3)
(4)
10,096 sq.ft
8,248 sq.ft
6,400 sq.ft
7,392 sq.ft
The next TWO (2) QUESTIONS are based on the following information:
The following are hypothetical data for residential real estate sales activity:
Year
1
2
3
4
5
16.
What is the percentage increase in housing prices from year 3 to year 5?
(1)
(2)
(3)
(4)
17.
Sales
12,125
12,852
14,008
14,709
15,445
10.3%
9.3%
5.3%
5%
Fill in the blanks.
The percentage change between year 1 and 2 is A
than between year 4 and 5, and the absolute
change between year 1 and 2 is B than between year 4 and 5.
(1)
(2)
(3)
(4)
18.
A = Larger, B = Larger
A = Smaller, B = Smaller
A = Smaller, B = Larger
A = Larger, B = Smaller
A $40,000 loan at 14% per annum, compounded monthly, is to be advanced on September 18th. The
first monthly payment is due November 1st. The interest adjustment charge on October 1st will be:
(1)
(2)
(3)
(4)
$ 214.12
$ 212.43
$ 198.79
$ 233.33
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
19.
14
Suppose a mortgage lender has a portfolio of investments which contains the following mortgage loans
at the following interest rates:
Value of Loans
($ millions)
57.5
110.2
40.3
Interest Rate
(%)
14.5
8.0
16.0
What is the weighted mean interest rate for this portfolio?
(1)
(2)
(3)
(4)
20.
12.83%
10%
14.5%
11.35%
An investor plans to purchase an existing leasehold interest in an income-producing property and expects
to receive income in accordance with the following schedule:
End of Month
6
9
12
15
Cash Flow
$17,125.00
$19,200.00
$19,800.00
$19,775.00
Given that the investor demands a minimum yield on investment of 15% per annum, compounded
monthly, his maximum bid price should be:
(1)
(2)
(3)
(4)
21.
$ 64,515.00
$ 67,671.51
$ 66,794.21
$ 66,535.02
An investor contemplates the purchase of an income producing property where annual payments of
$15,000.00 are expected for a period of 20 years. The first payment is to be received in 5 years time.
The present value of the investment, calculated at j1 = 12%, is:
(1)
(2)
(3)
(4)
$ 79,749.04
$ 112,041.65
$ 63,575.44
$ 71,204.50
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
15
The next TWO (2) QUESTIONS relate to the following equation for the price of a house:
PRICE=20,000 + 15,000ROOMS + 10,000BATHROOMS
22.
An increase in the number of bathrooms from 1 to 3 would increase the sale price by:
(1)
(2)
(3)
(4)
23.
A house with 4 rooms and 2 bathrooms would sell for:
(1)
(2)
(3)
(4)
24.
$20,000 less than a house with 2 rooms and 3 bathrooms.
$30,000 less than house with 6 rooms and 4 bathrooms.
The same as a house with 3 rooms and 3 bathrooms.
$5,000 less than a house with 3 rooms and 4 bathrooms.
Real estate appraisers often use the sale prices of 3 or 4 comparable properties in order to determine
the most likely sales price of the property they are appraising. The choice of comparable properties by
the appraiser is an example of what kind of sample?
(1)
(2)
(3)
(4)
25.
$ 10,000
$ 30,000
$ 20,000
$ 15,000
random sample
convenience sample
judgment sample
cluster sample
Mutual fund managers that are tactical asset allocators buy stocks when the market is in a downturn and
sell them when the market is on the upturn. These managers believe that the stock market exhibits what
kind of fluctuations?
(1)
(2)
(3)
(4)
Seasonal variation
Secular trend
Cyclical variation
Irregular variation
THE NEXT TWO (2) QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A loan in the amount of $216,000.00 bears interest at 18% per annum compounded monthly, not in advance.
Payments are made monthly. The loan has a 20 year amortization period and term. Payments are rounded up
to the next higher dollar.
26.
The monthly payments are:
(1)
(2)
(3)
(4)
$ 3,228
$ 3,227
$ 3,333
$ 3,334
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
27.
The outstanding balance after 120 payments is:
(1)
(2)
(3)
(4)
28.
16
$ 188,221.42
$ 186,321.42
$ 184,858.91
$ 182,741.65
The term of a bonused mortgage is increased from 5 years up to 15 years (while keeping the
amortization period at 20 years). This means the effective rate charged to the borrower on funds
advanced:
(1)
(2)
(3)
(4)
does not change.
increases.
decreases.
changes but the change cannot be determined with the information given.
THE NEXT TWO (2) QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Susy has decided to purchase her own rollerblade shop in Kitsilano. She has been advanced an interest only loan
of $215,500 by the Bank of Kara. The interest rate is 13.5% per annum, compounded quarterly, with annual
payments (not in advance). The term of the loan is 7 years.
29.
What is her yearly interest payment (rounded up to the next higher dollar)?
(1)
(2)
(3)
(4)
30.
$ 28,772
$ 29,093
$ 30,599
$ 30,075
Now assume that Susy's loan is interest accruing instead of interest only. How much would Susy owe
at the end of the 7 years?
(1)
(2)
(3)
(4)
$ 537,771.34
$ 545,860.80
$ 246,461.85
$ 246,098.73
END OF PART TWO
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
17
Formulae
BUSI 121 - Foundations of Real Estate Math
Mathematical Formula
1. Interest Rate
Conversion
= (1 + isa)2 = (1 + ia)1
2. Present Value of a Lump Sum
PV = FV × (1 + i)-n
FV
=
(1 % i)n
3. Future Value of a Lump Sum
FV = PV × (1 + i)n
4.
Present Value of an
Ordinary Simple Annuity
Present Value of a
5.
Simple Annuity Due
6.
Future Value of an
Ordinary Simple Annuity
7. Sinking Fund
PV = Pmt ×
1&(1%i)&(n)
i
Financial Notation
(1 + imo)12 = (1 + iq)4
PV = Pmt × aÚn, i%á
PV ' Pmt %
Pmt ×
1&(1%i)&(n&1)
i
FV = Pmt ×
(1%i)n&1
i
1
Pmt = FV ×
PV = Pmt + Pmt × aÚn-1, i%á
FV = Pmt × sÚn, i%á
Pmt = FV ×
(1%i)n&1
i
Pmt =
1
sÚn, i%á
FV
(1%i)n&1
i
Equation 8.14 APR = (100 x C)
(T x P)
Equation 9.3
Present Value Ratio =
Present Value of Net Cash Flows
Cost
Equation 9.4
Profitability Index =
Present Value of Positive Cash Flows
Present Value of Negative Cash Flows
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
Equation 10.1
Y = mX + b
Equation 10.2
A=L×W
18
Equation 10.3
c2 = a2 % b2
Equation 10.4
A=½×B×H
Equation 10.5
A = Br2
Equation 11.1
TOTAL = X1 + X2 + X3 + X4
Equation 11.2
j Xi ' X1 % X2 % X3 % ... % Xn
n
i'1
Equation 11.3
Percentage change = 100 ×
Equation 12.1
j Xi
Final value & Base value
Base value
n
i'1
j Xi
n
Equation 12.2
µ '
i'1
n
m
Equation 12.3
Equation 12.4
µ '
j fi Xi
i'1
n
m
j fi Xi ' f1 X1 % f2 X2 % f3 X3 % ... % fm Xm
i'1
Equation 12.5
m
j fi ' n
i'1
Equation 12.6
µ ' j
m
fi
i'1
n
(Xi)
n
Equation 12.7
µ '
j wi Xi
i'1
n
j wi
i'1
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
19
m
Equation 12.8
µ '
j fi Mi
i'1
n
n
Equation 12.9
F '
Equation 12.10
F '
2
j (Xi&µ)
i'1
n
n
1
2
(j Xi & nµ2)
n i'1
n
2
2
j (Xi & µ)
i'1
Equation 12.11
F '
Equation 12.12
σ2 '
Equation 12.13
Coefficient of variation =
Equation 13.1
n
1
2
(j Xi & nµ2)
n i'1
n
F
(100)
µ
n
n
n
i'1
i'1
i'1
nj XiYi & (j Xi) (j Yi)
r '
n
n
n
i'1
i'1
n
[n j Xi & (j Xi)2] [n j Yi & (j Yi)2]
2
i'1
Equation 13.2
Equation 13.3
Yi
2
i'1
' a % bXi
n
ˆ )2
S ' j (Yi & Y
i
i'1
Equation 13.4
ˆa '
Equation 13.5
ˆb '
ˆ
j Yi & bj Xi
n
nj XiYi & j Xij Yi
nj Xi & (j Xi)2
2
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
Equation 13.6
20
2
ˆ
j Yi & bj XiYi & ˆaj Yi
n & 2
SYX '
2
(1 & r 2)Fy n
Equation 13.7
SYX '
Equation 13.8
Yi ' a % bXi % cZi
Equation 14.1
I '
Equation 14.2
PI '
n & 2
Xt
(100)
Xt&n
Pn
Po
(100)
m
Equation 14.3
PI '
j Pn Qo
m
j
m
PI '
QI '
QI '
m
j
Qn
Qo
(100)
i i
Po Qn
(100)
i
i
i'1
m
j
m
QI '
i
j Qn Po
i'1
Equation 14.7
i
i'1
m
Equation 14.6
(100)
i i
Po Qo
j Pn Qn
i'1
Equation 14.5
i
i'1
i'1
Equation 14.4
i
j Qn Pn
i
i
i'1
m
j
i'1
(100)
i i
Qo Po
(100)
i i
Qo Pn
Copyright: 2011 UBC Real Estate Division Educational Testing Service
The University of British Columbia
BUSI 121 – Sample Final Examination
m
Equation 14.8
VI '
j Pn Qn
Fx '
i
i'1
m
j
i'1
Equation 15.1
i
21
(100)
i i
Po Qo
F
n
END OF EXAMINATION
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Copyright: 2011 UBC Real Estate Division Educational Testing Service