DISCLAIMER: Copyright: 2013

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1 DISCLAIMER: This publication is intended for EDUCATIONAL purposes only. The information contained herein is subject to change with no notice, and while a great deal of care has been taken to provide accurate and current information, UBC, their affiliates, authors, editors and staff (collectively, the "UBC Group") makes no claims, representations, or warranties as to accuracy, completeness, usefulness or adequacy of any of the information contained herein. Under no circumstances shall the UBC Group be liable for any losses or damages whatsoever, whether in contract, tort or otherwise, from the use of, or reliance on, the information contained herein. Further, the general principles and conclusions presented in this text are subject to local, provincial, and federal laws and regulations, court cases, and any revisions of the same. This publication is sold for educational purposes only and is not intended to provide, and does not constitute, legal, accounting, or other professional advice. Professional advice should be consulted regarding every specific circumstance before acting on the information presented in these materials. Copyright: 2013 by the UBC Real Estate Division, Sauder School of Business, The University of British Columbia. Printed in Canada. ALL RIGHTS RESERVED. No part of this work covered by the copyright hereon may be reproduced, transcribed, modified, distributed, republished, or used in any form or by any means graphic, electronic, or mechanical, including photocopying, recording, taping, web distribution, or used in any information storage and retrieval system without the prior written permission of the publisher.

2 LESSON 4 Mortgage Math Review Note: Selected readings can be found under "Online Readings" on your Course Resources webpage. Assigned Reading 1. UBC Real Estate Division BUSI 221 Course Workbook. Vancouver: UBC Real Estate Division. Lesson 4: Mortgage Math Review 2. UBC Real Estate Division Real Estate Finance in a Canadian Context. Vancouver: UBC Real Estate Division. Chapter 6: Mortgage Math Review Recommended Reading 1. UBC Real Estate Division. Real estate math self-assessment quiz. Review your knowledge of the math needed for this course, including both finance and investment analysis calculations. This quiz provides a review of this material and suggestions for additional review sources, if needed. 2. UBC Real Estate Division Foundations of Real Estate Mathematics. Vancouver: UBC Real Estate Division. Chapters UBC Real Estate Division. HP10BII: Introduction to the Calculator and Review of Mortgage Finance Techniques. 4. Hewlett-Packard. HP10BII Computer Based Training. 5. CGA Canada. Math Review Kit. An overview of math and algebra calculations. 6. Canadian vs US Mortgage Compounding Compound Interest and Mortgages Rule of 72 for compounding. beginnersinvest.about.com Learning Objectives After completing this lesson, students should be able to: 1. define and distinguish between simple and compound interest; 2. define and differentiate between nominal, periodic, and effective rates of interest; 3. calculate equivalent interest rates for different stated interest rates; 4.1

3 Lesson 4 4. calculate present value and future value for interest accruing loans; 5. define different types of annuities: simple annuity due, general annuity due, ordinary simple annuity, and ordinary general annuity; 6. calculate the elements of the future value and present value of an annuity: payment, amortization, interest rate, present value, and future value; 7. define and calculate mortgage constants; 8. calculate outstanding balances for a typical partially amortized mortgage; 9. calculate and discuss the principal and interest components of payments for a typical partially amortized mortgage; 10. calculate final payments for fully amortized mortgage loans; and 11. calculate and analyze the net present value, profitability index, and internal rate of return for an investment. Instructor's Comments For most people, real estate represents the single most significant investment they will ever make. Because of the significance of the scale of the investment and the complexity of the process, many will seek expert advice. Since it is essential that students are comfortable with the mechanics of real estate finance calculations, this lesson provides a review of key financial concepts: interest rate conversions, interest accruing loans, payment calculations, outstanding balance calculations, principal and interest splits, and final payments. In addition, the lesson reviews the basics of investment analysis, including net present value, profitability index, and internal rate of return calculations. Illustrations are used in the chapter to reinforce the important calculations. Mathematical steps are provided in the book chapter and students are encouraged to attempt to solve these illustrations on their own. To assist in solving these problems, we have provided detailed calculator steps for the HP10BII+ calculator at the back of the Course Workbook. We recognize that students may use a variety of business calculators to solve these problems. However, for consistency, the workbook solutions will use the HP10BII+ calculator. You are welcome to use any business calculator that you like, as long as it is not programmable; however, you will have to determine the calculator steps on your own. We use the HP10BII+ calculator in this course because it does all the necessary steps, is fairly inexpensive, and is not programmable. 4.2

4 Mortgage Math Review To emphasize the real-world nature of many of these calculations, we have provided spreadsheet applications for many of these, in Microsoft Excel format. You can find the Excel spreadsheet under "Online Readings" on the Course Resources webpage. After finishing this lesson, students have all the foundations necessary to advance to more comprehensive financial analysis. The remainder of this course examines practical applications of mortgage finance, including some highly technical, complex, and highly specialized techniques. It is expected that most students will have had prior experience with mathematical techniques related to mortgages through pre-requisite courses. This lesson provides an overview of these techniques, as a refresher, to prepare students for the more advanced problems in the remainder of the book. Because this chapter is intended only as a brief overview, it does not provide a comprehensive explanation of all details. If students experience difficulty with the material in this chapter, they may wish to review the more comprehensive coverage provided in Foundations of Real Estate Mathematics. If you are having trouble with the calculations in this chapter, the later more complex analysis will be much more difficult. Review and Discussion Questions 1. Which of the following interest rates represents the highest cost of borrowing? (a) j1 = 7.7% (b) j4 = 7.8% (c) j52 = 7.9% (d) j365 = 8% 2. Discuss the reasons why mortgage rates from Canadian banks are generally quoted with semi-annual compounding. 3. Mariah is currently in Whistler searching for residential real estate bargains. Her goal is to purchase a property, hold it for five years, and earn a return of 8% per annum, compounded annually over this time. Assuming that Mariah has just purchased a property for $500,000, at what price will she have to sell the property in five years in order to satisfy her return objective? 4. Kenyan has been collecting empty beer bottles and cashing them in for money at regular intervals. He receives 10 cents for each bottle and collects an average of 400 bottles per week. The market rate of interest is j52 = 10%. (a) How much will Kenyan have at the end of one year if he deposits his earnings into a savings account (earning j52 = 10%) at the end of each week? Kenyan would like to put aside some money into his savings account so that he can purchase a minivan to haul the bottles he collects. Assume that Kenyan sets aside $20 at the end of every week, and the savings account earns interest at j52 = 10%. (b) How much money will have accumulated in the savings account by the end of the 7 th year? 4.3

5 Lesson 4 5. Sharon and her husband Russ are borrowing money to purchase a home and must choose between three mortgage options. The three different loans are identical except for the rate of interest charged. Assuming they prefer the lowest rate, which mortgage loan should Sharon and Russ choose? (a) (b) (c) Loan A = 6.25% per annum, compounded daily Loan B = 6.5% per annum, compounded monthly Loan C = 7% per annum, compounded quarterly 6. Calculate the missing value for each of the following eight mortgage loans: Mortgage Amount Interest Rate Amortization (Years) Monthly Payment (to the next cent) $100,000 j 2 = 10% 25 years (a) (b) j 4 = 8% 20 years $ $500,000 i mo = (c) 25 years $4, $100,000 j 365 = 15% (d) $1, $75,000 j 52 = 10% 10 years (e) (f) j 1 = 12% 20 years $ $70,000 j 2 = (g) 10 years $ The following are all examples of annuities. Identify which specific category of annuity each example falls into. (a) A mortgage loan calling for bi-weekly payments, written at an interest rate of 8% per annum compounded semi-annually, not in advance. (b) (c) (d) An investor is projecting the effective annual yield which will be earned from his apartment building in the coming year, given the monthly rents which will be collected in advance on the first of each month. A winning lottery ticket which pays two semi-annual payments each year, at the beginning of the year (on January 1) and halfway through the year (on July 1). These payments are automatically deposited into a savings account which has an interest rate of isa = 3.5%. A bond which pays interest at the end of each quarter based on an interest rate of j4 = 6%. 4.4

6 Mortgage Math Review 8. Mr. and Mrs. Homeowner have just taken out a mortgage in the amount of $100,000. The interest rate charged is 5% per annum, compounded semi-annually. The loan is to be fully amortized over a 25-year period, with the payments to be made monthly. (a) (b) (c) (d) Calculate the amount of the monthly payments. What is the amount of principal and interest paid in the 85 th payment? What is the total principal and interest paid to date after the 85 th payment is made? What is the amount of the final payment? 4.5

7 Lesson 4 ASSIGNMENT 4 CHAPTER 6: Mortgage Math Review Marks: 1 per question 1. Gregory has just received $150 from his grandparents to buy a professional hockey stick for the upcoming season which starts in exactly 18 weeks. Fred's House of Pucks will have a professional hockey stick on sale at that time for $165. Gregory has deposited the money in an account bearing interest of 0.13% per week, compounded weekly. Will Gregory have enough to buy the professional hockey stick when the season starts? (1) Yes, he will have enough money. (2) No, he will be $7.84 short. (3) No, he will be $11.45 short. (4) No, he will be $8.84 short. 2. Which of the following statements are correct? A. A monthly periodic rate (imo) of 3% is equivalent to a nominal rate of j12 =6%. B. A daily periodic rate (id) of 0.02% is equivalent to a nominal rate of j365 =7.3%. C. A semi-annual periodic rate (isa) of 3.55% is equivalent to a nominal rate of j2 =7.1%. D. A quarterly periodic rate (iq) of 2.25% is equivalent to a nominal rate of j4 =9%. (1) Only statements A and D are correct. (2) Only statements B, C, and D are correct. (3) Only statements B and C are correct. (4) All of the above statements are incorrect. 3. Hakeem bet Shakeel $400,000 that he could shoot 10 left-handed free throws in a row while doing sit-ups. Shakeel lost the bet. Hakeem has arranged a $400,000 interest accruing loan for Shakeel, at an interest rate of 9% per annum, compounded annually. Shakeel has agreed to repay the full amount of the loan plus all interest at the end of 10 full years. What is the amount Shakeel must pay back at the end of the loan's 10-year term? (1) $1,037, (2) $980, (3) $621, (4) $946, ***Assignment 4 continues on next page***

8 Mortgage Math Review 4. Which of the following statements about equivalent interest rates is TRUE? (1) They have different effective annual rates. (2) They must have the same compounding frequency. (3) They will cause the same amount of interest to accrue on a given loan amount over a given length of time. (4) None of the above statements are true. 5. Rank the following nominal and periodic rates from highest to lowest in terms of their effective annual rate: A. id = 0.030% B. iq = 2.7% C. j2 = 10.4% D. j12 = 10.8% E. j52 = 10.5% (1) A, D, B, E, C (2) C, B, A, E, D (3) D, E, A, B, C (4) D, E, C, B, A THE NEXT FOUR (4) QUESTIONS REQUIRE YOU TO COMPLETE THE FOLLOWING TABLE: Loan Loan Amount Interest Rate (semi-annual compounding) Amortization Period (years) Monthly Payment (rounded up to the next higher cent) A $194,000 j2 = 5.5% 20 years? B $132,000 j2 = 9%? $1, C? j2 = 12.2% 18 years $ D $60,500? 25 years $ The monthly payment for Loan A is: (1) $1, (2) $1, (3) $1, (4) $1, The amortization period for Loan B is: (1) less than 20 years. (2) between 20 and 25 years. (3) more than 25 years. (4) 100 years. ***Assignment 4 continues on next page*** 4.7

9 Lesson 4 8. The loan amount for Loan C is: (1) $34, (2) $34, (3) $32, (4) $28, The nominal rate per annum, with semi-annual compounding, for Loan D (rounded to 6 decimal places) is: (1) % (2) % (3) % (4) % THE NEXT TWO (2) QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: A buoyant real estate market has resulted in extremely high prices in the Lower Mainland. After searching in North Vancouver for several months, newlyweds Tom and Katie bought their first house and financed it with a $475,000 mortgage. Fortunately, relatively low interest rates were available, and they were able to obtain a rate of 5.1% per annum, compounded semi-annually, fully amortized over 25 years. 10. What is the monthly payment, rounded up to the next higher dollar? (1) $ 20,057 (2) $ 20,068 (3) $ 2,805 (4) $ 2, Now suppose that the monthly payments are rounded up to the next higher 100 dollars. What would the final payment be? (1) $ (2) $ (3) $ 2, (4) $ 2, ***Assignment 4 continues on next page***

10 Mortgage Math Review 12. The following information describes a residential mortgage loan: Loan Amount: $250,000 Interest Rate: j2 = 7.5% Fully amortized over 25 years. Monthly payments are rounded to the next higher cent. Calculate the monthly payment and the interest portion of the first monthly payment. (1) $1, and $1,538.63, respectively (2) $1, and $284.98, respectively (3) $1, and $290.26, respectively (4) $1, and $290.26, respectively THE NEXT TWO (2) QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: A loan in the amount of $230,000 bears interest at 4.5% per annum, compounded semi-annually. The loan is fully amortized over 25 years and the monthly payments are rounded up to the next higher dollar. 13. Calculate the monthly payment. (1) $1, (2) $1, (3) $1, (4) $1, Assume the monthly payments on the loan were $1, Calculate the outstanding balance after 5 years. (1) $201, (2) $201, (3) $203, (4) $202, Two years ago, Fraser purchased a car wash as an income-generating investment. He financed most of the purchase price with a $1,500,000 mortgage loan, written at an interest rate of 6% per annum, compounded annually. The loan has a 20-year amortization period, 5-year term, and calls for monthly payments rounded to the next higher dollar. Fraser knows that interest paid on this mortgage is deductible from his income taxes. How much interest was paid during the third year of this mortgage? (1) $ 45, (2) $ 45, (3) $ 81, (4) $ 83, ***Assignment 4 continues on next page*** 4.9

11 Lesson Patience has just purchased a retail space for $200,000. Patience expects the following end of year after-tax cash flows: Year 1 $12,568 2 $14,650 3 $16,650 4 $19,546 After-Tax Cash Flow 5 $13,520 + $290,550 Patience would like to earn an after-tax return on equity of 9.5% per annum, compounded annually. What is the net present value (NPV) of her investment (rounded to the nearest dollar)? (1) NPV = $43,127 (2) NPV = $243,127 (3) NPV = $66,224 (4) NPV = $266, Consider the following hotel investments: Investment Present Value Cost Shady Tree Motel $300,000 $180,000 Essence B&B $1,000,000 $650,000 Ruby Inn $750,000 $300,000 Which hotel is the best investment according to the profitability index? (1) Essence B&B (2) Shady Tree Motel (3) Ruby Inn (4) Cannot be determined from the given information 4.10 ***Assignment 4 continues on next page***

12 Mortgage Math Review 18. You have been given the following information for the Rockefeller Centre. Year 1 $50,000 2 $55,650 3 $57,000 4 $58,000 Net After-Tax Cash Flow 5 $59,000 + $950,000 Calculate the internal rate of return (IRR), expressed as an effective annual rate, on the investment if the initial investment cost of the Rockefeller Centre was $900,000. (1) approximately 5.86% (2) approximately 7.15% (2) approximately 14.29% (3) approximately 15.59% 19. Callie's Coffee Corner leases space in a small strip mall. The lease calls for monthly payments of $3,000 to be made at the beginning of the month for which the payment applies. The interest rate charged on Callie's lease is 9% per annum, compounded monthly. What type of annuity is described by the above facts? (1) Ordinary Simple Annuity (2) General Annuity Due (3) Simple Annuity Due (4) Ordinary General Annuity 20. Calculate the annual mortgage constant for a loan with monthly payments over a 25-year amortization at an interest rate of 10% per annum, compounded semi-annually. (1) (2) (3) (4) Impossible to determine with information provided 20 Total Marks Planning Ahead Go to Project 1 and read what will be required for this project. Because this project addresses the readings required in Lessons 1-6, it is worthwhile to begin thinking of what you plan to do for this assignment well in advance. As well, the second part of this project requires you to carry out some research outside of the course materials, so you are advised to begin this research well in advance of the project's due date. This project is more comprehensive than the weekly assignments and will require considerable time and effort to avoid anxiety in completing this project, please begin this work ahead of time. ***Assignment 4 continues on next page*** 4.11