BUSI 221 MORTGAGE FINANCE REVIEW QUESTIONS REVIEW … › re_credit... · Mortgage Finance Review Questions 4 At the end of the one-year term, January 1, 2013, Susan prepaid 10% of

Mortgage Finance Review Questions 1

BUSI 221 MORTGAGE FINANCE REVIEW QUESTIONS Detailed solutions are provided at the end of the questions. REVIEW QUESTION 1 Gordon and Helen have recently purchased a 2,500 square foot home for $300,000. They have requested a loan for $225,000. Real property taxes are estimated at $2,000 per annum. The loan will be at a rate of 10% per annum, compounded semi-annually with monthly payments to fully amortize the loan over 20 years. A. If the lender sets a GDSR of 32%, what is the minimum income required by the purchasers to

qualify for the desired loan? [$86,546.50] B. Assume that Gordon and Helen earn $80,000 per annum, Gordon has a car loan of $275 per

month and Helen has a boat loan with payments of $500 per quarter. If the lender sets a GDSR of 32% and a TDSR of 40%, what is the maximum allowable loan? Assume all other facts remain as in the original question. [$206,655.83 under GDSR]

REVIEW QUESTION 2 Five years ago, David obtained a mortgage from the Helpful Credit Union. The loan was written for a 5-year term at an interest rate of 13% per annum, compounded semi-annually. Monthly payments of $1,375 were specified, sufficient to amortize the principal over a 20-year period. The term of the loan is up. All payments were made on schedule. Dave may renew the mortgage for another 5-year term at the current rate of 7.5% per annum, compounded semi-annually. After considering his overall financial position, Dave decides he would like to make weekly payments. However, the maximum payment he can afford is $225 per week. The credit union is willing to allow this change as loan as the loan is amortized at the current rate over the remaining 15 years. What immediate balloon payment must Dave make to reduce the outstanding balance to the point where the weekly payments will fully amortize the loan over 15 years? [$4,446.62] REVIEW QUESTION 3 Colleen has a $120,000 mortgage with ABC Trust Company at an interest rate of 8% per annum, compounded semi-annually, amortized over 25 years with a 4-year term. Monthly payments are rounded up to the next higher dollar. Colleen has made the first 40 payments on time and for the full amount. She then missed four payments and paid only $500 each on the final four payments. Calculate the OSB owing at the end of the term. [$118,227.97] REVIEW QUESTION 4 You are a loan officer for the Superior Insurance Company and are considering a loan secured by a luxury apartment building. You have been provided with an income statement for the building, which shows a rental schedule, which would generate $6,400,000 gross potential rental income per annum, if the units were continuously rented at rates advertised in the schedule. At full occupancy this building can also be expected to generate $230,000 per annum from parking fees


and $370,000 per annum from membership and usage fees in the Penthouse Club. The vacancy and bad debt loss estimate of 8% applies to each component of gross potential income. Expense items on the operating statement include $2,800,000 per annum depreciation of the building and equipment, $127,000 per annum to operate the Penthouse club, $880,000 per annum in property taxes, maintenance expenses of $440,000 per annum, utility costs of $362,000 per annum and property management fees of $678,000 per annum. Your guidelines include a net operating income capitalization rate of 9% per annum, a debt coverage ratio of 1.2 and a maximum loan-to-value ratio of 70%. The loan would require quarterly payments adequate to fully amortize the principal over 20 years at an interest rate of 12% per annum, compounded quarterly. What is the largest loan you can make? [$24,871,587] REVIEW QUESTION 5 You have been hired as a consultant to provide your expertise to Mr. and Mrs. Miller regarding mortgage information. The Millers have been shopping for a loan and are confused by the terminology. (a) The Millers have found three repayment schemes and ask you to briefly define them in

simple terms:

interest accrual loan; interest only loan; and constant payment repayment scheme.

(b) The Millers know they require $100,000 to help finance the purchase of their first home. They

call upon you, a mortgage finance expert, to calculate the size of the monthly payment and the outstanding balance owing at the end of a three-year term under each of the following scenarios:

(i) Bank A offers interest accrual loans at an effective annual rate of 12%. [no payment;

OSB = $140,492.80]

(ii) Credit Union B offers interest only loans at 11.75% per annum, compounded semi-annually. [payment = $956.02; OSB = $100,000 + $956.02]

(iii) Trust Company C offers a constant payment repayment scheme amortized over 20 years at a rate of 12.5% per annum, compounded monthly. [payment = $1,136.14; OSB = $95,899.05]

REVIEW QUESTION 6 Joanne and Fred recently arranged a $175,000 mortgage with the XYZ Trust Company. The mortgage contract is written at a rate of 10% per annum, compounded semi-annually, with a 25-year amortization period, a 5-year term and monthly payments rounded up to the next higher dollar. (a) Calculate the monthly payment required on this loan. [$1,566] (b) Calculate the outstanding balance owing at the end of the term and after two years.

[$164,435.92; $171,379.96]


(c) Calculate the interest and principal portions of the 1st, 36th and 60th monthly payments. Month Interest Principal 1 $1,428.85 $137.15 36 $1,383.69 $182.31 60 $1,344.40 $221.60

(d) Calculate the total interest paid and total principal repaid during the term. [Total interest =

$83,395.92; Total principal = $10,564.08]

(e) Suppose that 2 years after the loan is initiated (i.e., 24 payments have been made), the XYZ Trust Co. decides to liquidate their investment. At this point, if the trust company finds an investor who will pay $162,000, determine the following:

(i) What yield will XYZ Trust Co. earn on their investment, expressed, as an annual rate with

semi-annual compounding. [7.387689%] (ii) What yield, expressed as an annual rate with semi-annual compounding, will the investor

earn, and assuming the mortgage is held until the end of the term? [12.323743%] (iii) If the investor demands a return of 15% per annum, compounded semi-annually, on

his/her investment, what price would the investor is willing to pay today? How much of a discount, in dollars, has the investor received? [Price = $152,010.38; Discount = $19,369.58]

REVIEW QUESTION 7 Dorothy and Leonard have recently purchased their first home in Langley, a suburb of Vancouver, BC, for $300,000. They require a $285,000 mortgage loan. The total family income is $70,000 per annum and they have no debts outstanding. Net property taxes are estimated to be $2,800 per year. The ABC Credit Union will provide funds on an insured first mortgage with an interest rate of 6% per annum, compounded semi-annually, amortized over 25 years with monthly payments. ABC requires a 32% gross debt service ratio and a 95% loan-to-value ratio on this insured loan. Assume that a 3.6% premium will be added onto the loan amount. (a) What is the maximum first mortgage loan that Dorothy and Leonard qualify for?

[$255,284.95] (b) How large of a second mortgage is required in order for the borrowers to obtain the total

amount desired? [$39,975.05] (c) If the lender set a maximum total debt service ratio of 40%, will the borrowers qualify

under this constraint? The second mortgage rate will be 8% per annum, compounded semi-annually and payable with monthly payments over a 15-year amortization period. [Yes, max 2nd = $49,218.85 or TDSR is 38.5%]

REVIEW QUESTION 8 On January 1, 2012, Susan Shopper borrowed $95,000 as a mortgage loan. The interest rate on the loan was set at 10.5% per annum, compounded semi-annually, with level quarterly payments over a 15-year amortization period. The term was set for one year.


At the end of the one-year term, January 1, 2013, Susan prepaid 10% of the outstanding balance and renewed the remaining balance for a 3-year term at a rate of 9.5% per annum, compounded semi-annually. Payments were maintained at the original amount. On the anniversary of her mortgage, January 1, 2014, she again prepaid 10% of the outstanding balance and increased future payments by 20%. Due to a recent cash flow problem, Susan did not make the final two payments due (October 1st and January 1st) and the previous two were for $1,000 each (April 1st and July 1st). All other payments were on schedule and for the full amount. Calculate the amount owed to the lender on January 1, 2016, to fully repay the loan. [$65,323.35] REVIEW QUESTION 9 A potential borrower is analyzing several mortgage loan alternatives and wants you to calculate the payment obligation(s) under each option and the outstanding balance owing at the end of a two-year period. The loan amount required is $100,000. OPTIONS 1. A constant payment repayment scheme at 9% per annum, compounded semi-annually,

amortized over 25 years with monthly payments. [PMT = $827.98; OSB = $97,603.23] 2. A straight line principal reduction loan at 10% per annum, compounded semi-annually,

repaid over 10 years with semi-annual payments. [1st pmt = $10,000, 2nd pmt = $9,750 3rd pmt = $9,500, 4th pmt = $9,250; OSB = $80,000]

3. An interest accrual loan at a rate of 12% per annum, compounded semi-annually. [no

pmt; OSB = $126,247.70] 4. An interest only loan, repaid with quarterly payments at a rate of 12% per annum,

compounded semi-annually. [PMT = $2,956.30; OSB = $100,000 + $2,956.30] 5. A variable rate loan with the initial rate of 8% per annum, compounded semi-annually,

amortized over 25 years. Payments will be made annually. The interest rate adjustment will be made annually. Changes in interest rates will be reflected by changes in payments. The rate prevailing at the end of Year 1 will be 11% per annum, compounded semi-annually. Payments are to be calculated on the basis on the remaining amortization. [pmt year 1 = $9,496.24; pmt year 2 = $12,075.73; OSB = $97,739.50]

REVIEW QUESTION 10 J. Seinfeld arranged a $225,000 mortgage loan five years ago at an interest rate of 13.5% per annum, compounded semi-annually, with a 20-year amortization, a 10-year term, and quarterly payments. Interest rates have declined since the mortgage was arranged. Seinfeld has decided to refinance the loan and is considering two options. Option 1 Seinfeld can refinance his existing loan over an amortization of 15 years and a 5-year term at an interest rate of 9.75% per annum, compounded semi-annually. The lender is willing to waive the prepayment penalty, but will charge $250 for administration costs. Assume the fee is paid in cash at the time of refinancing. Payments will continue to be made quarterly.


Option 2 Seinfeld can refinance his existing loan with a new lender over a 15-year amortization and a 5-year term at an interest rate of 9% per annum, compounded semi-annually. Payments will be made quarterly. The prepayment penalty charge on the existing loan is 3 months' interest. Additional refinancing costs will total $1,000. Assume the fees will be paid in cash at the time of refinancing. 1) Calculate the actual cost of refinancing under both options. Express your final solution as

an annual rate with semi-annual compounding.

[Option 1 − 9.78% per annum, compounded semi-annually Option 2 − 10.07% per annum, compounded semi-annually]

2) On the basis of interest rates, which option should Seinfeld choose?

[Choose Option 1 since the rate is lower] 3) What other considerations should Seinfeld take into account before making a final

decision? REVIEW QUESTION 11 Cooperative Credit Union has agreed to lend $150,000 to Bob and Lydia in the form of a shared appreciation mortgage on a purchase price of $200,000. This loan is written at an interest rate of 9.25% per annum, compounded semi-annually with monthly payments over a 20-year amortization and 5-year term. In addition, Bob and Lydia must give the lender 45% of any capital gains realized (after selling costs) upon the sale of the house. Selling costs are estimated to be 8% of the selling price. If Cooperative has an expected return of 14% per annum, compounded monthly and Bob and Lydia plan to sell the house at the end of the term, what rate of house appreciation does Cooperative expect to accrue? Express your answer as an effective annual rate. [11.18%] REVIEW QUESTION 12 A lender has recently advanced a participation mortgage for $700,000 at a rate of 12% per annum, compounded monthly. The loan will be repaid with monthly payments over a 20-year amortization and a 5-year term. In addition, the lender will participate by taking 10% of the property's annual gross potential income. If the lender expects to earn 17% per annum, compounded annually overall, calculate the anticipated gross potential rental income. [$277,451]


DETAILED SOLUTIONS FOR BUSI 221 REVIEW QUESTIONS QUESTION 1 SOLUTION A. $225,000 = PMT × a[[240, j12 = 9.797815%]] PMT = $2,141.24 per month = $25,694.88 per year Press Display

10 NOM% 10 2 P/YR 2 EFF% 10.25 12 P/YR 12 NOM% 9.797815 225000 PV 225,000 240 N 240 0 FV 0 PMT −2,141.244963

Gross Income = PIT/GDSR

$24,694.88 $2,000Gross Income=

0.32

Gross Income = $86,546.50 B. GDSR

PMT $2,0000.32

$80,000

PMT = ($80,000 × 0.32) − $2,000 PMT = $23,600 per year = $1,966.67 per month PV = $1,966.67 × a[[240, j12 = 9.797815%]]

PV = $206,655.83

Press Display 10 NOM% 10 2 P/YR 2 EFF% 10.25 12 P/YR 12 NOM% 9.797815 1966.67 +/− PMT −1,966.67 240 N 240 0 FV 0 PV 206,655.827589

TDSR

PMT $2,000 ($275 12) ($500 4)0.40

$80,000

PMT = ($80,000 × 0.40) − $7,300 PMT = $24,700 per year = $2,058.33 per month PV = $2,058.33 × a[[240, j12 = 9.797815%]] PV = $216,287.37


Press Display 10 NOM% 10 2 P/YR 2 EFF% 10.25 12 P/YR 12 NOM% 9.797815 2058.33 +/− PMT −2,058.33 240 N 240 0 FV 0 PV 216,287.373886

The maximum loan is $206,655.83 under the GDSR. QUESTION 2 SOLUTION Original Loan PV = $1,375 × a[[240, j12 = 12.661289%]] PV = $119,822.54 OSB60 = $110,616.15

Press Display 13 NOM% 13 2 P/YR 2 EFF% 13.4225 12 P/YR 12 NOM% 12.661289 1375 +/− PMT −1,375 240 N 240 0 FV 0 PV 119,822.538077 119822.54 PV 119,822.54 60 INPUT AMORT PER 60-60 = = = 110,616.151428

New Loan PV = $225 x a[[780, j52 = 7.36801%]] PV = $106,169.53

Press Display 7.5 NOM% 7.5 2 P/YR 2 EFF% 7.640625 52 P/YR 52 NOM% 7.36801 225 +/− PMT −225 15 x 52 = N 780 0 FV 0 PV 106,169.531071

Repay $110,616.15 − $106,169.53 = $4,446.62


QUESTION 3 SOLUTION $120,000 = PMT × a[[300, j12 = 7.869836%]] PMT = $916 per month

Press Display 8 NOM% 8 2 P/YR 2 EFF% 8.16 12 P/YR 12 NOM% 7.869836 120000 PV 120,000 300 N 300 0 FV 0 PMT −915.856146

Option 1 FV of loan with no pmts less FV of pmts made FV of loan with no pmts FV = $120,000(1+ i)48 = $164,228.29

Press Display 8 NOM% 8 2 P/YR 2 EFF% 8.16 12 P/YR 12 NOM% 7.869836 120000 PV 120,000 48 N 48 0 PMT 0 FV −164,228.286049

FV of pmts made 1st 40 pmts: FV = 916 × s[[40, j12 = 7.869836%]](1+imo)

8 FV = $41,739.72(1+imo)

8

FV = $43,980.55 Final 4 pmts: FV = $500 × s[[4, j12 = 7.869836%]] FV = $2,019.76


Press Display 8 NOM% 8 2 P/YR 2 EFF% 8.16 12 P/YR 12 NOM% 7.869836 916 +/− PMT −916 40 N 40 0 PV 0 FV 41,739.715835 41739.72 PV 41,739.72 0 PMT 0 8 N 8 FV −43,980.549424 500 +/− PMT −500 0 PV 0 4 N 4 FV 2,019.760752

OSBterm = $164,228.29 − $43,980.55 − $2,019.76 = $118,227.98 Option 2 (Current Balance) OSB40 = $114,121.06 Find OSB44 FV = $114,121.06(1+imo)

4 = $117,144.35 New Loan with PV of $117,144.35 + $500 payment per month, find OSB 4 periods later = $118,227.97

Press Display 8 NOM% 8 2 P/YR 2 EFF% 8.16 12 P/YR 12 NOM% 7.869836 120000 PV 120,000 300 N 300 0 FV 0 PMT −915.856146 916 +/− PMT −916 40 INPUT AMORT PER 40-40 = = = 114,121.062751 114121.06 PV 114,121.06 0 PMT 0 4 N 4 FV −117,144.352559 117144.35 PV 117,144.35 500 +/− PMT −500 0 FV 0 4 INPUT AMORT PER 4-4 = = = 118,227.974738


QUESTION 4 SOLUTION NOI = $6,400,000 + $230,000 + $370,000 (total of $7,000,000) −8% Vacancy of $560,000 − expenses of $2,487,000* = $3,953,000 *Expenses = $880,000 + $440,000 + $362,000 + $678,000 + $127,000 = $2,487,000 Market Value = $3,953,000/0.09 = $43,922,222.22 Loan-Value = 70% × $43,922,222.22 = $30,745,556 DCR PMT = $3,953,000/1.2 PMT = $3,294,166.67 per year = $823,541.67 per quarter PV = $823,541.67 × a[[80, j4 = 12%]] PV = $24,871,587

Press Display 12 I/YR 12 4 P/YR 4 823541.67 +/− PMT −823,541.67 80 N 80 0 FV 0 PV 24,871,587.1635

Binding constraint is DCR, maximum loan is $24,871,587. QUESTION 5 SOLUTION

(a) interest accrual loan: no payment during loan; owe loan amount and accumulated interest over term interest only loan: periodic payments of interest only; owe principal at end constant payment: blended and equal payments; increasing principal and decreasing interest over time

(b) Interest Accrual

FV = $100,000(1+0.12)3 = $140,492.80 Press Display 12 I/YR 12 1 P/YR 1 100000 PV 100,000 3 N 3 0 PMT 0 FV −140,492.8 Interest Only $100,000 × imo = $956.02 Where imo = j12)12 and j12 = 11.472285%


Press Display 11.75 NOM% 11.75

2 P/YR 2 EFF% 12.095156 12 P/YR 12 NOM% 11.472285 ÷ 12 = 0.956024

% × 100000 = 956.023771 Constant Payment Repayment Scheme $100,000 = PMT × a[[240, j12=12.5%]] PMT = $1,136.14 OSB36 = $95,899.05

Press Display 12.5 I/YR 12.5 12 P/YR 12 100000PV 100,000 240 N 240 0 FV 0 PMT −1,136.14055 1136.14 +/− PMT −1,136.14 36 INPUT AMORT PER 36-36 = = = 95,899.053616

QUESTION 6 SOLUTION

(a) $175,000 = PMT × a[[300, j12=9.797815%]] PMT = $1,566 Press Display 10 NOM% 10 2 P/YR 2 EFF% 10.25 12 P/YR 12 NOM% 9.797815 175000 PV 175000 300 N 300 0 FV 0 PMT −1,565.352644 1566 +/− PMT −1,566

(b) OSB60 = $164,435.99

OSB24 = $171,379.96 (calculation continued) Press Display 60 INPUT AMORT PER 60-60 = = = 164,435.916103 24 INPUT AMORT PER 24-24 = = = 171,379.956182


(c) Month Interest Principal 1 $1,428.85 $137.15 36 $1,383.69 $182.31 60 $1,344.40 $221.60

(calculation continued) Press Display

1 INPUT AMORT = −137.151941 = −1,428.848059 36 INPUT AMORT = −182.3082 = −1,383.6918

60 INPUT AMORT = −221.596756 = −1,344.403244 (d) Total interest = $83,395.92; Total principal = $10,564.08

(calculation continued) Press Display 1 INPUT 60 AMORT = −10,564.083897

= −83,395.916103

(e) (i) $175,000 = $1,566 × a[[24, j12=?]] + $162,000(1+imo)-24

j12 = 7.276486% j2 = 7.387689% Press Display 12 P/YR 12

175000 PV 175,000 1566 +/− PMT −1,566 162000 +/− FV −162,000 24 N 24 I/YR 7.276486

EFF% 7.524134 2 P/YR 2 NOM% 7.387689

(ii) $162,000 = $1,566 × a[[36, j12=?]] + $164,435.92(1+imo)

-36

j12 = 12.018755% j2 = 12.323743%

Press Display 12 P/YR 12

162000 PV 162,000 1566 +/− PMT −1,566 164435.92 +/− FV −164,435.92 36 N 36 I/YR 12.018755

EFF% 12.703429 2 P/YR 2 NOM% 12.323743 (iii) PV = $1,566 × a[[36, j12=14.551655%]] + $164,435.92(1+imo)

-36

PV = $152,010.38


Press Display 15 NOM% 15 2 P/YR 2 EFF% 15.5625 12 P/YR 12 NOM% 14.551655

1566 +/− PMT −1,566 164435.92 +/− FV −164,435.92 36 N 36 PV 152,010.380609 Discount = $171,379.96 − $152,010.38 = $19,369.58 QUESTION 7 SOLUTION

(a) LTV 95% × $300,000 = $285,000 Premium = 3.6% × $285,000 = $10,260 Face Value = $285,000 + $10,260 = $295,260 GDSR PMT = (0.32 × $70,000) − $2,800 PMT = $19,600 per year = $1,633.33 per month PV = $1,633.33 × a[[300, j12 = 5.926346%]] Press Display 6 NOM% 6 2 P/YR 2 EFF% 6.09 12 P/YR 12 NOM% 5.926346 1633.33 +/− PMT −1,633.33 300 N 300 0 FV 0 PV 255,284.946976 $255,284.95 < face value of $295,260 Maximum loan = $255,284.95

(b) Maximum 2nd mortgage = $295,260 - $255,284.95 = $39,975.05

(c) TDSR PMT = (0.40 x $70,000) – ($1,633.33 x 12) - $2,800 PMT = $5,600.04 per year = $466.67 per month PV = $466.67 x a[[180, j12 = 7.869836%]]


Press Display 8 NOM% 8 2 P/YR 2 EFF% 8.16 12 P/YR 12 NOM% 7.869836

466.67 +/− PMT −466.67 180 N 180 0 FV 0

PV 49,218.852428 Maximum 2nd mortgage of $49,218.85 is greater than amount desired of $39,975.05; therefore, borrowers qualify for loan. Alternatively, find payment on desired amount and calculate TDSR $39,975.05 = PMT × a[[180, j12 = 7.869836%]] Press Display 8 NOM% 8 2 P/YR 2 EFF% 8.16 12 P/YR 12 NOM% 7.869836 39975.05 PV 39,975.05

180 N 180 0 FV 0

PMT −379.024615 TDSR = [($1,633.33 + $379.02)(12) + $2,800]/$70,000 TDSR = $26,948.20/$70,000 TDSR = 38.5% < maximum TDSR of 40%

QUESTION 8 SOLUTION

$95,000 = PMT × a[[60, j4 = 10.365691%]] PMT = $3,137.90 OSB4 = $92,188.86 (January 1, 2013) Press Display 10.5 NOM% 10.5 2 P/YR 2 EFF% 10.775625 4 P/YR 4 NOM% 10.365691

95000 PV 95,000 60 N 60 0 FV 0

PMT −3,137.896265 3137.9 +/− PMT −3,137.9 4 INPUT AMORT PER 4-4 = = = 92,188.862645 Prepay 10% = $92,188.86 x 10% = $9,218.89 “new” loan Jan 1, 2013 = $92,188.86 − $9,218.89


“new” loan = $82,969.97 $82,969.97 = $3,137.90 × a[[N, j4 = 9.38979%]] N = 41.779289 quarters OSB4 = $78,038.80 Jan 1, 2014 Press Display 9.5 NOM% 9.5 2 P/YR 2 EFF% 9.725625 4 P/YR 4 NOM% 9.38979

82969.97 PV 82,969.97 3137.9 +/− PMT −3,137.9 0 FV 0 N 41.779289 4 INPUT AMORT PER 4−4 = = = 78,038.797661 Prepay 10% = $78,038.80 × 10% = $7,803.88 “new” loan Jan 1, 2014 = $78,038.80 − $7,803.88 “new” loan Jan 1, 2014 = $70,234.92 New payment = $3,137.90 x 1.2 = $3,765.48 $70,234.92 = $3,765.48 × a[[N, j4 = 9.38979%]] N = 24.82385 quarters 8 payments remain to end of term: 4 in full and on time; 2 for $1,000 each and final two

payments not made Option 1 OSB at end assuming all payments made + FV of unpaid payments (deficit amount) = OSB as at Jan 1, 2016 + FV of unpaid payments OSB Jan 1, 2016 assuming all remaining payments made in full

Press Display 9.5 NOM% 9.5 2 P/YR 2 EFF% 9.725625 4 P/YR 4 NOM% 9.38979

70234.92 PV 70,234.92 3765.48 +/− PMT −3,765.48 0 FV 0 N 24.82385 8 INPUT AMORT PER 8−8 = = = 51,842.319001


FV of final two unpaid payments FV = $3,765.48 × s[[2, j4 = 9.38979%]] FV = $7,619.35 Press Display 9.5 NOM% 9.5 2 P/YR 2 EFF% 9.725625 4 P/YR 4 NOM% 9.38979 3765.48 +/− PMT −3,765.48 0 PV 0 2 N 2 FV 7,619.352664 FV of two payments for $1,000 (shortfall amount is $2,765.48 ($3,765.48 - $1,000)) FV = $2,765.48 × s[[2, j4 = 9.38979%]](1+iq)

2

FV = $5,595.88(1+iq))2

FV = $5,861.68 Press Display 9.5 NOM% 9.5 2 P/YR 2 EFF% 9.725625 4 P/YR 4 NOM% 9.38979 2765.48 +/− PMT −2,765.48 0 PV 0 2 N 2 FV 5,595.87819 5595.88 PV 5,595.88 0 PMT 0 2 N 2 FV −5,861.6843 OSB as at January 1, 2016 = OSB at end assuming all payments made + FV of unpaid payments (deficit amount) OSB Jan 1, 2016 = $51,842.32 + $7,619.35 + $5,861.68 OSB Jan 1, 2016 = $65,323.35 Option 2 Find the FV (OSB Jan 1, 2016) assuming no further payments are made and then subtract the FV of the payments actually made in the final year FV of loan (no further payments) FV = OSB Jan 1, 2015(1+iq)

4 FV = $61,465.08(1+iq)

4 OSB Jan 1, 2016 (no further payments) = FV = $61,465,08(1+iq)

4

FV = $67,442.94 (OSB Jan 1, 2016)



70234.92 PV 70,234.92 3765.48 +/− PMT −3,765.48 0 FV 0 N 24.82385 4 INPUT AMORT PER 4−4 = = = 61,465.080359 61465.08 PV 61,465.08 0 PMT 0 4 N 4 FV 67,442.943187 FV of payments made April 1, 2015 payment FV = $1,000(1+iq)

3 FV = $1,072.09 July 1, 2015 payment FV = $1,000(1+iq)

2

FV = $1,047.50


1000 PV 1,000 0 PMT 0 3 N 3 FV −1,072.089512 2 N 2 FV −1,047.5 OSB as at Jan 1, 2016 = $67,442.94 − $1,072.09 − $1,047.50 OSB as at Jan 1, 2016 = $65,323.35

QUESTION 9 SOLUTION

1. Constant payment repayment scheme $100,000 = PMT × a[[300, j12 = 8.835748%]] PMT = $827.98 OSB24 = $97,603.23


Press Display 9 NOM% 9 2 P/YR 2 EFF% 9.2025 12 P/YR 12 NOM% 8.835748 100000 PV 100,000 300 N 300 0 FV 0 PMT −827.977389 827.98 +/− PMT −827.98 24 INPUT AMORT PER 24−24 = = = 97,603.226677

2. Straight line principal reduction Principal amount of each payment = $100,000/20 = $5,000 per semi-annual period Total Payment = Interest + Principal Interest1 = $100,000 × 5% = $5,000 Payment1 = $5,000 + $5,000 = $10,000 Interest2 = ($100,000 - $5,000) × 5% Interest2 = $95,000 × 5% = $4,750 Payment2 = $4,750 + $5,000 = $9,750 Interest3 = ($95,000 - $5,000) × 5% Interest3 = $90,000 × 5% = $4,500

Payment3 = $4,500 + $5,000 = $9,500

Interest4 = ($90,000 - $5,000) × 5% Interest4 = $85,000 × 5% = $4,250 Payment4 = $4,250 + $5,000 = $9,250

OSB4 = $100,000 – ($5,000 x 4) = $80,000

3. Interest accrual loan

FV = $100,000(1 + 6%)4

FV = $126,247.70

Press Display 12 I/YR 12 2 P/YR 2 100000PV 100,000 0 PMT 0 4 N 4 FV −126,247.696

4. Interest only loan

Interest = $100,000 × iq

Interest = $100,000 × 2.956301% Interest = $2,956.30 End of Year 2, owe $100,000 + 8th payment of $2,956.30


Press Display 12 NOM% 12 2 P/YR 2 EFF% 12.36 4 P/YR 4 NOM% 11.825206 ÷ 4 = 2.956301 % 0.029563 × 100000 = 2,956.30141

5. Variable rate loan Year 1 $100,000 = PMT × a[[25, j1 = 8.16%]] PMT = $9,496.24 OSB1 = $98,663.76 Press Display 8 NOM% 8 2 P/YR 2 EFF% 8.16 1 P/YR 1 NOM% 8.16 100000 PV 100,000 25 N 25 0 FV 0 PMT −9,496.240892 9496.24 +/− PMT −9,496.24 1 INPUT AMORT PER 1−1 = = = 98,663.76

Year 2 $98,663.76 = PMT × a[[24, j1 = 11.3025%]] PMT = $12,075.73 OSB2 = $97,739.50

Press Display 11 NOM% 11 2 P/YR 2 EFF% 11.3025 1 P/YR 1 NOM% 11.3025 98663.76 PV 98,663.76 24 N 24 0 FV 0 PMT −12,075.734301 12075.73 +/− PMT −12,075.73 1 INPUT AMORT PER 1−1 = = = 97,739.501474

QUESTION 10 SOLUTION

Original Loan $225,000 = PMT × a[[80, j4 = 13.279566%]] PMT = $8,060.86 OSB20 = $208,589.74



225000 PV 225,000 80 N 80

0 FV 0 PMT −8,060.863237 8060.86 +/− PMT −8,060.86 20 INPUT AMORT PER 20-20 = = = 208,589.741184 Option 1 $208,589.74 = PMT × a[[60, j4 = 9.633983%]] PMT = $6,608.58 OSB20 = $168,479.78 $208,589.74 − $250 = $6,608.58 × a[[20, j4 = ?]] + $168,479.78(1+iq)

-20 j4 = 9.667014% j2 = 9.783828% Press Display 9.75 NOM% 9.75 2 P/YR 2 EFF% 9.987656 4 P/YR 4 NOM% 9.633983

208589.74 PV 208,589.74 60 N 60

0 FV 0 PMT −6,608.582447 6608.58 +/- PMT −6,608.58 20 INPUT AMORT PER 20-20 = = = 168,479.783663 208589.74 − 250 = 208,339.74 PV 208,339.74 6608.58 +/− PMT −6,608.58 168479.78 +/− FV −168,479.78 20 N 20 I/YR 9.667014 EFF% 10.023137 2 P/YR 2 NOM% 9.783828 Option 2 $208,589.74 = PMT × a[[60, j4 = 8.900966%]] PMT = $6,332.37 OSB20 = $166,574.97 Penalty = $208,589.74 × (13.135263%÷12) × 3 = $6,849.70 Where 13.135263% is the j12 equivalent for the original contract rate of j2 = 13.5%


$208,589.74 − $6,849.70 − $1,000 = $6,332.37 × a[[20, j4 = ?]] + $166,574.97(1+iq)-20

j4 = 9.949601% j2 = 10.073345% Press Display 9 NOM% 9 2 P/YR 2 EFF% 9.2025 4 P/YR 4 NOM% 8.900966

208589.74 PV 208,589.74 60 N 60

0 FV 0 PMT −6,332.367749 6332.37 +/− PMT −6,332.37 20 INPUT AMORT PER 20−20 = = = 166,574.973109 208589.74 − 6849.70 − 1000 = PV 200,740.04 4 P/YR 6332.37 +/− PMT −6,332.37 166574.97 +/− FV −166,574.97 20 N 20 I/YR 9.949601 EFF% 10.327025 2 P/YR 2 NOM% 10.073345 QUESTION 11 SOLUTION $150,000 = PMT × a[[240, j12 = 9.076624%]] PMT = $1,356.99 OSB60 = $133,190.98 $150,000 = $1,356.99 × a[[60, j12 = 14%]] + $133,190.98(1+imo)

-60 + SHARE(1+imo)-60

FV = $183,875.55 − $133,190.98 FV = $50,684.57 (Share)



150000 PV 150,000 240 N 240

0 FV 0 PMT −1,356.989685 1356.99 +/− PMT −1,356.99 60 INPUT AMORT PER 60-60 = = = 133,190.981603 14 I/YR 14 12 P/YR 12 150000 PV 150,000 1356.99 +/− PMT −1,356.99 60 N 60 FV −183,875.546116 Share = $183,875.55 - $133,190.98 = $50,684.57 Total Gain = $50,684.57/0.45 = $112,632.38 NSP = $112,632.38 + $200,000 = $312,632.38 GSP = $312,632.38/0.92 = $339,817.80 $200,000(1+ia)

5 = $339,817.80 ia = j1 = 11.18% Press Display 200000 PV 200,000 339817.8+/− FV −339,817.8 5 N 5 0 PMT 0 1 P/YR 1 I/YR 11.184238 QUESTION 12 SOLUTION $700,000 = PMT × a[[240, j12 = 12%]] PMT = $7,707.60 OSB60 = $642,210.54 $700,000 = $7,707.60 × a[[60, j12 =15.803533%]] + $642,210.54(1+imo)

-60 + Participation Pmt x a[[5 years, j1=17%]] $700,000 = $611,233.85 + Participation Pmt x a[[5 years, j1=17%]] $88,766.15 = Participation Pmt × a[[5 years, j1=17%]] Participation Pmt = $27,745.09 Gross Potential Rental Income = $27,745.09 ÷ 0.10 = $277,451


Press Display 12 I/YR 12

12 P/YR 12 700000 PV 700,000 240 N 240

0 FV 0 PMT −7,707.602935 7707.6 +/− PMT −7,707.6 60 INPUT AMORT PER 60-60 = = = 642,210.541613 17 NOM% 17 1 P/YR 1 EFF% 17 12 P/YR 12 NOM% 15.803533

7707.6 +/− PMT −7,707.6 642210.54 +/− FV −642,210.54 60 N 60 PV 611,233.851456 17 I/YR 17 1 P/YR 1 700000 − 611233.85 = 88,766.15 PV 88,766.15 5 N 5 0 FV 0 PMT −27,745.090867 ÷ 0.1 = −277,450.90867

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