Chapter 5 Review Problems, with worked solutions

Chapter 5 Practice Problems

 

Calculate the future value of the ordinary annuity in questions 1-4.

Size of Payment	Term of the Annuity	Nominal Interest Rate	Payment and Conversion Period
1.$2,100	10 years	9.50%	6 months
2.$4.25	12 years	9.00%	1 day
3.$750	8 years	10.00%	1 month
4.$3,500	27 months	12.00%	3 months

B/E	Py	Cy	N	IY	PV	PMT	FV
E	2	2	20	9.5	0	-2,100	67,631.83
E	365	365	4380	9	0	-4.25	33,511.96
E	12	12	96	10	0	-750	109,636.81
E	4	4	9	12	0	-3,500	35,556.87

Note: #2 might explain how much an “average” smoker could save in 12 years (assuming a constant, year 2000, price for cigarettes and a fixed interest rate).

5. $500.00 is deposited at the end of every six months for nine years in an account paying 10.0% compounded semi-annually. Calculate the accumulated value of the deposits.

B/E	Py	Cy	N	IY	PV	PMT	FV
E	2	2	18	10	0	-500	$14,066.19

 

6. Calculate the amount of interest included in the accumulated value of $600.00 deposits made at the end of each month for 5 years. The interest rate is 13.5% compounded monthly.

B/E	Py	Cy	N	IY	PV	PMT	FV
END	12	12	60	13.5	0	-600	$51,021.08

7. Bill Holden is preparing retirement plans for his employees. He requires each employee to deposit $265.00 at the end of each month for 9 years. The interest rate is 8.75% compounded monthly.

a) How much money will be in each employee’s account at the end of 9 years?

B/E	Py	Cy	N	IY	PV	PMT	FV
END	12	12	108	8.755	0	-275	$43,307.02

b) How much will each employee have actually contributed?

c) How much of the amount will be interest?

Contribution		108*265 =		$28,620.00
Interest		43,307.02 – 28,620		$14,687.02

8. Corinne Smith made $2,750 deposits every 6 months into a registered retirement savings plan paying 11.25% compounded semi-annually. Just after making the 16th deposit, the interest rate changed to 10.00% compounded quarterly. If neither deposits nor withdrawals were made during the next five years how much would Ms. Smith then have in her account?

B/E	Py	Cy	N	IY	PV	PMT	FV
END	2	2	16	11.25	0	-2750	$68,459.66
END	4	4	20	10	-68,459.66	0	$112,179.12

For questions 9-12, calculate the present value of the ordinary annuity.

Size of	Term of	Interest	Payment and Conversion
Payment	the Annuity	Rate	Period
9.$2,100	10 years	9.50%	6 months
10.$4.25	12 years	9.00%	1 day
11.$750	8 years	10.00%	1 month
12.$3,500	27 months	12.00%	3 months

B/E	Py	Cy	N	IY	PV	PMT	FV
END	2	2	20	9.5	$26,734.40	-2100	0
END	365	365	4380	9	$11,382.03	-4.25	0
END	12	12	96	10	$49,426.12	-750	0
END	4	4	9	12	$27,251.38	-3500	0

13. You wish to take two years off work to attend school and also wish to receive $950.00 at the end of every month for the 2 years. If you were able to deposit money into an account paying 10.00% compounded monthly:

a. How much should be deposited when you take the time off?

b. How much interest will you receive in the two years?

B/E	Py	Cy	N	IY	PV	PMT	FV
END	12	12	24	10	$ 20,587.31	-950	0

Interest Earned = (950 * 24) – 20,587.31 = $2,212.69

14. A micro-computer system was bought by paying $750 down and an installment contract with payments of $85 at the end of each month for 2½ years. If the interest was calculated at 16.9% compounded monthly:

a) What was the equivalent cash price of the system?

b) How much was the cost of the financing?

B/E	Py	Cy	N	IY	PV	PMT	FV
END	12	12	30	16.9	$ 2,068.12	-85	0

Cost = 2,068.12 + 750 = $2,818.12

Interest Earned = (85 * 30) – 2,068.12 = $481.88

15. Peter Van Dusen opened a trust account to fund his son’s education. The account paid 10.25% compounded quarterly. His son is expected to require four years of quarterly payments of $2,000 with the first payment occurring 10 years 3 months from today. How much must Mr. Van Dusen deposit now so that his son will be able to receive the 4 years of payments? This is an example of a deferred annuity.

Amount required in 10 years and 3 months’ time

B/E	Py	Cy	N	IY	PV	PMT	FV
END	4	4	16	10.25	$25,983.46	-2000	0

Amount to Deposit Now

B/E	Py	Cy	N	IY	PV	PMT	FV
END	4	4	40	10.25	-$9,443.94	0	-$25,983.46

16. To purchase a new trawler-type yacht for chartering, Henry Skipper signed an agreement to borrow the entire amount and to make payments of $2,750 at the end of every month for seven years.

a) What was the purchase price of the yacht if money was worth 15.5% compounded monthly?

B/E	Py	Cy	N	IY	PV	PMT	FV
END	12	12	84	15.5	$140,461.30	-2,750	0

b) In his third year of operation, an economic downturn caused Mr. Skipper to miss payments 25 and 26. What payment was required at the time that payment 27 was due in order to bring the contract up to date?

2,750 * (1+(0.155/12))² = $2,821.50

2,750 * (1 +(0.155/12)) = $2,785.52

2.750$2,750

Payment on Month 27 = $8,357.02

c) Upon receipt of payment 27, the mortgage company wished to invoke a contractual clause and cancel the mortgage. How much (in addition to the payment calculated in part b above) would Mr. Skipper have to pay in order to fully pay out the mortgage?

B/E	Py	Cy	N	IY	PV	PMT	FV
END	12	12	27	15.5	-140,461.30	-2,750	$110,460.68

OR

B/E	Py	Cy	N	IY	PV	PMT	FV
END	12	12	84	15.5	-140,461.30	-2,750	$0

AMORT P1 = 27   P2 = 27BAL = $110,460.68

For questions 17-20, calculate the payment.

Future Value	Present Value	Interest Rate	Payment and Conversion Period	Time
17.	$17,750	10.5%	1 quarter	4 years, 6 months
18.$12,000		16.5%	6 months	20 years
19.$6,500		8.4%	3 months	9 years, 3 months
20.	$12,500	10.5%	1 month	8 years

B/E	Py	Cy	N	IY	PV	PMT	FV
E	4	4	18	10.5	-17,750	$1,250.01	0
E	2	2	40	16.5	0	$43.36	-12,000
E	4	4	37	8.4	0	$117.926	-6500
E	12	12	96	10.5	-12,500	$193.00	0

21) A used Corvette can be bought for $15,000 cash or for equal payments at the end of each quarter for 5 years. Calculate the size of the quarterly payments at 10% compounded quarterly.

B/E	Py	Cy	N	IY	PV	PMT	FV
E	4	4	20	10	-15,000	$962.21	0

22) A Pentium micro-computer system priced at $4,600 can be purchased for $1,900 down and the balance paid by 36 equal monthly payments at 13.8% compounded monthly. Calculate the size of the monthly payments.

• 4,600 – 1,900 = $2,700

B/E	Py	Cy	N	IY	PV	PMT	FV
E	12	12	36	13.8	-2,700	$92.02	0

For questions 23-26 calculate the term of the annuity:

B/E	Py	Cy	N	IY	PV	PMT	FV
E	4	4	15.58 q = 16	10.75	-8,500	675	0
E	12	12	40.43 q = 42	8.25	0	491.25	-23,500
E	52	52	9.72 weeks = 10	10	-962	100	0
E	2	2	18.5 semi = 19	13	0	2500	-85,000

27) Charlie Horseshoe invested his $250,000 lottery winnings in a term deposit paying 8% compounded monthly for 10 years. For how long can $3,500 be withdrawn from the account at the end of each month starting at the end of the term deposit? Does your answer make sense? If not, why not?

B/E	Py	Cy	N	IY	PV	PMT	FV
E	12	12	120	8	-250,000	0	$554,910
E	12	12	n = ∞	8	554,910	-3,500	0

With a payment of $3,500 per month, you will never spend more than you earn

Perpetuity PMT = Principal * (I) = 554,910 * 0.08/12 = $3,699.40

28) $1,000 is deposited at the end of each month for 5 years. Find the nominal rate of interest compounded monthly at which the deposits will accumulate to $80,000. (Answer in% using 2 places of decimals.)

B/E	Py	Cy	N	IY	PV	PMT	FV
E	12	12	60	11.2269	0	-1,000	80,000

You will need to earn a nominal interest rate of 11.23%

29) A camperized van can be purchased for $27,500 plus 14% GST and Provincial Sales Tax. The dealer will finance the balance owing after the payment of the sales tax and a downpayment of 20% of the price (without the taxes). Payments will be $330.30 at the end of each month for 7 years. What nominal rate of interest compounded monthly is being charged?

• 27,500 *0.14 = $3,850 $27,500 * 0.02 = $5,500
• Pay $3,850 (Taxes) + $5,500 (DP)Still Owing = $22,000

B/E	Py	Cy	N	IY	PV	PMT	FV
E	12	12	84	6.8380805	-22,000	330.3	0

The nominal interest rate is 6.83808%

30) A micro-computer system valued at $2,800 can be bought for 25% down and monthly payments of $84.60 for two and one-half years. What effective rate of interest is being charged?

B/E	Py	Cy	N	IY	PV	PMT	FV
E	12	12	30	15.222	-2100	84.6	0

NOM = 15.222Cy = 12EFF = 16.33%

3l.A loan of $30,000 is to be repaid with monthly payments over a period of 20 years. Calculate the total savings in interest if the loan is financed at 12.25% compounded monthly rather than 13% compounded monthly.

B/E	Py	Cy	N	IY	PV	PMT	FV
E	12	12	240	12.25	-30,000	$335.57	0
E	12	12	240	13	-30,000	$351.47	0

At 12.25% PM2nd Amort P1 = 1P2 = 240INT = $50,536.66

At 13% PM2nd Amort P1 = 1P2 = 240INT = $54,353.45

Interest Saving = $54,353.45 – $50,536.66 = $3,816.79

32) Judith Leisure-Lee is able to set aside $1,500 every 3 months from her income. She plans to buy a studio condominium at Hemlock Valley Ski Area when she has accumulated at least $50,000. How long will it take her if she can invest her savings at 9.5% converted quarterly? (State the answer in months.)

B/E	Py	Cy	N	IY	PV	PMT	FV
E	4	4	24.84	9.5	0	-1500	50,000

She can invest after 75 months (25 quarters)

33) A firm believer in technological education wishes to provide an educational institution with $6,000 bursaries to be awarded at the end of each year for the next 10 years. If the institution can invest money at 8.75% effective, how much should the philanthropist donate, one year prior to the first award, to set up the fund for the 10 bursaries?

B/E	Py	Cy	N	IY	PV	PMT	FV
	1	1	10	8.75	38,933.32	-6000	0

Need to invest $38,933.32 now.

34) An agreement for sale contract carries payments of $4,500 at the end of every six months for 10 years. How much should you be willing to pay for the contract if you require a return of 12.25% compounded semi-annually on your money?

B/E	Py	Cy	N	IY	PV	PMT	FV
	2	2	20	12.25	51,094.94	-4500	0

Pay $51,094.94 for the contract

35) Samuel Hardy received $48,650 as a severance settlement when his position was terminated. Harvey had just celebrated his 41st birthday. He immediately and prudently invested the money in an account paying a guaranteed 9.6% compounded semi-annually until his 60th birthday last year. At that time, he converted the existing balance into an ordinary annuity paying $3,750 per month with interest at 10% compounded monthly. For how long will the annuity run until all the funds have been paid out?

B/E	Py	Cy	N	IY	PV	PMT	FV
	2	2	38	9.6	-48,650	0	288,942.61
	12	12	123.81	10	288,942.61	-3,750	0

The annuity will end after 10 years and 4 months (123.81 months)

36) Judith Leisure-Lee purchased a small studio condominium at Whistler for $115,000. She paid $40 ,000 down and agreed to make equal payments at the end of every month for 25 years. The interest rate was 13.25% compounded monthly.

Mortgage = 115,000 – 40,000 = $75,000

a) What size payment is Judith making each month?

B/E	Py	Cy	N	IY	PV	PMT	FV
E	12	12	300	13.25	-75,000	860.0532	0

Monthly Payments rounded up will be $860.03

N = 299.987

b) After 10 years of payments, how much will Judith still owe?

2^nd Amort P1 = 1 P2 = 120BAL = $67,097.29

c) How much will she have paid, in total, over the 25 years?

• 2^nd Amort P1 = 1 P2 = 300BAL = $11.00

860.03 *299 + 860.03 – 11) = 257,148.97 + 849.03 = $257,998

d) How much total interest will she have paid after 25 years of payments?

257,998 – 75,000 = $182,998

37) A well-used car, priced at $1,250 was sold on “easy terms” for a downpayment of $250 and $50 per month for two years. What effective interest rate is being charged?

B/E	Py	Cy	N	IY	PV	PMT	FV
E	12	12	24	18.157	-1000	50	0

NOM = 18.157Cy = 12EFF = 19.7469%

38) Dogwood Holdings financed a factory expansion by borrowing $325,000 at 10% compounded semi-annually for 10 years. Payments are to be made at the end of every six months.

• Calculate the size of the payment.

B/E	Py	Cy	N	IY	PV	PMT	FV
E	2	2	20	10	-325,000	$26,078.84	0

• How much of the 4th payment is interest?

2^nd Amort P1 = 4 P2 = 4INT= $14,700.73

• Calculate the outstanding balance after the 4th payment.

2^nd Amort P1 = 4 P2 = 4BAL= $282,636.47

• Construct an amortization schedule for the first 4 payments.

Payment	Amount	Interest	Principal	Balance
0				325,000.00
1	26,078.85	16,250.00	9,828.85	315,171.15
2	26,078.85	15,758.56	10,320.29	304,850.86
3	26,078.85	15,242.54	10,836.31	294,014.55
4	26,078.85	14,700.73	11,378.12	282,636.43

• Save-On-Auto Parts borrowed $120,000 to purchase a fleet of seven vans. They intend to repay by making monthly payments of $2,400. Interest is at 16% compounded monthly.

• How many full payments will Save-On-Auto Parts have to make?

B/E	Py	Cy	N	IY	PV	PMT	FV
End	12	12	82.944	16	-120,000	2,400	0

Make 82 full payments and 1 partial payment

• Calculate the size of the final payment, to be made one month after the last full $2,400 payment, which will fully amortize the debt.

2^nd AmortP1 = 1P2 = 83BAL = $133.52

Final Payment = $2400 – 133.52 = $2,266.48

• How much interest is included in the 40th payment?

2^nd AmortP1 = 40P2 = 40INT = $1,059.00

• What percentage of the loan will have been repaid by the first 48 monthly payments?

2^nd AmortP1 = 48P2 = 48BAL = $66,691.36

(120,000 – 66,691.36) / 120,000 = 44.424% of the loan is paid after the first 48 payments.

• How much total interest will be paid?
• 2^nd AmortP1 = 1P2 = 83INT = $79,066.48

• A New York lottery offers a choice to the winner of $1,000,000 cash or $8,500 per month for 15 years. Which alternative should the winner select if money is worth:

• 6.0% compounded monthly?
• 6.25% compounded monthly?

B/E	Py	Cy	N	IY	PV	PMT	FV
E	12	12	180	6	$1,007,279.88	-8,500	0
E	12	12	180	6.25	$991,342.82	-8,500	0

• Find the present value and future value for an annuity whose periodic payments of $1,000.00 are made at the beginning of every quarter for 7 years. The rate of interest is 13% compounded quarterly.

B/E	Py	Cy	N	IY	PV	PMT	FV
BGN	4	4	28	13	$18,794.90	-1,000	0
BGN	4	4	28	13	0	-1,000	$46,021.60

• Find the present value and future value for an annuity whose periodic payments of $3,000 are made at the beginning of each six months for 15 years. The rate of interest is 8.5% compounded semi-annually.

B/E	Py	Cy	N	IY	PV	PMT	FV
BGN	2	2	30	8.5	$52,476.38	-3,000	0
BGN	2	2	30	8.5	0	-3,000	$182,913.49

• Amby Dextrous bought a grand piano and agreed to a series of monthly payments of $225 for 4 years starting the date of sale. The rate of interest is 13% compounded monthly.

• Calculate the cash price. $8,477.78
• How much will be paid over the term of its financing? $10,800
• How much of the total payments will be interest? $2,322.22

B/E	Py	Cy	N	IY	PV	PMT	FV
BGN	12	12	48	13	$8,477.78	225	0

$225 * 48 = $10,800 – 8,774.78 = $2,322.22

• Find the cash value of a three-year service contract for monthly payments of $1,600 paid at the beginning of each month. The value of money is 14% compounded monthly.

B/E	Py	Cy	N	IY	PV	PMT	FV
BGN	12	12	36	14	$47,360.41	-1,600	0

• If an annuity with a present value of $50,000 has periodic payments at the beginning of each quarter for 10 years, find the size of the periodic payment. The rate of interest is 12% compounded quarterly.

B/E	Py	Cy	N	IY	PV	PMT	FV
BGN	4	4	40	12	-50,000	$2,100.12	0

• An annuity whose present value is $50,000 is extinguished by payments of $650.00 made at the beginning of each month for 12 years. Find the nominal rate of interest compounded monthly. Calculate the effective rate.

B/E	Py	Cy	N	IY	PV	PMT	FV
BGN	12	12	144	11.993%	-50,000	650	0

NOM = 11.993Cy = 12EFF = 12.6747%

• Payments of $200 were made into a stock ownership plan at the beginning of each quarter for 15 years. They now have a net value of $50,000. What has been the nominal rate of return on the investment?

B/E	Py	Cy	N	IY	PV	PMT	FV
BGN	4	4	60	16.10%	0	200	-50,000

• $100 is deposited into a retirement plan at the beginning of every month for 20 years. One month after the last deposit, money is withdrawn in equal monthly payments for 15 years. If interest is 8.5% compounded monthly, find the size of the monthly withdrawals.

B/E	Py	Cy	N	IY	PV	PMT	FV
BGN	12	12	240	8.5	0	-100	$63,144.02
BGN	12	12	180	8.5	-63,144.02	$617.43	0

• If $75 is deposited into an education fund at the beginning of each month for 18 years and one month after the final deposit monthly withdrawals of $310.56 a month are made until the fund is exhausted, find the term of the annuity of withdrawals. Interest is 7.5% compounded monthly. (Try this problem both as an annuity due and an ordinary annuity.)

B/E	Py	Cy	N	I/Y	PV	PMT	FV
BGN	12	12	18*12	7.50	0	-75.00	$34,308.14
BGN	12	12	186	7.50	-34,308.14	$310.56	0

The annuity lasts for 15.5 years (15 years and 6 months)

• Find the present value of a deferred annuity whose periodic payment is $550 at the beginning of each year for 20 years, with the first payment following a two-year period of deferment. The interest rate is 3.6% compounded annually.

B/E	Py	Cy	N	I/Y	PV	PMT	FV
BGN	1	1	20	3.6	$8,025.44	550	0
End	1	1	2	3.6	$7,477.38	0	8,025.44

• Find the present value of a deferred annuity whose periodic payment of $360.00 is made at the end of every semi-annual period for 18 years after a deferment period of six months. The interest rate is 10% compounded semi-annually.

B/E	Py	Cy	N	I/Y	PV	PMT	FV
END	2	2	36	10	$5,956.87	-360	0
END	2	2	1	10	$5,673.21	0	5,956.87

• A deferred annuity has a present value of $15,000 and periodic payments are made at the beginning of each quarter for 10 years after a deferral period of 8 years. The rate of interest is 6% compounded quarterly. Find the size of the periodic payment.

B/E	Py	Cy	N	I/Y	PV	PMT	FV
END	4	4	32	6	-15,000	0	$24,154.86
BGN	4	4	40	6	-24,154.86	$795.49	0

• Tri-City Holdings borrows $500,000 to fund the expansion of the firm into its eastern market region. If the loan is to be repaid by making equal payments at the end of each quarter for 8 years beginning after a deferral period of 2 years, find the size of the periodic payments. Interest rate is 11% compounded quarterly.

B/E	Py	Cy	N	I/Y	PV	PMT	FV
END	4	4	8	11	-500,000	0	$621,190.28
END	4	4	32	11	-621,190.28	$29,439.84	0

• Find the present value of a perpetuity whose periodic payments of $4,000 are made at the end of each quarter. Interest is at 6.8% compounded quarterly.

B/E	Py	Cy	N	I/Y	PV	PMT	FV
END	4	4	4000	6.8	$235,294.12	-4,000	0

• An institute lecturer position in Math of Finance is being funded by a perpetual fund. The fund earns interest at I0% compounded annually and is to pay $60,000 at the end of each year, with the first payment two years from the date of the fund being set up. Find the size of initial funding required.

B/E	Py	Cy	N	I/Y	PV	PMT	FV
END	1	1	1000	10	$600,000	-60,000	0
END	1	1	1	10	$545,454.55	0	-600,000

• You want to have accumulated $4,000 for your European trip four years from now. If interest is 6.4% compounded quarterly, find the size of quarterly deposits required for 4 years if deposits are made:

• at the beginning of each quarter.
• at the end of each quarter.

B/E	Py	Cy	N	I/Y	PV	PMT	FV
BGN	4	4	16	6.4	0	$217.86	-4,000
END	4	4	16	6.4	0	$221.35	-4,000

• A three-year car lease has a present value of $16,600. If money is worth 13.5% compounded monthly, find the equivalent monthly lease payments, payable in advance for 3 years.

B/E	Py	Cy	N	I/Y	PV	PMT	FV
BGN	12	12	36	13.5	-16,600	$557.06	0

• A Caribbean holiday tour package may be financed by making monthly payments of $300 at the beginning of each month for 2 years. Interest is 15% compounded monthly.

• Find the purchase price now. $6,264.61
• What will be the total paid in installments over the term? $7,200
• How much interest will be paid over the term of the financing? $935.39

B/E	Py	Cy	N	I/Y	PV	PMT	FV
BGN	12	12	24	15	$6,264.61	300	0

Total Paid = $300 * 24 = $7,200

Interest Paid = 7,200 – 6,264.61 = $935.39

• A building will produce net monthly incomes of $2,000 at the beginning of each month indefinitely. What is the maximum purchase price if one can get 14% compounded monthly on one’s money?

B/E	Py	Cy	N	I/Y	PV	PMT	FV
BGN	12	12	12,000	14	$173,428.57	-2,000	0

• $400 is deposited into a retirement fund at the end of each quarter for 10 years and interest is paid at a rate of 9.6% compounded quarterly.
• Find the accumulated balance at the end of the 10 years. $26,370.83
• How much of that balance is interest? $10,370.83

B/E	Py	Cy	N	I/Y	PV	PMT	FV
END	4	4	40	9.6	0	-400	$26,370.83

Interest Earned = 26,370.83 – (40 * 400) = $10,370.83

• A recreational property is purchased for $54,000 with a downpayment of 10% and the balance secured by a mortgage. amortized by equal monthly payments over 20 years. Interest is 16% compounded monthly.
• Loan = $54,000 * 0.9 = $48,600
• Find the size of the monthly payments. $676.16
• Find the balance after 5 years. $46,036.44
• How much will have been paid for the property in total over the 20-year term of the financing? $162,261.80
• How much of the total payments will represent interest? $113,661.80

B/E	Py	Cy	N	I/Y	PV	PMT	FV
END	12	12	240	16	-48,600	676.1503869	0
	12	12	239.975	16	-48,600	676.16	0

Round up to the next penny and determine the new N.

2^nd AMORTP1 = 1P2 = 60BAL = $46,036.44

P1 = 1P2 = 240BAL = $16.60

Total Paid = (240 * 676.16) – 16.60 = $162,261.80

Interest Paid = $162,261.80 – 48,6000 = $113,661.80

• To finance a current $2,000,000 redevelopment, the Jericho Club is selling 2,000 $1,000 debentures to its members. These debentures bear interest at 8% payable yearly. To redeem all these debentures in 5 years, a sinking fund with guaranteed interest at 9% compounded annually is being set up with a bank. The fund requires regular payments to be made at the end of every year.
• How much is the yearly payment to the sinking fund? $334,184.92
• Draw up a sinking fund schedule.
• What is the effective rate of interest being paid by the club to amortize this expenditure? 7.489%

B/E	P/Y	C/Y	N	I/Y	PV	PMT	FV
END	1	1	5	9%	0	$334,184.92	-2,000,000

Payment	Amount	Interest	Principal	Balance
0				$0.00
1	$334,184.92	0	$334,184.92	$334,184.92
2	$334,184.92	30,076.64	$364,261.56	$698,446.48
3	$334,184.92	62,860.18	$397,045.10	$1,095,491.59
4	$334,184.92	98,594.24	$432,779.16	$1,528,270.75
5	$334,184.88	137,544.37	$471,729.25	$2,000,000.00

Total Annual Payments =	Amount paid to Sinking Fund			$334,184.92
	Amount of Interest Payment to Bond Holders		(2,000,000 * 0.08)	$ 160,000.00

Total Annual Payments = $494,184.92

B/E	P/Y	C/Y	N	I/Y	PV	PMT	FV
END	1	1	5	7.489%	2,000,000.00	-494,184.92	0

• An agreement for sale contract carries payments of $4,500 at the end of every six months for 10 years. How much should you be willing to pay for the contract if you require a return of 12.25% compounded monthly on your money? (see #34)

ICONV NOM = 12.25Cy = 12EFF = 12.96

NOM = 12.5669Cy = 2EFF = 12.96

B/E	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	2	2	20	12.5669	$50,447.62	-4,500	0

• John Leisure purchased a small studio condominium at Whistler for $115,000. He paid $40,000 down and agreed to make equal payments at the end of every month for 25 years. The interest rate was 8.75% compounded semi-annually. (see #36)
• Loan = $115,000 – 40,000 = $75,000
• What size payment is John making each month? $608.72
• After 10 years of payments, how much will John still owe? $61,465.88
• How much will he have paid, in total, over the 25 years? $182,606.20
• How much total interest will he have paid after 25years of payments? $107,606.20

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	300	8.75	-75,000	608.71065	0
End	12	2	299.9838	8.75	-75,000	608.72	0

2nd AMORTP1 = 1P2 = 120BAL = $61,465.88

2nd AMORTP1 = 1P2 = 300BAL = $9.80

Total Payments = (300 * 608.72) – 9.80 = $182,606.20

Total Interest = $182,606.20 – 75,000 = $107,606.20

• You want to have accumulated $4,000 for your European trip four years from now. If interest is 6.4% compounded monthly, find the size of the quarterly deposits required for 4 years if deposits are made:
• at the beginning of each quarter.
• at the end of each quarter.

B/E	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	4	12	16	6.40	0	$217.70	-4000
END	4	12.00	16	$6.40	0	$221.20	-4000

Supplementary Problems

• Starting today, Mrs. Robinson will put $500 into her RRSP every month for 20 years. If her RRSP earns 6% compounded monthly, how much will she earn in interest over the 20 years?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	12	12	240	6	0	-500	$232,175.55

Interest Earned = $232,175.55 – (240 * 500) = $112,175.55

• Mrs. Watson wants to save $52,450 for a down payment on a house. She will save $2,000 per quarter, starting today.

• If her invested funds earn 6% compounded quarterly, how long will it take her to reach her goal?
• How much interest will she earn during that time?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	4	4	22	6	0	2,000	-52,450

22 quarters = 5.5 years

Interest Earned = $52,450 – (22 *2,000) = $8,450

• You have just graduated from BCIT. You owe $15,238.98 in student loans. You will be charged 7% interest compounded monthly. You can afford to make monthly payments of $300 starting today.

• How long (in years) will it take you to repay your student loans?
• How much interest will you have paid?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	12	12	60	7	-15,238.98	300	0

60 months = 5 years

Interest Earned = (60 *300) – $15,238.98 = $2,761.02

• How long will it take to save $100,000 if you start saving $1,597 every 3 months, starting today? Assume an interest rate of 6%, compounded quarterly.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	4	4	44	6	0	-1,597	100,000

44 quarters = 11 years

• You borrow $50,000 from the bank to consolidate your credit card debt and student loans. The bank charges you 12% interest compounded monthly. The first payment is one month from now.

• i.How long will it take to pay off the loan if you pay $504.25 per month?

ii.What is the cost of financing?

480 months = 40 years

• How long will it take to pay off the loan if you pay only $500 per month?

Cost of Financing (Interest Paid) = (480 * 504.25) – 50,000 = $192,040

$500 per month will cover the interest cost only, the amount owing will remain at $50,000

• Mr. Eskanderian contributes $1,000 into his RRSP at the end of every quarter for 10 years. If his RRSP earns 10% compounded quarterly, how much interest will he earn in the 10 years?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	4	4	40	10	0	-1,000	$67,402.55

Interest Earned = 67,402.55 – (40 * 1,000) = $27,402.55

Deferred Annuities

• Judith transfers $25,000 into an RRSP today. She plans to let the RRSP accumulate earnings at the rate of 8.75% compounded annually for exactly 10 years and then immediately purchase a 15-year annuity. The first withdrawal will start 3 months after she purchases the annuity. The annuity earns 9% compounded quarterly. What size of payment will she receive every 3 months?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
END	1	1	10	8.75	-25,000	0	$57,840.58
END	4	4	60	9	-57,840.58	$1,766.18	0

• Katherine has recently received an inheritance. She wants to set aside part of the inheritance to put it into an RRSP to save for her retirement. She anticipates that she will need to receive $1,200 per month for 15 years with the first withdrawal starting exactly 10 years from today. The invested funds will earn 7% compounded monthly for the entire

25 years. What amount must she contribute to her RRSP today?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	12	12	180	7	$134,285.94	-1,200	0
END	12	12	120	7	$66,820.18	0	-134,285.94

• Barry wants to set up an annuity that will pay him $3,000 per month for 20 years beginning when he turns 65 years of age. If his current age is 50 years and the invested funds will earn 6.5% compounded monthly, what amount must he invest today?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	12	12	240	6.5	$404,554.54	-3,000	0
BGN	12	12240	6.5		$152,996.91	0	-404,554.54

• Samuel recently inherited money from his grandfather’s estate. He wants to purchase an annuity that will pay $5,000 every 3 months between age 60 (when he plans to retire) and age 65 (when his permanent pension will begin). The first withdrawal is to be 3 months after he reaches 60, and the last is to be on his 65th birthday. If Sam is currently 50-1/2 years old, and the invested funds will earn 6% compounded quarterly, what amount must he invest today?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
END	4	4	20	6	$85,843.20	-5,000	0
END	4	4	38	6	$48,752.43	0	-85,843.20

ll.It is time to start saving for your retirement. You are 40 years old and want to retire when you turn 60. You will deposit $1,000 into an RRSP at the beginning of every month for 20 years. You will then use the accumulated funds to purchase a 15-year annuity with the first withdrawal one

month after your 60th birthday. Assume that the RRSP earns 7% compounded monthly, and the funds invested in the annuity earn 5% compounded monthly.

• Find the size of the monthly withdrawals.
• You decide that you will need at least $5,000 per month to live on when you retire at age 60. How much extra money must you contribute to your RRSP each month so you can withdraw $5,000 every month for 15 years?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	12	12	240	7	0	-1,000	$523,965.40
END	12	12	180	5	523,965.40	$4,143.48	0
END	12	12	180	5	$632,276.21	5,000.00	0
BGN	12	12	240	7	0	$1,206.71	632,276.21

To withdraw $5,000 per month you need to invest an extra $206.71 ($1,206.71 – $1,000)

• Starting today, Giselle Lafleur will deposit $200 in her RRSP each month for 20 years. One month after the last deposit, she will withdraw the money in equal monthly withdrawals for 10 years.

• Find the size of the monthly withdrawals if the invested funds earn j12 = 9%.
• How much interest will Miss Lafleur earn over the next 30 years?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	12	12	240	9	0	-200	$134,579.20
BGN	12	12	120	9	134,579.20	$1,692.10	0

Interest Earned = (1,692.10 * 120) – 200 * 240) = $155,052.20

• You have just celebrated your 20th birthday. You want to retire when you turn 65. Starting today, you are going to make monthly contributions to your RRSP, so that when you retire you can withdraw $2,250 per month for 15 years. The first withdrawal is made when you turn 65. You anticipate the invested funds will earn 7% compounded monthly.

• How much money must you contribute each month to your RRSP to achieve your goal?
• How much interest did you earn during the entire time?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	12	12	180	7	$251,786.14	-2,250	0
BGN	12	12	540	7	0	$66	251,786.14

Interest Earned = (180 * 2,250) – (540 * 66) = $369,360

• Jean is thinking about retiring in five years. He would like to have $50,000 in an account when he retires. He decides to make monthly deposits (at the beginning of each month) in his local credit union where he can earn 7.0% compounded quarterly.

• What would be the required monthly deposit to accumulate $50,000 in five years?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN/END	4	12	60	7	0	$695.09	-50,000

• How much interest does John earn over the 5 years?

Interest Earned = 50,000 – (60 * 695.09) = $8,294.60

• Six years from now, when you turn 55, you are planning to retire. You want to set aside some money today so you can receive $2,500 at the end of every quarter for 15 years with the first withdrawal 3 months after you turn 55. The invested funds earn 9% compounded semi-annually.

• What amount must you invest today?
• Today you need to invest $48,559.18
• How much interest will you earn during the 21 years?

Interest Earned = (60 * 2,500) – 48,559.18 = $101,440.82

Perpetuities and General Annuities

• You have become wealthy beyond your wildest dreams and would like to create a scholarship at BCIT. You would like to give $2,000 per year to a student studying business math. You would like your scholarship to continue in perpetuity.

• How much money should you set aside today if the first payment is in one year and the interest rate is 8% compounded annually?
• How much should you set aside if the first scholarship is today?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	1	1	1,000	8	$27,000	-2,000	0

• You are a single man who recently has been having trouble getting dates. You think the problem is your car, an older model 4-door sedan. You decide you need to buy a sportier car to attract women. You take out a loan to buy a black Honda S2000 convertible sports car with leather interior. The bank requires that you put $9,000 down followed by payments of $925 at the end of every month for 5 years.

• If the bank charges you 12% interest compounded monthly, what is the selling price of your car?
• Cost of Car = 41,583.41 + 9,000 = $50,583.41

• If the bank charges you 12% interest compounded semi­ annually, how much would your monthly payments be? Note: You borrow the same amount as found in part (a).

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	60	12	-41,583.41	$918.92	0

• Mr. Bean borrows $7,500 today. He will repay the loan with quarterly payments at the end of every three months for three years. The interest rate charged is 9% compounded monthly. Find the size of the payment.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	4	12	12	9	-7,500	$720.87	0

• A corporation donates $11,000 to Langara. The funds are invested at 10% compounded annually per year. The interest is paid out each year as a scholarship. How much will be paid out each year if the first scholarship is paid immediately after the donation is received? Note: It’s easier to use the calculator.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	1	1	1,000	10	-11,000	$1,000	0

• You have just won the Set for Eternity Lottery; the lottery will pay you and your descendants $5,000 per month forever with the first payment in one month. Instead of receiving $5,000 per month forever you would like to receive the cash now. What is the cash value of the prize if the interest rate is j₁₂ = 6%?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	12	12,000	6	$1,000,000	-5,000	0

Note: this is a perpetuity

• The TRIUMF research lab at UBC recently received a donation from a private individual. The funds were matched two-to-one by the government. Each year they plan to pay out a scholarship of $8,596.59 as they anticipate earning 11% compounded quarterly on the invested funds. What amount did the private individual donate? (The first scholarship will be one year later). Round answer to the nearest dollar.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	1	4	1,000	11	$75,000	-8,596.59	0

The Investor put up $25,000 and the government put up $50,000

• You have decided to purchase preferred shares of Plutonium Fuel Cells Inc. that pays a semi-annual dividend of $1.25 per share.

• What would you be willing to pay per share if you want to earn at least 5% compounded semi-annually on your investment and the next dividend is to be paid in 6 months?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	2	2	2,000	5	50	-1.25	0

bYou are short of cash and need to sell your shares of Plutonium Fuel Cells Inc. Unfortunately, interest rates have risen to 8% compounded semi-annually. Calculate your gain or loss per share if the next dividend is due in six months – and the dividend per share remains the same.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	2	2	2,000	8	31.25	-1.25	0

Cost = $50.00Selling Price = $31.25LOSS = 50 – 31.25 = $18.75

• You purchase 500 preferred shares of Yardmucks Coffee that pays a quarterly dividend of $.26 per share. The next dividend is due in 3 months. Current interest rates are 8% compounded quarterly.

• What would you be willing to pay for the 500 shares?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	4	4	4,000	8	13	-0.26	0

500 Shares * $13.00 = $6,500

• If interest rates drop to 6.5% compounded quarterly, how much would you expect to gain or lose if you sell all of your shares? Note: the next dividend is due in 3 months and the dividend per share is still $0.26.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	4	4	4,000	6.5	16	-0.26	0

500 Shares * $16.00 = $8,000

Sold at $8,000, purchased at $6,500 Profit = 8,000 – 6,500 = $1,500

• The Winfall lottery offers you two choices for its grand prize. Either a cash prize of $1,500,000 today or $7,000 per month forever (with the first installment one month from now).

• Which choice should you select if interest is 6% compounded monthly?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	12	12,000	6	1,500,000	-7,500	0

Take the Cash now.

• You choose the $1,500,000 today and deposit it into an account earning 6% compounded monthly. How much could you withdraw each month forever? Assume the first withdrawal is one month from now.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	12	12,000	6	1,500,000	-7,500	0

You can withdraw $7,500 per month forever.

• You borrow $50,000 and agree to make monthly payments for 15 years starting one month later. Calculate the size of your payments if the interest rate is 9% effective.
• EFF = 9Cy = 12NOM = 8.6488

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	1	180	9	-50,000	496.74	0

• You borrow $20,000 today from a moneylender called, The Money Branch. As a result of your bad credit The Money Branch charges you an interest rate of 28.8%. You will repay the loan with equal monthly payments over four years with the first payment one month from now.

• Find the size of the monthly payment if the interest is
• 28.8% compounded monthly.
• 28.8% compounded semi-annually.
• How much less interest would you pay if they compound the interest semi-annually, instead of monthly?

Saving of interest cost = (706.23 *48) – (688.03 *48) = $873.60

• The University of Edmonton received a donation from a wealthy individual. Some of the donated money will be set aside to create a scholarship fund that will pay out $10,000 at the end of every 6 months, in perpetuity. If the invested funds can earn 8% compounded semi-annually, instead of 5% compounded semi-annually, how much less money must they set aside today to pay out a $10,000 scholarship?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	2	2	2,000	8	250,000	-10,000	0
End	2	2	2,000	5	400,000	-10,000	0

• The difference in investment required is $400,000 – 250,000 = $150,000
• You purchase 200 preferred shares of Black Bear Brewing that pays a dividend of $1 per share every 3 months. Current interest rates are 12% compounded monthly. The next dividend is due in 3 months.
• How much would you be willing to pay for the 200 shares?
• You would be willing to pay $33.00 per share or $6,600 for 200 shares
• If interest rates fall to 8% compounded quarterly, how much would you expect to gain or lose if you sell all of your shares? Note: the next dividend is due in 3 months and the dividend per share is unchanged.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	4	4	4,000	8	50	-1.00	0

Sell shares at $50 per share ($10,000 for 200 shares)

Gain of $50 – 33 = $17 per share, or 200 * 17 = $3,400 total gain

Mortgages

• a. What is the interest rate per month if the rate is
• 12% compounded monthly?
• 12% / 12mo = 1% per month
• 12% compounded semi-annually?
  Periodic Rate = (NOM / Cy) = 12% / 2 = 6%
• Rate Per Month = (1 + Periodic Rate)^{1 / months in compounding period} -1
  = (1 + 0.06)^{(1 / 6)} – 1 = 0.009758794 = 9.758794%

b. If you borrow $200,000, how much interest is paid in the first month? Use both rates and compare your answers.

Monthly = $200,000 * 0.01 = $2,000

Semi = $200,000 * 0.009758794 = $1,951.76

• A debt of $1,200 is repaid by monthly payments of $350 at the end of each month. The interest rate is 12% compounded monthly. Construct the complete amortization schedule without using the AMRT keys. Then go back and verify all
  
  
  
  
  the answers using the AMRT and Pl/P2 Ikeys .

PeriodPMTINTPRN PaidBAL

000 -1,200

135012338-862

23508.62341.38520.62

33505.2062344.7938175.8262

41.758 175.830

Final Payment is $1.76 + 175.83 = $177.59

• You purchase a house in Surrey for $220,000 and put 25% down so you could receive a conventional mortgage. You decide to get your mortgage at Superstore since they give you free groceries each year as an incentive. A 5-year term mortgage (i.e., the interest rate is fixed for 5 years) is negotiated with President’s Choice Financial Group (Superstore) in which the balance is amortized over 20 years (repaid with equal payments over 20 years) at 6.70% interest compounded semi-annually.

• Calculate your monthly payment. The bank rounds up the payment to the next dollar.

Amount Borrowed = $220,000 * (1-0.25) = $165,000

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	240	6.7	-165,000	1,240.74	0
			239.90			1,241

• COMP n to verify that n is slightly smaller than 240. If the value of n = -100 then you forgot to make the payment negative. If the value of n is exactly 240 then you forgot to re-enter the payment.

• Find out how the first payment is distributed between interest and principal. Compare this to the results for the 60th payment.

2^nd AMORTP1 =1P2 = 1PRN = $332.35INT = $908.65

2^nd AMORTP1 =60P2 = 60PRN = $459.53INT = $781.47

• How much interest did you pay in the first year? By how much was the balance outstanding reduced in the first year?

2^nd AMORT P1 =1 P2 = 12BAL = 160,888.74PRN = $4,111.26INT = $10,780.74

Reduction in PRN = 165,000 – 160,888.74 = $4,111.26

• What percent of the original mortgage was paid off in the first two years?

2^nd AMORT P1 =24 P2 = 24BAL = 156,497.40

% Reduction = (165,000 – 156,497.40) / 165,000 = 5.153%

• How much principal was paid off in the first five years? How much interest did you pay in the first five years?

2^nd AMORT P1 =1 P2 = 60BAL = 141,445.61PRN = $23,554.39INT = $50,905.61

• How much interest did you pay in the fifth year of the mortgage? How much principal was repaid in the fifth year?

2^nd AMORT P1 =49 P2 = 60BAL = 141,445.61PRN = $5,351.30INT = $9,540.70

• How much will you still owe after you have made five years of payments?

Still owing = 165,000 – 23,554.39 = $141,445.61

• Calculate the value of the final payment assuming that the interest rate never changes during the 20 years.

2^nd AMORT P1 =49 P2 = 240BAL = 127.52

Final Payment = 1,241 – 127.52 = $1,113.48

• The Archibald’s are eligible for a Canada Mortgage and Housing Corp. insured mortgage allowing them to qualify for a mortgage of up to 95% of the selling price of the house. They are also subject to the 30% rule: no more than 30% of their gross income can go towards paying the mortgage and property taxes.

• What is the maximum mortgage they qualify for if their gross monthly income is $5,000 and they want to amortize the mortgage over 25 years? Assume that the property taxes on the house are $1,800 per year (after the homeowner’s grant). Current mortgage rates are 6.80% compounded semi-annually. Round answer to the nearest $100.

MonthlyAnnualAnnualMonthly

Income5,00060,000* 30%$18,000

Taxes 150 1,800- 1,800

Available for Mortgage PMT$16,200 / 12$1,350

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	300	6.8	196,189	-1,350	0

Rounded to nearest $100$196,200

The Archibald’s take out a mortgage for $195,000 with Citizen’s Bank amortized over 25 years at 6.8% interest compounded semi-annually for a 5-year term.

• What is the size of the monthly mortgage payment

(round up to the next dollar)?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	300	6.8	-195,000	1,341.82	0
End	12	2	299.89	6.8	-195,000	1,342

• How much interest did they pay in the first five years of the mortgage?

2^nd AMORT P1 =1 P2 = 60BAL = 177,071.48PRN = $17,928.52INT = $62,591.48

• How much money would they still owe on this mortgage after five years of payments?
• Owing = $177,071.48
• When they renew their mortgage in five years time, mortgage rates have fallen to 6.0 % compounded semi­ annually for a five-year term. They have saved $15,000 and will use it to reduce the size of the mortgage. Find the size of their new monthly payments assuming the

balance outstanding is amortized over the remaining time. The bank rounds up the payment to the next dollar.

New Mortgage Amount = 177,071.48 – 15,000 = $162,071.48

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	240	6	-162,071.48	1,154.25	0
End	12	2	239.704	6	-162,071.48	1,155

• What is the size of the final payment of the renewed mortgage assuming that the interest rate does not change during the remaining 20 years?

2^nd AMORT P1 =1 P2 = 240BAL = 341.53

Final Payment = 1,155 – 341.53 = $813.47

• Banks normally use the 30% rule: no more than 30% of your gross income can go towards paying your mortgage and property taxes.

• If your gross monthly income is $4,000 per month and your property taxes are $2,400 per year (after the $470 homeowners grant), what is the largest mortgage a bank would authorize if the mortgage is amortized over 25 years and the rate of interest is 7.45% compounded semi-annually. Round answer to the nearest $100. (Assume monthly payments).

MonthlyAnnualAnnualMonthly

Income4,00048,000* 30%$14,400

Taxes 2,400- 2,400

Available for Mortgage PMT$12,000 / 12$1,000

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	300	7.45	137,283.97	-1,000	0

Rounded to nearest $100$137,300

You purchase a fixer-upper house in downtown Mission for $164,000. Your down payment is 25% and you negotiate a first mortgage for the balance. The mortgage is at 7.45%, compounded semi-annually, amortized over 25 years for a 3-year term.

Size of Mortgage = 164,000 – (164,000 * 0.25) = $123,000

• How large a monthly payment is required? The bank rounds payment up to the next dollar.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	300	7.45	-123,000	895.95	0
End	12	2	299.955	7.45	-123,000	$896

• How much interest did you pay in the first three years of the mortgage?

2^nd AMORT P1 =1 P2 = 36BAL = 117,222.31PRN = $5,777.69INT = $26,478.31

• What percent of the original mortgage have you paid off in the first three years?

You have paid off $5,777.69 of the mortgage after 3 years.

% Paid = 5,777.69 /123,000 = 4.973%

• How much interest did you pay in the third year only?

2^nd AMORT P1 =25 P2 = 36BAL = 117,222.31PRN = $2,068.36INT = $8,683.64

• You have paid $8,683.64 in interest after 3 years.

• In three years the term of your mortgage is up and you wish to renew. Interest rates have increased to 9.25% compounded semi-annually for a three-year term mortgage. At this time you make a lump-sum payment of $10,000 to reduce the size of your mortgage. Calculate the size of the new monthly payments assuming the balance outstanding is amortized over the remaining time. Round up to the next dollar.
• New Mortgage Amount = 117,222.31 – 10,000 = 107,222.31
• What is the size of the final payment of the renewed mortgage assuming that the interest rate does not change during the remaining 22 years?

2^nd AMORT P1 =1 P2 = 254BAL = 394.42

Final Payment = 940 – 394.42 = $545.58

• Suppose you take out a mortgage amortized over 25 years for $179,940 at j2 = 12.5%.
• Find the size of the monthly payment. Round payment to nearest penny.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	300	12.5	-179,940	1,920	0

2^nd AMORT P1 =1 P2 = 300BAL = 0PRN = $179,940INT = $396,060.19

• How much time and money would you save if you make 26 bi-weekly payments (equal to half of the monthly payment) instead of twelve monthly payments each year? Assume that the interest rate never changes during the 25 years.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	26	2	447.874	12.5	-179,940	960	0

2^nd AMORT P1 =1 P2 = 300BAL = 0PRN = $179,940INT = $250,019.51

Time Saved = 25 yrs – (447.874 / 26) = 25 – 17.226 = 7.774 yrs = 7 yrs and 9 months

Interest Saving = 396,060.19 – 250,019.51 = $146,040.68

• In March 2002 the Beckers purchased a house in Delta for $512,500. They made a down payment of exactly 20% and took out a mortgage with TD Canada Trust for the balance at an interest rate of 7.5% compounded semi-annually, for a 5-year term, amortized over 25 years.

Size of Mortgage = 512,500 *0.8 = $410,000

• What is the size of the monthly payment? The bank rounds the payment up to the next dollar.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	300	7.5	-410,000	2,999.375	0
End	12	2	299.82	7.5	-410,000	3,000

• How much of the 30th payment is· interest?

2^nd AMORT P1 =30 P2 = 30BAL = 394,348.12PRN = $569.47INT = $2,430.53

• How much interest did the Beckers pay in the 3rd year of the mortgage?

2^nd AMORT P1 =25 P2 = 36BAL = 390,856.93PRN = $6,856.18INT = $29,143.82

• Today, (March 2007), they have decided to increase the size of their mortgage and use the money for house renovations. How much extra money can they borrow if they want to keep the same monthly payment as before but still pay off the mortgage by March 2027 (twenty years from now)? The interest rate has fallen to 5.1%, compounded semi-annually for a five-year term.
• 2^nd AMORT P1 =1 P2 = 60BAL = 375,532.99

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	240	5.1	–452,806.23	3,000	0

Increase in Mortgage = 452,806.23 – 375,532.99 = $77,273.24

• The Beckers increase their mortgage to $450,000 and amortize it over 20 years at 5.1%, compounded semi­ annually. What is the size of their monthly payment? (The bank rounds up the payment to the next dollar.)

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	240	5.1	-450,000	2,981.41	0
End	12	2	239.918	5.1	-450,000	$2,982

• What is the size of the final payment assuming that the interest rate stays the same over the remaining twenty years?

2^nd AMORT P1 =1 P2 = 240BAL = 244.74

Final Payment = 2982 – 244.74 = $2,737.26

• The Blacks are considering purchasing a three bedroom townhouse in the Killarney area of Vancouver. Their gross monthly income is $12,000 per month. The property taxes on the townhouse are $3,000 per year, and they also have to pay a monthly maintenance fee of $150 per month for the upkeep on the townhouse complex. Banks normally use the 30% rule: no more than 30% of your gross income can go towards paying your mortgage, property taxes, and monthly maintenance fees. What is the largest mortgage a bank would authorize if the mortgage is amortized over 25 years and the rate of interest is 5.6%, compounded semi-annually? (Assume monthly payments.)
• MonthlyAnnualAnnualMonthly
• Income12,000144,000* 30%$43,200
• Taxes 3,000- 3,000
• Maint Fees1501,800- 1,800
• Available for Mortgage PMT$38,400 / 12$3,200
• Rounded to nearest $100$519,300

• Five years ago, the Smiths purchased a home in North Vancouver for $650,000. They made a down payment of exactly 20% and mortgaged the balance with Westminster Savings Credit Union. The interest rate was 5.6%, compounded semi-annually, for a 5-year term, amortized over 25 years.

Size of Mortgage = 650,000 * 0.8 = $520,000

• Calculate the size of the monthly payment required. The credit union rounds the payment up to the next dollar.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	300	5.6	-520,000	3,204.37	0
End	12	2	299.873	7.5	-410,000	$3,205

• What percentage of the original mortgage was paid off in the first 3 years?

2^nd AMORT P1 =1 P2 = 36BAL = 488,507.65PRN = $31,492.35INT = $83,887.65

• How much interest did the Smiths pay in the fourth year of the mortgage?

2^nd AMORT P1 =37 P2 = 48BAL = 476,796.09PRN = $11,711.56INT = $26,748.44

• The Smiths, after having made 5 years of payments, made a lump-sum payment to reduce the outstanding balance to $450,000. What was the amount of the lump-sum payment?

2^nd AMORT P1 =1 P2 = 60BAL = $464,419.50PRN = $55,580.50INT = $136,719.50

Lump Sum = 464,419.50 – 450,000 = $14,419.50

• After making the lump sum payment, the Smiths renew their mortgage for another 5-year term, amortized over the remaining time, at 6.8%, compounded semi-annually. Calculate the new monthly payment. Round the payment up to the next dollar.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	240	6.8	-450,000	3,409.77	0
End	12	2	239.97	7.5	-410,000	$3,410

• Assuming the interest rate remains the same over the remaining time, what is the size of the final payment?

2^nd AMORT P1 =1 P2 = 240BAL = 112.97

Final Payment = 3,410 – 112.97 = = $3,297.03

• In October 2002 the Reids obtained a mortgage for $380,000 at 6.8% interest compounded semi-annually, for a 5-year term, amortized over 25 years. (The bank rounds the payment up to the next dollar.) Today, (October 2007), they have decided to increase the size of their mortgage and use the money for house renovations. How much more money can they borrow if they want to keep the same monthly payment as before but still pay off the mortgage by October 2027 (twenty years from now)? The interest rate has fallen to 4.3%, compounded semi-annually for a five-year term.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	300	6.8	-380,000	2,614.82	0
End	12	2	239.95	6.8	-410,000	$2,615

2^nd AMORT P1 =1 P2 = 60BAL = 345,075.12

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	240	4.3	-421,860.24	2,615	0

Increase in Loan = 421,860.24 – 345,075.12 = $76,785.12

• You purchase a new car. The dealer offers you terms of 20% down and the remainder financed over five years at an interest rate of 8% compounded monthly.

• Find the size of your monthly payment if your first payment is due at the end of the month and the price of the car was $33,906.25 including GST, PST, documentation fee, and government environmental levies.
• Loan Amount = 33,906.25 * 0.8 =

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	12	60	8	-27,125	$550	0

• What is the cost of financing (how much interest will you pay over the life of the loan)?

Interest Cost = (550 * 50) – 27,125 = $5,875

• You would like to save for your retirement by making monthly deposits of $200 into an account. What nominal interest rate of interest, compounded monthly, must you earn to accumulate $1,000,000 in thirty years? Assume that the first payment will be at the end of the month.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	12	360	13.5894%	0	-200	1,000,000

• You are considering quitting smoking due to the high cost of a pack of cigarettes. You smoke 1 pack a day at a cost of $7:50. If you put the $7.50 you would have spent on cigarettes, into a savings account earning 5.75% interest compounded daily, how much would you have in the bank at the end of ten years?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	365	365	3650	5.75	0	-7.5	$36,994.34

• Upon graduation, you have a student loan of $15,000. The most you can afford to pay is $550 per month. How long will it take you to repay the loan with payments of $550 per month starting in one month if the interest rate is 6% compounded monthly?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	12	29.39	6	-15,000	550	0

It will take 30 months to pay off the loan

• Kent sold his car to Carolyn for $1,000 down and monthly payments of $120.03 at the end of every month for 3½ years. The interest rate charged is 12%, compounded monthly. What was the selling price of the car?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	12	42	12	-4,100	120.03	0

Cost of Car = $1,000 + 4,100 = $5,100

• Terry bought a car with $5,000 down followed by equal monthly payments of $783.41 at the end of every month for 2 years at 16% compounded monthly.

• What is the selling price of the car?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	12	24	16	–16,000	783.41	0

Cost of Car = 5,000 + 16,000 = $21,000

• What is the cost of financing?

Cost of Finance = (24 * 783.41) – 16,000 = $2,801.84

• Fred paid $14,000 to buy a used car. He made a down payment followed by equal monthly payments of $249.50 at the end of every month for 4 years at 8% compounded monthly.

• What is the size of the down payment?
• Down Payment = 14,000 – 10,220 = $3,780
• What is the cost of financing?

Cost of Finance = (48 * 249.50) – 10,220 = $1,756

• For $42,000 an individual can purchase a 5-year annuity from Continental Life and receive monthly payments of $871.85 for 5 years with the first payment one month from now. What effective rate of interest does this investment earn?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	1	60	9.3806%	-42,000	871.85	0

Deferred Perpetuities

• A scholarship fund is to be set up. The fund will pay out a scholarship of $20,000 every year with first scholarship paid out two years after the fund is set up. Find the size of the donation needed. Assume the fund will earn 10% effective.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	1	1	1,000	10	-220,000	20,000	0
End	1	1	2	10	$181,818.18	0	220,000

• A bursary fund for BCIT honor students is to be funded by a perpetual fund. The fund earns interest at 10% compounded annually and is to pay $30,000 each year, with the first payment four years after the fund is set up. Find the size of the initial funding that is required.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	1	1	1,000	10	-330,000	30,000	0
End	1	1	4	10	$225,394.44	0	330,000

• An alumnus wants to donate a sum of money to his Alma Mater that will provide a scholarship of $750.00 every six (6) months in perpetuity. If money can be invested at 6% compounded semi-annually and the first $750.00 is to be awarded at the end of one year, how much must he donate to the school today?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	2	2	2,000	6	-25,700	-750	0
End	2	2	2	6	$24,271.845	0	25,700

• You are considering purchasing shares of New Wave Technology Corp. The company has stated that they will pay dividends of $.72 per share every three (3) months with the first dividend paid exactly four (4) years from today. If the current interest rates are 8% compounded quarterly, what would you be willing to pay for one share today?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	4	4	4,000	8	-36.72	-0.72	0
End	4	4	16	8	$26.75	0	36.72

Using All 5 Keys

• You just celebrated your 40th birthday. You dream about retiring when you turn 55. You currently have $80,000 accumulated in your retirement plan. You decide to make deposits each month into a retirement plan for exactly 15 years, starting today. You want to purchase an annuity which will pay you $6,000 per month for l0 years, with the first withdrawal starting one month after your 55th birthday. The retirement plan and the annuity earn 6% compounded monthly.

• How much must you deposit each month into your retirement plan?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	12	120	6	-540,440.72	6,000	0
BGN	12	12	180	6	-80,000	-$1,177.37	540,440.72

• How much interest will you earn over the ENTIRE 24 years?

Interest Earned = (120 * 6000) – ((180 * 1,177.37) + 80,000) = 720,000 – 291,926.60 = $428,073.40

• Barney just celebrated his 40th birthday. He currently has $52,034 accumulated in his retirement plan and he plans to continue making equal monthly deposits into a savings account for 15 years, starting today. Two months after his last deposit, he intends to withdraw $4,000 per month for his living expenses for a period of 5 years. The invested funds earn 6%, compounded monthly, for the entire 20 years.

• Find the size of the monthly deposits.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	12	60	6	-206,902.24	4,000	0
BGN	12	12	180	6	-52,034	-$271	206,902.24

• How much interest will he earn over the entire 20 years?

Interest Earned = (60 * 4000) – ((180 * 271) + 52,034) = 240,000 – 100,814 = $139,186

• You have $50,000 in your RRSP today. For the next 12.5 years you will contribute $500 per month into your RRSP with the first deposit made one month from now. How much will you have in your RRSP at the end of 12.5 years if you earn 6.9598% compounded monthly on your RRSP?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	12	150	6.9598	-50,000	-500	$238,086.105

• You take out a loan for $50,000 and make monthly payments of $500 with the first payment made one month from now. If the interest rate on the loan is 6.9598% compounded monthly, how long (in months) will it take you to pay off the loan?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	12	150	6.9598	-50,000	500	0

It will take 12.5 yrs to pay off the loan

• Starting today, you will contribute $695.09 per month into an RRSP for a period for 5 years. You will then use the accumulated funds to purchase an annuity with monthly payments paid out over 12.5 years. What will be the size of the monthly withdrawals if the first withdrawal is made 2 months after the last deposit? Assume you earn 6.9598% compounded monthly the entire time.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	12	12	60	6.9598	0	695.09	$50,000
End	12	12	150	6.9598	-50,000	-$500	0

• What amount will be in an RRSP after 20 years if monthly contributions of $300 are made for the first 15 years and then contributions of $600 per month are made for the subsequent 5 years? The first deposit is made one month from now and the funds invested in the RRSP earn 7% compounded monthly.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	12	180	7	0	300	$95,088.69
End	12	12	60	7	95,088.69	600	$177,755.87

• Starting today, you contribute $1,200 every 3 months into your RRSP for five years. The interest rate was 10% compounded quarterly for the first 2 years and 9% compounded quarterly for the last 3 years. How much will you have in your RRSP at the end of 5 years?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	4	4	8	10	0	1,200	-$10,745.42
BGN	4	4	12	9	10,745.42	1,200	$30,723.99

• Herb has made contributions of $2,000 to his RRSP at the end of every 6 months for the past 8 years. The RRSP has earned 9.5% compounded semi-annually. Today, he moved the funds to a different play, paying 8% compounded quarterly. He will now contribute $1,500 at the end of every 3 months. How much will he have in his RRSP 7 years from now?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	2	2	16	9.5	0	2,000	-$46,365.73
End	4	4	28	8	46,365.73	1,500	$136,300.67

• The Alumni Association of BCIT would like to set up a scholarship that will pay out $2,000 every 6 months forever. The funds will be deposited in an account which earns 8.16% effective. The first scholarship will be awarded 2 years after the funds are deposited. How much money do they need to set aside today?

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	2	1	2,000	8.16	52,000	2,000	0
End	2	1	4	8.16	$44,449.82	0	$52,000

• You plan to contribute $900 every 6 months into your retirement plan for a period of 15 years. How much INTEREST would you earn over the 15 years if your RRSP earns 7% compounded monthly? The first contribution is made 6 months from now.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	2	12	30	7	0	900	$46,855.69

Interest Earned = 46,855.69 – (30 * 900) = 240,000 – 100,814 = $19,855.69

• Barney just celebrated his 25th birthday. Starting today he will make contributions every month into his retirement plan for a period of 30 years. How much must Barney contribute every month to his retirement plan so he can withdraw $5,000 per month for a period of 20 years with the first withdrawal starting two months after his last deposit? Assume Barney earns 8.5%, compounded quarterly, the entire time.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	4	240	8.5	$578,663.12	-5,000	0
BGN	12	4	360	8.5	0	$352.39	$578,663.12

• Marika has already accumulated $18,000 in her RRSP . If she contributes $2,000 at the end of every 6 months for the next 10 years, and $300 per month for the subsequent 5 years, what amount will she have in her plan at the end of the 15 years? Assume that her plan earns 9%, compounded semi­ annually, for the first 10 years and 9% compounded monthly for the next 5 years.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	2	2	20	9	-18,000	-2,000	$106,153.70
End	12	12	60	9	-106,153.70	-300	$188,830.07

• How much larger will the value of an RRSP be at the end of 25 years if the contributor makes month-end contributions of $300 instead of year-end contributions of $3,600? In both cases the RRSP earns 8.5%, compounded semi-annually.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
End	12	2	300	8.5	0	-300	$302,244.75
End	1	2	25	8.5	0	-3,600	$290,846.96

Difference = 302,244.75 – 290,846.96 = $11,397.79 more for the monthly saver

• What will be the amount in an RRSP after 25 years if contributions of $2,000 are made at the beginning of each year for the first 10 years and contributions of $4,000 are made at the beginning of each year for the subsequent 15 years? Assume that the RRSP earns 8%, compounded quarterly.

BGN/END	P/Y	C/Y	N	I/Y	PV	PMT	FV
BGN	1	4	10	8	0	-2,000	$31,725.99
BGN	1	4	15	8	31,725.99	-4,000	$23,904.53