BUSI 121 Foundations of Real Estate Mathematics

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1 Real Estate Division BUSI 121 Foundations of Real Estate Mathematics SESSION 2 By Graham McIntosh Sauder School of Business University of British Columbia

2 Outline Introduction Cash Flow Problems Cash Flow Keys Annuities Annuities Due Deferred Annuities Perpetual Annuities 2

3 HP 10BII + Calculator 3

4 Types of Cash Flow Problems Time Present Value Future Value Cash Flows Even Uneven Interval Regular Irregular 4

5 Types of Cash Flow Problems Even cash flows occurring at regular intervals are characteristic of annuity-type of problems Use the annuity formulas and financial calculator keys Examples: solving standard mortgage problems When cash flows are uneven and or irregular it is better to use the cash flow keys 5

6 Types of Cash Flow Problems Time Diagram: Present Value PV = PV 1 + PV 2 + PV 3 + PV PV n-1 + PV n Present Value PV =? CF 1 CF 2 CF 3 CF 4 CF n n-1... CF n n PV 1 = CF 1 (1 + i) -1 PV 2 = CF 2 (1 + i) -2 PV 3 = CF 3 (1 + i) -3 PV 4 = CF 4 (1 + i) -4 PV n-1 = CF n-1 (1 + i) - n-1 PV n = CF n (1 + i) -n 6

7 Types of Cash Flow Problems Time Diagram: Future Value FV = FV 1 + FV 2 + FV 3 + FV FV n-1 + FV n FV =? 0 CF 1 CF 2 CF 3 CF 4 CF n n-1 n... CF n CF n (1 + i) n-n = FV n CF n-1 (1 + i) n-(n-1) = FV n-1 CF 4 (1 + i) n-4 = FV 4 CF 3 (1 + i) n-3 = FV 3 CF 2 (1 + i) n-2 = FV 2 CF 1 (1 + i) n-1 = FV 1 7

8 Types of Cash Flow Problems EXAMPLE: PRESENT VALUE OF UNEVEN CASH FLOWS AT REGULAR INTERVALS An investor has an opportunity to purchase a property. The investor expects to hold the property for 3 years and estimates it to produce the following net cash flows (benefits or revenues costs or expenses ) at the end of each year: End of Year Net Cash Flow 10,000 35, ,000 If the investor desires to earn a yield of j 1 = 11% (compounded annually), how much should the investor pay for the property today (present value)? 8

9 Types of Cash Flow Problems Time Diagram: PV = PV 1 + PV 2 + PV 3 PV = -10,000(1 +.11) ,000(1 +.11) ,000(1+.11) -3 Present Value of uneven Cash Flows PV =? CF 1 -$10,000 CF 2 $35,000 CF 3 $100, PV 1 = CF 1 (1.11) -1 PV 2 = CF 2 (1.11) -2 PV 3 = CF 3 (1.11) -3 9

10 Types of Cash Flow Problems Solution Method 1: Using the Financial Keys Press Display 1 P/YR 1 11 NOM% 11 1 N 1 0 PMT /- FV -10,000 PV 9, M 9, N FV 35,000 PV -28, M+ -28, N 3 100,000 FV 100,000 PV -73, M+ -73,

11 Types of Cash Flow Problems In order for the investor to earn j 1 = 11% on their investment, the maximum amount they would be willing to pay would be $92, Cumbersome process using the financial keys Much easier using the cash Flow Keys for this type of problem 11

12 Types of Cash Flow Problems Method 2: Using the Cash Flow Keys (highly recommended!) 12

13 Types of Cash Flow Problems NOTATION PV = n -t t t=1 k = discount rate C t = Net Cash Flow in period t n = Last compounding period or cash flow n t=1 [CF x (1 + k) ] = sum of all cash flows at time = 0 SHORTHAND WAY OF WRITING: PV = [CF 1 (1 + k) -1 + CF 2 (1 + k) CF n (1 + k) -n ] or PV = CF1 CF2 CFn + + (1+k) (1+k) (1+k) 1 2 n 13

14 Net Present Value (NPV) Solution Using a HP 10B II+ Financial Calculator: Turn ON and Clear Screen DCF set up Blue Shift Key Access Blue Functions 2. Clear Mem Clear Memory Key 3. 0 Clear Cash Flow Keys (cflo clr) 3

15 Net Present Value (NPV) Solution Using a HP 10B II+ Financial Calculator: 1 P/YR set P/YR = 1 11 I/YR set discount rate = 11% 0 CF j CF 0 0 (cost = 0 we are calculating a PV) /- CF j CF 1 = -10,000 (cash flow group 1) CFj CF 2 = 35,000 (cash flow group 2) CFj CF 3 = 100,000 (cash flow group 3) NPV 92, NPV NPV > 0 Therefore accept the investment Note: 92, is also stored in PV Note: far fewer steps Note: DCF Keys can also be used to solve annuity problems and multi cash flow problems (using N j )

16 Types of Cash Flow Problems Problems that involve irregular or uneven cash flows, solve with the cash flow keys This topic will be revisited in Chapter 9 16

17 What is a mortgage? Financial Definition: A mortgage is a loan secured by real estate Legal Definition: mort = dead gage = pledge The borrower pledges their real estate to the lender as security (collateral) for the loan The pledge (mortgage) is registered on the certificate of title Once the loan is repaid, the mortgage is removed from the title

18 Mortgage Terminology The borrower is referred to as the mortgagor The lender is referred to as the mortgagee Their relationship is subject to a mortgage contract 18

19 Mortgage Terminology Fully Amortized Loan: when the amortization period equals the term Partially Amortized Loan: when the term is less than the amortization period 19

20 Classification of Mortgage Loans Type of property (residential vs non residential) Use of Mortgage Default Insurance conventional < 80% L/V high ratio >80% L/V Sources of Mortgage Funds Institutional Lenders: banks, credit unions, trust co, mortgage loan companies, life insurance co, pension funds Private Lenders Priority of mortgage on title (first, second, etc.)

21 Constant Blended Payments Equal payments occurring at equal intervals of time (an annuity) Payments are a blend of principal and interest The blend of principal changes as the loan is repaid, gradually more principal and less interest constitutes each payment 21

22 Periodic Payment on a Constant Mortgage Principal and Interest Split Constant Payment Mortgage Monthly Payments ($) Interest Principal Amortization Period Outstanding Balance on a Constant Payment Mortgage Outstanding Balance ($) Amortization Period

23 Standard Mortgage Calculations General Steps: 1. Read the Question Carefully. 2. Read it again! 3. Summarize the Facts. 4. Calculator Steps 5. Record the Answer 23

24 Standard Canadian Mortgages This is an application of the general annuity Characteristics 1. Equal payments occurring at regular intervals of time. 2. Payment frequency usually does not match the interest rate compounding frequency. 24

25 Standard Canadian Mortgages 3. The term does not match the amortization period, called a partially amortized loan. Solution Before calculating the payment on this type of mortgage, an interest rate conversion must be conducted. 25

26 EXAMPLE 1: Standard Canadian Mortgages Interest rate conversion and mortgage payment calculation. A $300,000 mortgage is negotiated with the following terms: interest rate j 2 = 9% 25 year amortization monthly payments not in advance What is the size of the required monthly payments? 26

27 EXAMPLE 1: Standard Canadian Mortgages Equations for Annuities PV = PMT 1 (1 + i) i n Solve for PV OR PMT = PV 1 (1 + i) i n Solve for PMT NOTE: These equations can only be used if the payment and compounding frequencies match. 27

28 EXAMPLE 1: Standard Canadian Mortgages Solution STEP 1: Convert j 2 = 9% to the equivalent nominal rate (i.e., j 12 ) to match the payment frequency. STEP 2: Calculate the required monthly payment. 28

29 EXAMPLE 1: Standard Canadian Mortgages Step 1: Interest Rate Conversion (using the interest rate conversion keys) Press Display Comments 9 NOM% 9 stated nominal rate 2 P/YR 2 stated compounding frequency EFF% effective rate 12 P/YR 12 desired compounding frequency NOM% j 12 equivalent rate (automatically stored in I/YR ) 29

30 EXAMPLE 1: Standard Canadian Mortgages Step 2: Payment Calculation Press Display Comments = 300 N 300 number of payment period 300,000 PV 300,000 loan amount 0 FV 0 PMT -2, monthly payment NOTE: Always round payments to the next highest cent. PMT = 2,

31 Recall Function RCL Key Use in conjunction with other TVM keys to check data entered RCL N RCL I/YR RCL PV RCL PMT RCL FV - check number of periods - check nominal interest rate - check present value - check payment - check future value RCL P/YR - check compounding frequency per year 31

32 Recall Function RCL Key Last example: RCL N 300 (months) RCL I/YR (j 12 rate) RCL PV 300,000 (loan amount) RCL PMT -2, (pmt) RCL FV 0 RCL P/YR 12 (compounding frequency per year) 32

33 Standard Mortgage Questions Table Complete the following table Mortgage Interest Amortization Payment Payment Amount Rate (years) Frequency $500,000 j 4 = 8% 15 Monthly?? j 2 = 5% 20 Monthly $2500 $100,000 j 2 = 7%? Monthly $ $200,000 j 2 =? 25 Monthly $

34 Step 1: Interest Rate Conversion Question 1 Solving for the Payment Convert j 4 = 8% j 12 Rate Press Display Comments 8 NOM% 8 4 P/YR 4 EFF% P/YR 12 NOM% j 12 Rate j 4 = 8% Automatically stored in I/YR 34

35 Question 1 Solving for the Payment Step 2: Solve for the Monthly PMT Press Display Comments = 180 N 180 amortization period 500,000 PV 500,000 loan amount 0 FV 0 PMT -4, Monthly PMT = $4,

36 Mortgage Questions Table Complete the following table Mortgage Interest Amortization Payment Payment Amount Rate (years) Frequency $500,000 j 4 = 8% 15 Monthly $ ? j 2 = 5% 20 Monthly $2500 $100,000 j 2 = 7%? Monthly $ $200,000 j 2 =? 25 Monthly $

37 Question 1a Solving for the Interest Only Payment Step 2: Solve for the Interest Only Monthly PMT Press Display Comments = 180 N 180 Payment Schedule 500,000 PV 500,000 loan amount 500,000 +/- FV -500,000 no amortization PV = FV PMT -3, Monthly Interest Only PMT = $3,

38 Question 1a Solving for the Interest Only Payment Step 2: Solve for the Interest Only Monthly PMT Another Method: Press Display Comments RCL I/YR j12 rate as a percentage 12 = imo as a percentage % E -3 imo as a decimal X 500,000 = -3, Monthly Interest Only PMT Monthly Interest Only PMT = $3,

39 Question 1a Solving for the Interest Only Payment Step 2: Solve for the Interest Only Monthly PMT Press Display Comments 1 N 1 N = any value > 1 500,000 PV 500,000 loan amount 500,000 +/- FV -500,000 no amortization PV = FV PMT -3, Monthly PMT = $3,

40 Step 1: Interest Rate Conversion Question 2 Solving for the Mortgage Amount Convert j 2 = 5% j 12 Rate Press Display Comments 5 NOM% 5 2 P/YR 2 EFF% P/YR 12 NOM% j 12 Rate j 2 = 5% Automatically stored in I/YR 40

41 Question 2 Solving for the Mortgage Amount Step 2: Solve for the Mortgage Amount PV Press Display Comments = 240 N 240 2,500 +/- PMT -2,500 0 FV 0 PV 380, Mortgage Amount = $380,

42 Mortgage Questions Table Complete the following table Mortgage Interest Amortization Payment Payment Amount Rate (years) Frequency $500,000 j 4 = 8% 15 Monthly $ $ 380, j 2 = 5% 20 Monthly $2500 $100,000 j 2 = 7%? Monthly $ $200,000 j 2 =? 25 Monthly $

43 Step 1: Interest Rate Conversion Question 3 Solving for the Amortization Period Convert j 2 = 7% j 12 Rate Press Display Comments 7 NOM% 7 2 P/YR 2 EFF% P/YR 12 NOM% j 12 Rate j 2 = 7% Automatically stored in I/YR 43

44 Question 3 Solving for the Amortization Period Step 2: Solve for the Amortization Period (in Years) N Press Display Comments 100,000 PV 100, /- PMT FV 0 N Amortization Period in Months 12 = Amortization Period = 20 Years 44

45 Question 4 Solving for an Interest Rate Step 1: Solve for I/YR Press Display = 300 N ,000 PV 200, /- PMT FV 0 12 P/YR 12 I/YR j 12 rate automatically stored in NOM% 45

46 Mortgage Questions Table Complete the following table Mortgage Interest Amortization Payment Payment Amount Rate (years) Frequency $500,000 j 4 = 8% 15 Monthly $ $ 380, j 2 = 5% 20 Monthly $2500 $100,000 j 2 = 7% 20 Monthly $ $200,000 j 2 =? 25 Monthly $

47 Step 2: Convert j 12 j 2 Rate Question 4 Solving for an Interest Rate Press Display Comments EFF% P/YR 2 NOM% j 2 Rate j 12 =

48 Mortgage Questions Table Complete the following table Mortgage Interest Amortization Payment Payment Amount Rate (years) Frequency $500,000 j 4 = 8% 15 Monthly $ $ 380, j 2 = 5% 20 Monthly $2500 $100,000 j 2 = 7% 20 Monthly $ $200,000 j 2 = % 25 Monthly $

49 Standard Mortgage Questions Mathematical Method Example 1: j 2 = 9% Mortgage = $300, year Amortization Monthly payments 49

50 Standard Mortgage Questions Mathematical Method Example 1: Step 1:Convert j 2 = 9% to i mo Calculator Steps Press Display Comments y x enter i sa 6 1/x raise to power of 1/6 = i mo as a % -1 = i mo as a decimal M i mo store in memory 50

51 Standard Mortgage Questions Mathematical Method Question 1: Step 2: Find Payment using the annuity equation i mo = n = 300 PV =$

52 Standard Mortgage Questions Mathematical method Question 1: Step 2: Find Payment Calculator Steps Press Display Comments RM + 1 = i mo + 1 y x 300 +/ E-1 raised to n = / E-1 subtract from 1 RM = divided by i mo 1/x E-3 take the reciprocal of X 300, ,000 multiply by PV = 2, monthly PMT 52

53 Annuities Due PMT s Occur at the beginning of each period Use BEG/END Function A $300,000 lease is negotiated with the following terms: interest rate j 2 = 9% 25 year amortization monthly lease payments in advance What is the size of the required monthly payments (due)? 53

54 EXAMPLE: Annuities Due Equation: Annuities Due PV = PMT 1 (1 + i) i n X (1+i) Solve for PV PMT = PV 1 (1 ) n + i Solve for PMT i X (1+i) NOTE: multiplying by (1+i) moves the PMTs to the beginning of the period NOTE: These equations can only be used if the payment and compounding frequencies match (interest rate conversion may be required). 54

55 EXAMPLE: Annuities Due Solution STEP 1: Convert j 2 = 9% to the equivalent nominal rate (i.e., j 12 ) to match the payment frequency. STEP 2: Calculate the required monthly lease payment due 55

56 EXAMPLE: Annuities Due Step 1: Interest Rate Conversion (using the interest rate conversion keys) Press Display Comments 9 NOM% 9 stated nominal rate 2 P/YR 2 stated compounding frequency EFF% effective rate 12 P/YR 12 desired compounding frequency NOM% j 12 equivalent rate (automatically stored in I/YR ) 56

57 EXAMPLE: Annuities Due Step 2: Payment Due Calculation Switch to Begin Mode: Press BEG/END Press Display Comments = 300 N 300 number of payment period 300,000 PV 300,000 loan amount 0 FV 0 PMT -2, monthly lease payment NOTE: Round payments to the next higher cent. PMT = 2,

58 Deferred Annuities Annuities where the payments begin at some point in the future. PV =? PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT Period of Deferral Beginning of Annuity End of Annuity 58

59 Deferred Annuities ANALYSIS 2 Methods: (1) Two present values calculations (i) PV of PMT at beginning of the annuity enter as FV (ii) PV of FV at beginning of Deferral period (2) Difference between two present values (i) PV of PMT of the entire period LESS (ii) PV of PMT of the deferral period 59

60 Deferred Annuities EXAMPLE: An investor has an opportunity to receive 10 payments of $1000 per year, however, the payments will not begin for 4 years. How much should the investor pay for the payment stream if the desired yield is j 1 = 10%? 60

61 Deferred Annuities ANALYSIS: Data: PMT = $1000 per annum N = 10 j 1 = 10% Deferment = 4 years PV =? 61

62 Deferred Annuities Method 1: Calculate 2 PV s. Time Diagram: PV 3 = 1000 a(10,0.10) PV 3 PV 0 = FV 3 ( ) PV 0 FV

63 Deferred Annuities Equations: PV 3 = PMT a [n,i] PV 0 = FV 3 (1 + i) -3 OR PV = [PMT a [n,i]] (1 + i) -d (-d = deferment period) NOTE: PV 3 = FV 3 a[n,i] is an abbreviation of the PV of an annuity equation 63

64 Deferred Annuities Solution : Calculate PV 3, then discount PV 3 to find PV 0 Press Display Comments 1 P/YR 1 10 I/YR 10 j 1 = 10% /- PMT annuity payments 10 N 10 number of payments 0 FV 0 PV 6, PV 3 of 10 payments of $1000 +/- FV - 6, enter PV 3 as FV 3 0 PMT 0 3 N 3 deferral period PV 4, PV 0 The investor should pay no more than $4,

65 Deferred Annuities METHOD 2: Difference between two Present values. Time Diagram: PV =? Years in the future ( $100) PV = PMT a 13, j 1 = 10% PV =? MINUS ( $100) Years in the future PV = PMT a 3, j 1 = 10% EQUALS Years in the future ( $100)

66 Deferred Annuities Equations: PV = PMT a[13, i = 10%] PMT a 3, [i = 10%] OR PV = [ PMT a[n + d, i]] [PMT a [d,i]] d = deferment period 66

67 Deferred Annuities Solution: Press Display Comments 1 P/YR 1 10 I/YR 10 j 1 = 10% /- PMT payments 13 N 13 entire period 0 FV 0 PV 7, PV of entire period M 7, stored in memory 3 N 3 deferment period PV 2, PV of deferred annuity M+ +/- -2, add to value in memory RM 4,

68 Deferred Annuities Cash Flow Method: Press Display Comments 1 P/YR 1 10 I/YR 10 0 CF j CF CF j CF N j n 1 3 deferment period 1000 CF j CF annuity payments 10 N j 10 number of payments NPV 4,

69 Perpetual Annuities Used in Real Estate investment analysis and appraisal and other cost/benefit situations where the cash flows are expected to endure for a very long period of time. In real estate appraisal and investment analysis the "Capitalization Rate" or Cap Rate is a direct application of perpetual annuities 69

70 Perpetual Annuities Equation: Perpetual Annuites Note: this equation can only be applied if: 1) The payments remain constant 2) The interest rate remains constant Note: PMTs occur at the END of each period 70

71 Perpetual Annuities Example: PMT = $3,250,000 per annum Interest rate = 9% per annum Compounding Frequency = annual (P/YR =1) Calculate the Present Value when the number of PMTs = a) n = 50 b) n = 120 c) n = 999 d) n = (infinity) 71

72 Perpetual Annuities Equation: 72

73 Perpetual Annuities Time Diagram: Perpetual Annuities PV =? Qa Qb Qc Qd PMT PMT PMT PMT PMT PMT PMT

74 Perpetual Annuities Calculation Press Display Comments 1 P/YR 1 comp frq yr 9 I/YR 9 interest rate yr 3,250,000 +/- PMT 3,250,000 PMT yr 0 FV 0 reversionary value 50 N 50 # of PMTS (50) PV 35,625, = N 120 N =120 PV 36,109, = N 999 N =999 PV 36, 111, =999 74

75 Perpetual Annuities If n = - (really small!) then use the PV annuity equation becomes the PV perpetuity equation: The term (1+i) - becomes = 0 75

76 Perpetual Annuities = $36,111, same as N =999 years 76

77 Perpetual Annuities Summary Table N = PMT Interest PV Rate/yr 50 3,250,000 9% $35,625, PV 120 3,250,000 9% $36,109, $484, ,250,000 9% 36,111, $1, ,250,000 9% 36,111, $0 77

78 Perpetual Annuities PV vs N with PMT = $3.25M, I/YR =9% Present Value ($ millions) (25, 31.9m) (50, 35.6m) (120, 36.1m) (999; 36.11m) (5, 12.6m) (1, 3m) Time (Years) 78

79 Perpetual Annuities TVM: Time Value of Money Money received further in the future is worth much less today. After about 100 periods, the value of any future cash flow is effectively 0 hence we do not have to worry about reversionary values. 79

80 Perpetual Annuities Note: this property also explains that for given PMT amount at a certain interest rate, why extending mortgage amortization period beyond about 50 periods leads to relatively small increase to amount that can be borrowed (PV). 80

81 Perpetual Annuities Due Equation: Perpetual Annuities Due PMT PV = 1+ i ( i) Note: this equation can only be applied if: 1) The payments remain constant 2) The interest rate remains constant Multiplying by (1+i) moves the PMTs to the beginning of each period 81

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