1

WEEK 2TUTORIAL QUESTIONS

Chapter 5

Question 1.

For each of the following compute the future value

Present InterestValue Rate

$20,000 8 10%$40,000 8 10%$20,000 8 20%$20,000 16 10%

Years

FV = 20 000(1.10)8 = $42 871.78 FV = 40 000(1.10)8 = $85 743.55 FV = 20 000(1.20)8 = $85 996.34FV = 20 000(1.10)16 = $91 899.46

10 I/Y8 N

20000 +/- PVCOMP FV

= 42,871.78

0.1 8

FV = 20000 1 + ------------1

= $42,871.8

Question 2.

For each of the following, compute the present value:

Future InterestValue Rate$4,995 13 4%$4,782 8 6%$8,277 15 7%$14,305 7 25%

Years

PV = 4 995/(1.04)13 = $2 999.87PV = 4 782/(1.06)8 = $3 000.29PV = 8 277/(1.07)15 = $2 999.97PV = 14 305/(1.25)7 = $2 999.98

2

4 I/Y13 N

4995 +/- FVCOMP PV

= 2,999.87

1 + 13

2,999.87$ PV =

4995

0.04

Question 3.

A local bank is offering 4.75 per cent compounded daily on savings accounts. If you deposit $2,000 today, how much will you have in two years? How much will you have in four years?

Daily rate = 4.75/365 = 0 .013014%

2 years: FV = 2 000(1.00013014)2×365 = $2 199.314 years: FV = 2 000(1.00013014)4×365 = $2 418.48

4.75 365 I/Y730 N2000 +/- PV

COMP FV= 2,199.30

Question 4.

Assume that the cost of a university education will be $150,000 when your children enter university in 15 years' time. You presently have $40,000 to invest. What rate of interest must be simply earned on your investment to cover the cost of a university education 15 years from now?

150,000 = 40 000(1 + r)15 3.75 = (1 + r)15

3.751/15 = 1+ r1.0921 = 1 + rr = 9.21%

15 N40000 +/- PV150000 FVCOMP I/Y

= 9.21

3

Question 5.

You are considering an investment that has the following cash flows. If the discount rate is 8 per cent, what is the present value of these flows? What is the present value at 10 per cent? What is the present value if the interest rate is zero?

8%: PV = 200/(1.08) + 400/(1.08)2 + 800/(1.08)3

= 185 + 343 + 635= $1 163

10% PV = 200/(1.10) + 400/(1.10)2 + 800/(1.10)3

= 182 + 331 + 601= $1 114

0% = 200 + 400 + 800= $1 400

Must be in NORMAL Mode0 +/- Data

200 Data400 Data800 Data

on/C2ndF CASH 2ndF CA

8 ENTCOMP

= 1163.19

Question 6.

What is the future value of $8,000 in 15 years, assuming a rate of 12 per cent compounded monthly?

EAR = (1.01)12 − 1 = 12.6825%FV = 8 000(1.126825)15 = $47 966.40

12 x 15 N12 12 I/Y

8000 +/- PVCOMP FV

= 47,966.42

x

YearInterest

Rate1 $2002 $4003 $800

4

0.12 15 x 12FV = 8000 1 + ------------

12= $47,966.4

Question 7.

You have just joined the investment advisory firm Skyhigh & Co. They have offered you two very. different salary arrangements. You can have $250,000 per year for the next five years; or $145,000 per year for the next five years, along with a $400,000 signing bonus today. If the interest rate is 6 per cent compounded quarterly, which do you prefer? For simplicity, assume the salaries are to be paid at the end of each year.

EAR = (1.015)4 – 1 = 6.136355%Option 1: PV = 250 000/(1.06136355) + 250 000/(1.06136355)2 + 250 000/(1.06136355)3 + 250 000/(1.06136355)4 + 250 000/(1.06136355)5

= $1 049 196

Option 2: PV = 400 000 + 145 000/(1.06136355) + 145 000/(1.06136355)2 + 145 000/(1.06136355)3 + 145 000/(1.06136355)4 + 145 000/(1.06136355)5

= $1 008 534Select Option 1. This is assuming the salary is received at year end.

Question 8.

A bank is offering the following rates for term deposits: a) 6 per cent per annum paid annually, b) 5.91 per cent per annum paid half-yearly, c) 5.87 per cent per annum paid quarterly and d) 5.84 per cent per annum paid monthly.

What is the EAR for each of these rates?

a EAR = 6%b EAR = (1.02955)2 − 1 = 5.997%c EAR = (1.014675)4 − 1 = 6%d EAR = (1.004866667)12 − 1 = 5.999%

2 (x,y)5.91 2ndF EFF

= 6.0000

0.0591 22

Effective Annual Rate = 0.0600

Effective Annual Rate = 1 + - 1

5

Question 9.

Find the NIR or stated rate in each of the cases below:

a 1.05 = [1 + (NIR/2)]2

NIR = 4.94 %b 1.06 = [1 + (NIR/4)]4

NIR = 5.87%c 1.07 = [1 + (NIR/12)]12

NIR = 6.78%d 1.08 = [1 + (NIR/365)]365

NIR = 7.70%

2 (x,y)5 2ndF APR= 4.9400

1Nominal Interest Rate = EAR + 1 ------ - 1 x n

n

10.0494 = 0.05 + 1 ------ - 1 x 2

2Question 10.

Namby Bank charges 7 per cent, compounded daily, on its personal loans. Combi Bank charges 7.1 per cent, compounded semi-annually. As a potential borrower, which do you prefer?

Namby EAR = (1 + 0.07/365)365 − 1 = 7.25%Combi EAR = (1 + 0.071/2)2 − 1 = 7.23%

As a borrower, you would prefer Combi.

When Effective Compounded RateSemi-Annually 5%

Quarterly 6%Monthly 7%

Daily 8%

6

Question 11.

Investment A pays $2,000 per year for three years. Investment B pays $1,600 per year for four years. Which of these cash flow streams has the higher PV if the discount rate is 10 per cent? If the discount rate is 40 per cent?

A: PV = 2000 x PVIFA(3, 10%) = $4 973.70B: PV = 1600 x PVIFA(4, 10%) = $5 071.78At a 10% discount rate, investment B has a higher present value.

A: PV = 2000 x PVIFA(3, 40%) = $3 177.84B: PV = 1600 x PVIFA(4, 40%) = $2 958.77At a 40% discount rate, investment A has a higher present value.

Question 13.

An investment will pay $100,000 in 10 years. If the appropriate discount rate is 8 per cent, continuously compounded, what is the PV?

PV = 100 000/e(0.08x10) = $44,923.90

PresentValue

PresentValue

0.0800 x 10

PresentValue

= 44,937$

Future Value

100,000

=

2.718 r t

=

2.718

Question 14.

The population of the country Moldavia is growing at 3 per cent per year. The current population is 24 million. What will the population be in eight years? How long until the population exceeds half a billion?

FV = 24 million(1.03)8= 30.402 million people500 million = 24 million(1.03)t

20.8333 = (1.03)t

t = Log 20.8333 / Log 1.03t is close to 102.73 years

500000000 +/- FV24000000 PV

3 I/YCOMP N

= 102.73

7

Future Present Rate PeriodsValue Value per period

8

n

n

log 20.833333 = log

n = 102.7293

1.03

500000000

20.833333 = 1 + 0.03

+

= x 1 +24000000 0.03

30402482

= x 1

= x 1 + 0.0324000000

Question 15.

You are considering buying a Greek investment that requires you to invest $7,000 today in exchange for $12,533.71 in two years. What is the rate of return on this investment?

12 533.71 = 7 000(1 + r)2

1.79053 = (1 + r)2

1.790531/2 = 1 + rr = 33.92%

2 N7000 +/- PV

12533.71 FVCOMP I/Y

= 33.81

TotalFuture Present R PeriodsValue Value Compounding

Periods pa

R 21

R 21

12 R

1

R1

0.3381 = R1

1 +

=1.3381 1 +

1.79053 1 +=

1.79053 =

= 1 +

12533.71 = 7000 1 +

8

Question 16.

You are comparing two investments. Both require a $2,500 initial investment. Investment A returns $4,600 in eight years. Investment B pays $5,250 in 12 years. Which of these investments has the higher return?

A: 4600 = 2 500(1 + r)8

r = 7.92%B: 5250 = 2 500(1 + r)12

r = 6.39%Investment A has a higher return.

8 N2500 +/- PV4600 FV

COMP I/Y= 7.92

12 N2500 +/- PV5250 FV

COMP I/Y= 6.39

Question 18.

A credit card rate is 1.75 per cent per month. Credit cards are legally required to report interest rates. If this was the NIR, what rate should they report? What is the effective annual rate?

NIR = 1.75 x 12 = 21%EAR = (1.021)12 − 1 = 23.14%

12 (x,y)21 2ndF EFF= 23.14

0.21 1212

Effective Annual Rate = 0.2314

Effective Annual Rate = 1 + - 1

Question 19.

An investment offers $125 per quarter for 10 years. If the required return is 7 per cent per annum compounding quarterly, what is the value of the investment? What would the value be if the term were 25 years? Forever?

t = 10years or 40 quarters: PV = 125 x PVIFA(40, 1.75%) = $3 574.28t = 25years or 100 quarters: PV = 125 x PVIFA(100, 1.75%) = $5 882.68tt = : PV = 125/0.0175 = $7 142.86

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Question 20.

If you cashed in your share market investment of $50,000 today and purchased an 8 per cent, 10-year annuity, what will the annual cash flow be?

Payment = 50 000/PVIFA(10, 8%) = $7 451.47

8 I/Y10 N

50000 +/- PVCOMP PMT

= 7,451.47

Present 1 - 1 + Discount Factor - Term

Value

1 - 1 + 0.08-10

500006.710081

6.710081399

Payment =

Payment =

= PaymentDiscount Factor

= Payment0.08

7,451.47$

50000

50000 = Payment

Question 21

Solve for the unknown interest rate in each of the following cases:

Present FutureValue Value$1,100 $3,357 5$1,000 $7,000 15$8,000 $16,609 7$10,625 $25,750 13

Time

3 357 = 1 100(1 + r)5

r = 25%7 000 = 1 000(1 + r)15

r = 13.85%16 609 = 8 000(1 + r)7

r = 11%25 750 = 10 625(1 + r)13

r = 7.05%

10

Question 22.

You have determined that your company can afford a $300,000 annual payment for the next 10 years. A new computer system including software costs $1,800,000 in total. If you can borrow the money at 9 per cent, can you afford the new system?

Payment = 1,800,000/PVIFA(10, 9%) = $280 476.16Yes, the company can afford the computer system.

9 I/Y10 N

1800000 +/- PVCOMP PMT

= 280,476.16

9 I/Y10 N

300000 PMTCOMP PV

= 1,925,287.31

10.09 0.09 1 + 0.09 10

PV Annuity = $1,925,297.31

1 - 1 + -10

PV Annuity = $1,925,297.31

PVAnnuity

= 300000 0.090.09

PVAnnuity

= 300000 - 1

ALTERNATIVELY

Question 23.

What is the relationship between the value of an annuity and the level of the interest rate? What would happen to the value of an annuity if the interest rate were to suddenly increase? Illustrate your answer by calculating the present value of a 10-year annuity of $100 per year at 5, 10 and 15 per cent.

There is an inverse relationship between the value of an annuity and the level of the interest rate. The value of the annuity would drop if the interest rate suddenly increased.

5%: PV = 100 x PVIFA(10, 5%) = $772.1710%: PV = 100 x PVIFA(10, 10%) = $614.4615%: PV = 100 x PVIFA(10, 15%) = $501.88

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Question 24.

If you borrow $120,000 at 6 per cent NIR for 60 months, what will your monthly payment be? What is the effective interest rate on this loan?

Monthly rate = 0.06/12 = 0.5%Payment = 120 000/PVIFA(60, .5%) = $2 319.94EAR = (1.005)12 − 1 = 6.17%

6 12 I/Y60 N

120000 +/- PVCOMP PMT

= 2,319.94

x

Present 1 - 1 + Discount Factor - Term

Value

1 - 1 + 0.005-60

12000051.72556

51.72556075

Payment =

Payment =

= PaymentDiscount Factor

= Payment0.005

2,319.94$

120000

120000 = Payment

6 (x,y)12 2ndF EFF= 12.62

Question 25.

You have just concluded the purchase of a new factory. To finance the purchase, you have arranged for a 10-year mortgage for 80 per cent of the $1400 000 purchase price. The monthly payment will be $100 000. What is the NIR on the loan? The effective annual rate?

Loan amount = 0.8(1 400 000) = $1 120 0001 120 000 = 100 000 x PVIFA(120, monthly rate)11.2 = PVIFA(120, monthly rate)Try 8%PVIFA = [ 1 – 1/(1.08)120 ]/0.08 = 12.5 Try 9%PVIFA = [ 1 – 1/(1.09)120 ]/0.09 = 11.11 Therefore r lies between 8% and 9%, and close to 9%.r is 8.9283%

12

1120000 +/- PV100000 PMT

120 NCOMP I/Y

= 8.93

1 - 1 + -10

PV Annuity = $1,120,000

PV $1,120,000 = 100000 0.08919850.0891985

NIR 8.9283 × 12 = 107.14%EAR (1.089283)12 – 1= 179.05%

Question 26.

One of your customers is having trouble paying her bills. You agree to a repayment schedule of $3,000 per month. You charge 1 per cent per month interest on late accounts. If the current account balance is $200,000, how long will it take until the debt is fully paid?

200 000 = 3 000 × PVIFA(t, 1%)66.66667 = PVIFA(t, 1%)It will take about 110 months.

200000 +/- PV3000 PMT120 I/Y

COMP N= 110.41

Question 27.

If you deposit $3,000 at the end of each of the next six years in an account paying 5 per cent interest, how much will you have in six years? If you left that money in the account for an additional four years, how much will you have in 10 years?

6 years: FV = 3000 × FVIFA(6, 5%) = $20 405.7410 years: FV = 20405.74 × FVIFA(4, 5%)(1.05)4 = $24 803.30

Question 28.

Suppose you deposited $3,000 each year into an account earning 8 per cent interest. The first deposit was made today. If you make 10 deposits in all, how much would you have in 10 years' time?

FV = 3 000 × FVIFA(10, 8%)(1.08) = $46 936.46

13

BGN 10 NMode 8 I/Y

3000 PMTCOMP FV

= 46936

1 + 0.08 10 - 1

1 + 0.08 11 - 1ALTERNATIVELY

46936.46 = 3000 - 30000.08

46936.46 = 3000 1 + 0.080.08

Question 29.

A local loan shark offers 'five for ten on pay-day'. This means you get $5 today and you must repay $10 in 30 days when you get your next pay cheque. What is the effective annual interest rate on this loan?

10 = 5(1 + r)30

2 = (1 + r)30

21/30 = 1 + r 1.02337 = 1 + r daily rate = 2.3374%

5 +/- PV30 N10 FV

COMP I/Y= 2.34

EAR = (1.023374)365 − 1 = 4 596.6%

Question 30.

You think that the value of a piece of beach real estate you just purchased will increase by 5 per cent per year. You paid $950,000 for the property and plan to sell when you can make a $200,000 profit. How long will you wait if the value does increase by 5 per cent per year?

1 150 000 = 950 000(1.05)t

1.2105 = (1.05)t

t = 3.92 years

14

Future Present Rate PeriodsValue Value per period

8

n

n

log 1.2105263 = log

n = 3.915856

0.05

1403583

= x 1

= x 1 + 0.05

+

= x 1 +950000

950000

1.05

1150000

1.2105263 = 1 + 0.05

Question 31.

If a loan has an NIR of 12 per cent, what is the effective annual rate (EAR) assuming the loan calls for semi-annual payments? If the loan calls for monthly payments?

Semi-annual: EAR = (1.06)2 – 1 = 0.1236 = 12.36%Monthly: EAR = (1.01)12 – 1 = 0.1268 = 12.6825%

2 (x,y)12 2ndF EFF= 12.36

12 (x,y)12 2ndF EFF= 12.68

0.12 22

Effective Annual Rate = 0.1236

Effective Annual Rate = 1 + - 1

Question 32.

What will $1000 amount to in four years' time if the interest rate is 8 per cent per annum compounded quarterly? If interest is paid on daily balances?

Quarterly: FV = 1000 x (1.02)16= 1 372.79Daily: FV = 1000 x (1 + .08/365)4 x 365 = 1 377.08

950000 +/- PV5 I/Y

1150000 FVCOMP N

= 3.92

15

1000 +/- PV8 365 I/Y4 365 N

COMP FV= 1,377.08

x

0.08 4 x 4FV = 1000 1 + ------------

4= $1,372.8

0.08 4 x 365FV = 1000 1 + ------------

365= $1,377.1

Question 33.

It is estimated that a fund has a superannuation liability of $12 million to be paid in 24 years. To assess the value of the fund, financial analysts want to discount this liability back to the present. If the discount rate is 5 per cent, what is the present value of this liability?

PV = 12 000 000/(1.05)24 = $3 720 814.92

12000000 +/- FV5 I/Y24 N

COMP PV= 3,720,814.92

1 + 24

3,720,814.92$ PV =

12000000

0.05

Question 34.

Jane has to borrow funds at 8 per cent to purchase an expensive ruby ring. She has a contract to sell the ring for $90,000. Payment is to be received in two years' time. The ring costs $78,000 today. She likes this arrangement as she does not have to give up the ring for two years and by then fashions will possibly have changed. Does Jane have a good arrangement from a financial point of view? How much is it costing her to have the ring for two years? What is the borrowing rate at which she would break even?

2 years’ time Jane must repay the loan:FV = 78 000 (1.08)2 = $90 979.20However, she receives $90 000 from the sale of the ring.Therefore, it costs her 90 000 – 90 979.20 = –979.20

16

78000 +/- PV8 I/Y2 N

COMP FV= 90,979.20

Breakeven for Jane for having the ring is:78 000 = 90 000/(1 + r)2

(1 + r )2= 90 000/78 000r = (90 000/78 000)1/2 – 1r = 7.42%

Question 35.

You have won the lottery and lottery officials offer you the choice of the following alternative pay-outs:Alternative 1: $500 000 one year from nowAlternative 2: $960 000 five years from now

Which should you choose if the discount rate is:a. 0 per cent?b. 10 per cent? c. 15 percent?

a 0% Alt. 1: PV = 500 000/(1.0)1 = $500 000Alt. 2: PV = 960 000/(1.0)5 = $960 000

Choose Alternative 2.b 10% Alt. 1: PV = 500 000/(1.10)1 = $454 545.45

Alt. 2: PV = 960 000/(1.10)5 = $596 084.47Choose Alternative 2.

c 15% Alt. 1: PV = 500 000/(1.15)1 = $434 782.61Alt. 2: PV = 960 000/(1.15)5 = $477 289.67

Choose Alternative 1.

78000 +/- PV2 N

90000 FVCOMP I/Y

= 7.42

17

Question 36.

With a 6 per cent interest rate, calculate the present value of the following streams of payments:

a. $1200 per year forever with the first payment todayb. $9 500 per year forever, with the first payment two years from today c. $30430 per year forever, with the first payment three years from today.

a PV = 1 200 + 1 200/0.06 = $21 200b PV = [9 500/0.06]/(1.06) = $149 371c PV = [30 430/0.06]/(1.06)2 = $451 377

Question 37.

What is the present value of cash flows of $1,200 per year, with the first cash flow received five years from today and the last one 24 years from today (a total of $24000)? Use a 7 per cent interest rate.

PV = [1 200 x PVIFA(20, 7%)]/(1.07)4 = $9 698.55

0 +/- Data0 (x,y) 4 Data

1200 (x,y) 20 Dataon/C

2ndF CASH 2ndF CA7 ENT

COMP= 9698.55

10.07 0.07 1 + 0.07 20

PV Annuity = $12,712.82

1 - 1 + -20

PV Annuity = $12,712.82

PVAnnuity

= 1200 0.070.07

PVAnnuity

= 1200 - 1

ALTERNATIVELY

1 + 4

9,698.55$ PV =

12712.82

0.07

18

Question 38.

Given an interest rate of 5 per cent per year, what is the value at date t = 7 of a perpetual stream of $1000 payments coming at dates t = 12, t = 13, t = 14 and so on to infinity?

Value at t = 7: [1 000/0.05]/(1.05)4 = $16 454.04

Question 39.

You have recently won the Set for Life jackpot in the lottery, with a payoff of $4,960,000. On reading the fine print, you discover that you have the following two options:

You could receive 31 payments of $180,000 per year with the first payment today. The income would be taxed at an average rate of 30 per cent. Assuming that the appropriate interest rate for you is 10 per cent, what is the present value of the after-tax cash flows? (Taxes are withheld when the funds are transferred to your account.)

You could receive $2 000 000 now. This would be taxed at an average rate of 30 per cent. With the after-tax amount of $1400 000, you could take $500 000 now and buy a 30-year annuity with the remaining $900 000. The annuity would pay $132 000 at the end of each year, which would be taxed at an average rate of 30 per cent. Once again using a 10 per cent interest rate, what is the present value of the cash flows under this scenario?Which option would you take?

Option a: After-tax cash flow = 180 000(1 – 0.30)

= $126 000PV = 126 000 + (126 000 × PVIFA(30, 10%))

= $1 313 791.22Option b: After-tax cash flow = 132 000(1 – 0.30)

= $92 400PV = 500 000 + (92 400 × PVIFA(30, 10%))

= $1 371 046.90You should choose option b.

2ndF BGN/END BGNMode

31 N10 I/Y

126000 PMTCOMP PV

= 1313791

19

1 - 1 + -30

PV Annuity =

Alternatively1 - 1 + -30

PV Annuity = $1,313,791.22

0.1PV

Annuity= 126000 0.1 + 1

$1,313,791.22

PVAnnuity

= 126000 + 1260000.1

0.1

Question 41.

What is the present value of receiving a 10-year annuity of $18,000 each year, with the first receipt starting immediately? The interest rate is 10 per cent per annum compounding quarterly.

PV = 18 000 × PVIFA(9, 10.38%) + 18 000 = $120 109.80

Question 42.

You are considering a one-year loan of $206,000. The interest rate is quoted as 5.5 per cent but it compounds daily. What is the effective annual interest rate?

EAR = (1 + 0.055/365)365 − 1 = 5.653%

365 (x,y)5.5 2ndF EFF= 5.65

0.055 365365

Effective Annual Rate = 0.0565

Effective Annual Rate = 1 + - 1

4 (x,y)10 2ndF EFF= 10.38

2ndF BGN/END BGNMode

10 N10.38 I/Y18000 PMTCOMP PV

= 120115

20

Question 46.

A local finance company quotes a 20 percent interest rate on one-year loans. So, if you borrow the interest for the year will be $2,000. Since you will pay a total of $12,000 the finance company requires you to pay $1,000 per month over the next 12 months. Is this a 20 per centWhat rate would legally have to be quoted? What is the effective annual rate?

$10 000 = 1 000 × PVIFA(12, monthly rate)PVIFA = 10Monthly rate = 2.92%NIR = 2.92(12) = 35.07%EAR = (1.0292)12 – 1 = 41.30%35.07% is the rate that would legally have to be quoted.

Question 47.

Your friend is celebrating her 35th birthday today and wants to start saving for her anticipated retirement at age 65. She wants to be able to withdraw $10,000 from her savings account on each birthday for 10 years following her retirement; the first withdrawal will be on her 66th birthday. Your friend intends to invest her money in the local savings bank, which offers 7 per cent interest per year. She wants to make equal, annual payments on each birthday in a new savings account she will establish for her retirement fund.

a. If she starts making these deposits on her 36th birthday and continues to make deposits until she is 65 (the last deposit will be on her 65th birthday), what amount must she deposit annually to be able to make the desired withdrawals on retirement?

b. Suppose your friend has just inherited a large sum of money. Rather than making equal payments, she has decided to make one lump-sum payment on her 36th birthday to cover her retirement needs. What amount would she have to deposit?

PV at age 65 = 10 000 x PVIFA(10, 7%) = $70 236a 70 236 = PMT × FVIFA(30, 7%)

PMT = $743.54b PV = 70 236/(1.07)29 = $9 872.54