HS300 Chapter 5

Present value of an annuity due (PVAD)

Problems if the payments are made at the beginning of the year

22. Using a compound annual interest rate of 12%, calculate the future value of: A. An annuity of $100 per year for 5 years B. An annuity due of $100 per year for 4 years

**if the screen displays the word BEGIN, it's set for the beginning of the year payments. ⏏️ BEG/END and the displays show no word, then it is safe for end of the year payments.** A. The FVA of $100 for 5 years at 12 percent compound annual interest is $635.28 HP calculator= ⏏️ C ALL; ⏏️ BEG/END (if begin display);100 +/- PMT; 12 i/yr; 5 N; FV B. The FVAD of $100 for 4 years at 12 percent compound annual interest is 535.28 HP calculator= ⏏️ C ALL; ⏏️ BEG/END (if begin does not display); 100 +/- PMT; 12 i/yr; 4 N; FV

Future value of an annuity (FVA)

A series of equal payments made at the end of each period or specific number of periods

Future value of a single sum (FVSS)

A single sum is an individual cash flow and not in an annuity or series of cash flows. Requires compounding, or increasing, the present value at some interest rate for specified number of years.

32. Which would you prefer to have: $10,000 today in a lump sum or $1,000 per year for 13 years, beginning one year from now, if your opportunity cost is A. 4% B. 6%

A. $10,000 in lump sum now is preferable to the present value of the annuity at $9,985.65 HP calculator=⏏️ C ALL; ⏏️ BEG/END (if only BEG display); 1000, +/-, PMT; 13 N; 4 i/yr; PV B. $10,000 in lump sum is even more preferable at 6% discount rate compared with the present value of the annuity at $8,852.68 HP calculator= ⏏️ C ALL; ⏏️ BEG/END (If BEG display), 1000, +/-, PMT; 13 N; 6 I/yr; PV

28. You have just started a program of deposit $2000 at the beginning of each year in education fund account for your newborn son. How much will be in that account: A. After 11 years, if it earns 8.5% compound annual interest? B. After 13 years, if it earns 8.5%, compound annual interest rate

A. After 11 years at 8.5 percent, there will be $37,098.50 in the account. HP Calculator= ⏏️ C ALL; ⏏️ BEG/END (BEG does not display); 2000, +/-, PMT; 11 N; 8.5 i/yr; FV B. After 13 years at 8.5 percent, there will be $48,197.73 in the account HP calculator= ⏏️ C ALL; ⏏️ BEG/END (BEG must not display); 2000,+/-,PMT; 13 N; 8.5 i/yr; FV

2. To what amount would $1,000 grow by the end of 3 years if it earned: A. 10% simple interest B. 10% compound interest

A. At 10% simple interest, $1,000 would grow to $1,300 by the end of 3 years as shown below: Year 1= $1,000 x 10% is $100 x 3 years = $300.00 B. At 10% compound interest $1,000 would grow to $1,331 by end of 3 years as shown: (HP-10BII keystrokes)

Frank received $7,020 from his grandmother on his 10th birthday. The money was invested and has been earning 10% for the last eight years. A. How much money should be in Frank's account? B. How much interest did Frank earn from his 10% investment over the 8 years? C. How much money would be in Frank's account if the money had been deposited in a 5% savings account? D. How much interest would have be earned at the 5% interest rate? E. Was the amount of interest earned exactly twice as much when the interest rate was 10% rather than that 5%?

A. Frank should have $15,047.99 For HP calculator= ⏏️ and C ALL; 7020, +/-, and PV; 10 and i/yr;8 and N; FV B. Frank earned $8,027.99 ($15,047.99-$7,020= $8,027.99) C. The savings account would have grown to $10,371.74 For HP calculator= ⏏️ and C ALL;7020,+/-, and PV;5 and i/yr;8 and N;FV D. A 5 percent investment would have produced $3,351.74 of interest ($10,371.74-$7,020 = $3,351.74) E. No. Doubling the interest rate increases the interest earned by more than twofold because of the compounding. (2x$3,351.74<$8,027.99) ($6,703.48< $8,027.99) not double

20. You hope to accumulate $45,000 as a down payment on a vacation home in the near future A. If you can set aside $38,000 now in account that will be credit with 8% compounded annual interest, how long would it take until you have the needed down payment? B. What if you can get 9% per year on your money?

A. It will take 2.2 years to accumulate the down payment HP calculator= ⏏️ C ALL; 38000 +/- PV; 45000 FV; 8 i/yr; N B. HP Calculator = ⏏️ C ALL; 38000 +/- PV; 45000 FV; 9 i/yr; N

23. Sally is contributing $2,000 each year to her individual retirement account (IRA). She makes her deposits at the end of the year. How much will she have in the IRA after A. 10 years, if the account earns 8 percent compound interest every year? B. 8 years, if the account earns 7 percent compound interest every year?

A. Sally will have $28,973.12 after 10 years at 8 percent compound interest HP calculator= ⏏️ C ALL; ⏏️ Begin/End (If Begin display); 2000 +/- PMT; 8 i/yr; 10 N; FV B. Slay will have $20,519.61 after 8 years at 7 percent compound HP Calculator= ⏏️ C ALL; ⏏️ Begin/END (If BEG display); 2000 +/- PMT; 7 i/yr; 8 N; FV

35. A bank is willing to lend you $15,000 to make some home improvements. The loan is to be repaid in five equal annual installments, beginning one year from now. If the interest rate on the loan is 8 percent, A. What will be the size of the annual payment B. How much of the second payment would be interest C. How much of the final payment would be principal

A. The annual payment on the $15,000 loan will be $3,756.85 HP calculator= ⏏️ C ALL; ⏏️ BEG/END (If BEG display), 15000 PV, 5 N; 8 i/yr, PMT B. $995.45 of the second payment is allocated to interest. HP calculator= ⏏️ C ALL; ⏏️ BEG/END (If BEG display), 15000 PV; 5 N; 8 i/yr;PMT; 2, INPUT,, and AMORT; =;=. C. $3,478.57 of the final payment will be principal. HP calculator= ⏏️ C ALL; ⏏️ BEG/END (If BEG display), 15000 PV; 5 N; 8 i/yr; PMT; 5, INPUT,,AMORT;=.

5. How is the future value of a single sum (FVSS) affected by A. The interest rate used in the calculation? B. The number of years used in the calculation?

A. The future value of a particular single sum increase as the interest rate used in the calculation increases, and it decreases as the interest rate decreases. B. The future value of a particular single sum increases as the number of years used in the calculation increases, and it decreases as the number of years decreases.

7. Lyle has an opportunity to work overseas for five years. He is trying to decide whether to keep his house or sell it and invest the money. If he sells the house, he will have the net sales proceeds of $100,000 to invest after paying $10,000 in selling expense on the sale A. The current market value of the house is $110,000 and it's value is expected to increase 4% each year over the next five years. What will the house sell for after 5 years if 4% growth is correct? B. How much will the $100,000 net sale proceeds be worth after 5 years if they're invested at 9% compound interest?

A. The house will be worth $133,831.82 if it's price increases by 4 percent in each of the years. HP calculator = ⏏️ and C ALL; 110000, +/-, and PV;4 and i/yr; 5 and N; FV B. The $100,000 investment should grow to $153,862.40 HP calculator= ⏏️ and C ALL; of 100000, +/-, and PV; 9 and i/yr; 5 and N;FV

16. Steve invested the proceeds from sale of his business in an investment that pays no in current income to him. The investment will provide him with $60,000 when it matures in nine years. A. What is the present value of $60,000 when discount it at 8% for the 9 years? B. Immediately after making the investment, Steve decide to purchase another business. However, he needs the funds he just invested. His friend Paul will buy the investment a price that yield 10% on his current investment. How much will Paul pay for the investment?

A. The present value of Steve's future $60,000 fund is $30,014.94. HP calculator= ⏏️ and C ALL; 60000 and FV; 8 and i/yr; 9 and N; PV B. Paul would pay $25,445.86 for the investment HP Calculator= ⏏️ C ALL; 60000 and FV; 10 and i/yr; 9 and N; PV

15. How is the present value of a single sum (PVSS) affected by: A. The discount rate used in the calculation? B. The number of years using a calculation?

A. The present value of a particular single sum decreases as the discount rate used in the calculation increases, and it increases as the discount rate decreases. B. The present value of a particular single sum decreases as the number of years used in the calculation increases, and it increases as the number of years decreases.

26. You have decided that, beginning one year from now, are you going to deposit your $1200 annual dividend check in your savings account at the credit union to build up a retirement fund. That account will be credited with 6% compounded annual interest. A. If you plan to retire 18 years from now, how much will be in the account at that time? B. If you should decide to retired three years earlier than that how much would be in their account? C. How much would be in the account if you contributed at the beginning of each year instead of waiting a year to start contributing to the account and still retired three years earlier?

A. There will be $37,086.78 in the account at the end of 18 years. HP calculator = ⏏️ C ALL; ⏏️ BEG/END (if BEG display); 1200, +/-, PMT; 6 i/yr; 18 N; FV B. At the end of 15 years, the account balance will be $27,931.16. HP calculator= ⏏️ C ALL; ⏏️ BEG/END (if BEG display); 1200, +/-, PMT, 6 i/yr, 15 N; FV C. If the contribution at the beginning of each year, the account will have $29,607.03 at the end of 15 year HP calculator= ⏏️ C ALL; ⏏️ BEG/END (if BEG does not display ); 1200, +/-, PMT; 6 i/yr; 15 N; FV

25. Tracy wants to buy a house for her grandson when she finished college 10 years from now. In the following situations, calculate how much money Tracy will have to set aside each year, beginning one year from now, in order to have $30,000 for a down payment when her grandson graduate. A. The annual contributions and 8% interest each year B. Tracy assumes a 5% interest rate

A. Tracy will need to contribute $2,070.88 each year to accumulate the$30,000 fund in 10 years. HP calculator = ⏏️ C ALL; ⏏️ BEG/END (If BEG display); 30000 FV; 10 N; 8 i/yr; PMT B. Tracy will need to contribute $2,385.14 each year to accumulate the$30,000 fund in 10 years. HP calculator = ⏏️ C ALL; ⏏️ BEG/END (must show BEG); 30000 FV; 10 N; 5i/yr; PMT

27. The round the world trip you and your spouse intend to take on your 25th wedding anniversary, 6 years from now, will cost $22,000: A. How much should you set aside each year, beginning today, to reach the objective if you can earn 9% compound annual interest on your money? B. How will the size of the annual deposit be affected if you can only earn 8% compound annual interest rate?

A. You should set aside $2,682.78 now and at the beginning of each of the next 5 years in order to reach the target amount. HP calculator= ⏏️ C ALL; ⏏️ BEG/END (If BEG does not display); 22000 FV; 9 i/yr; 6 N; PMT B. If only 8 percent compound annual interest is earned, the size of the needed annual deposit rises to $2,776.79 HP calculator= ⏏️ C ALL; ⏏️ BEG/END (If BEG does not display); 22000 FV; 8 i/yr; 6 N; PMT

31. A company leases an office building you own for $25,000 each year. The next rental payment is due in one year. A. For what lump sum amount would you today sell the next three payments if you could invest the proceeds at a 12.5 percent compound annual rate of return? B. If the next rental payment is due later today, what amount would you sell the payments for? ***Definitely need to check with professors on this answer***

A. You would be willing to sell the next 3 payments for $59,533.61 today. HP calculator= ⏏️ C ALL; ⏏️ BEG/END (If BEG does not shows up), 25000, +/- PMT, 12.5 i/yr; 3 N;PV B. In this case the selling price would be higher at $66,975.31 HP Calculator= ⏏️ C ALL; ⏏️ BEG/END( only If BEG display ), 25000, +/-, PMT; 3 N; 12.5 i/yr; PV

13. Although you have made no deposit or withdrawals from emergency fund savings account at the bank, the account balance has risen during the past 3 years from $15,000-$17,613.62. A. What has been the compound annual interest rate that the bank has been credited to your account? B. At that rate, how many more years will be needed until your account balance which is $20,000?

A. Your account has been credited with 5.5 percent compound annual interest. HP calculator= ⏏️ and C ALL; 15000, +/-, and PV; 17,613.62 and FV; 3 and N; i/yr B. Your account balance will reach $20,000 in 2.37 HP calculator= ⏏️ C ALL;17613.62 -/+ PV; 5.5 and i/yr; 20000 FV; n

3. Draw a line depicting each of the following problems: A. What amount will a deposit of $X made at the end of the year one grow to buy to by the beginning of year 7? B. What is the present value at the beginning of year one of a sum of $Y due to be received at the end of year 4

A. |——|—-|—-|—-|—-|—-|—-|—-| _____1__X_2__3__4___5___6____?__7 B. |——-|——|——|——|——| ______1_____2___3____4_y__5

21. Draw a time line depicting a problem in which you are to calculate the future value of: A. An annuity of $100 per year for for 5 years B. An annuity due of $200 per year for 4 years

A. |—————-|————|————|————|—————| ___________1_________2_________3________4________5_______6__ $100__________________$100____$100____$100____$100 Start off first $100 at the beginning of the year and then end of year 2 and onwards ____? B. |————|————|———-|————|———-|———| |____1____|____2____|___3___|____4____|___5___|___6__| $200______$200____$200__$200 The money is deposited at the beginning of each year.

12. According to the rule of 72, approximately how long will it take for some of money to double in value if it earns a compound annual interest rate of 4%?

According to the Rule of 72, it will take about 18 years for a sum of money to double in value if it earns a compound annual interest rate of 4 percent (72/4=18)

Opportunity cost

An activity (in this case waiting to receive the money) is the value of the lost opportunity to engage in the best alternative activity (spending or investing the money now) with the same resources (the specified sum of money). It is typically expressed as a percentage rate of return.

Debt service problem

Can be solved in a manner analogous to that used for solving sinking funds problems as explained earlier, though here we are dealing with the PVA or the PVAD rather than the FVA or the FVAD

Present value of an annuity (PVA)

Compute the present value of a series of level future payments. Problems if the payments are made at the end of the year

Simple Interest

Computed by applying an interest rate to only the original principal.

Compound Interest

Computed by applying an interest rate to the sum of the original principal and the interest credited to it in previous periods. Interest earning interest.

Deferred annuity

Deferred annuity problem is to divide it into two separate parts and use the answer from solving the first part as an input in solving the second part of the problem. The first part is computing the PVA; the second part involves treating the PVA as the FVSS and computing its PVSS

34. Supposed that a bank lend you $10,000 if you agree to repay $4,000 at the end of each of the next three years. What compound annual interest rate is the bank charging you?

HP calculator= ⏏️ C ALL; ⏏️ BEG/END (If BEG display), 10000, PV; 4000, +/-, PMT; 3 N; i/yr

FVA formula

FVA=annual deposit x [(1+i)n-1]/i

FVAD formula

FVAD= FVA formula x (1+i) FVAD= annual deposit x {[(1+i)^n-1] / i}x (1+i) =$100 x [(1.05^4-1)/.05] x (1+.05) =$100 x 4.3101 x 1.0500 =$452.56

FVSS formula

FVSS = PVSS x (1+i)n Solve =single sum of $500 will grow in 7 yrs at 9% ⏏️, C ALL; 500,+/-, PV;9 and i/yr;7 and N; FV Answer is $914.02

FVSS factor

FVSS = PVSS x FVSS factor = PVSS x (1+i)n = $500 x 1.09(7) i= interest N= period of years

Risk premium

Higher degree of risk, the higher the interest rate you should require. Example: market value of the investment may decline. Inflation may erode the purchasing power of your principal.

36. If you deposit $1100 in your bank account today, and add deposit of $600 to it at the end of each of the next 9 years, and if all your deposit and 6% compound annual interest, how much will be in your account immediately after you make the last deposit?

If immediately after you make the last deposit, your account will have $8,753.22 HP calculator= ⏏️ C ALL; ⏏️ BEG/END (If BEG display), 1100, +/- PV; 600, +/-, PMT, 9 N, 6 I/yr; FV

Amortization schedule

In any installment loan, as previously indicated, A portion of each payment is used to pay interest while the rest is applied to reduce the loan principal. Over the term of the loan, the portion each payment used to pay interest declines while the portion used to reduce loan principal increases. And amortization schedule shows the portion of each payment that is applied toward interest and principal.

9. Approximately what rate of compound annual interest must be earned in order for $100 single sum to grow to $300 in 10 years?

In order for $100 to grow to $300 in 10 years, it must earn a compound annual interest rate of 11.61 percent. HP Calculator= ⏏️ and C ALL; 100 +/-PV; 300 and FV;10 and N;i/yr

Risk-free rate

Investing in something that is perfectly safe. A reasonable measure of this minimum opportunity cost is the rate of interest. Example is 3-month US Treasury bills. These bills are always available.

Annuity

Is a finite stream of equal periodic payments

Annuity due

Is a series of equal payments made at the beginning of each period for a specified numbers of periods.

Perpetuity

Is an infinite stream of equal periodic payments

Interest

It is a way of quantifying the opportunity cost incurred by waiting to received money or by giving up the opportunity to delay payment

Time value of money (TVM)

It reflects the idea that a dollar today is worth more than a dollar recieved in the future.

18. How many years would it take $10,000 to grow to $25,000 and a rate of 9%?

It will take 10.63 years for $10,000 to grow to $25,000 at a compound annual interest rate of 9 percent HP= ⏏️ C ALL; 25,000 FV; 10000 +/- PV; 9 i/yr; N

8. At a 5% compound annual interest rate, approximately how long will it take for $1000 single sum to grow to $2000 according to the rule of 72?

It will take approximately 14.4 years for an amount to double if the rate is 5% 72/i=n (72/5=14.4)

1. You have received a bill for services rendered, and the invoice requests that you pay within 30 days. Should you pay the bill immediately on receipt or wait until the end of 30 days?

It would be in your best interest tonpay the bill at the end of the 30 day period. By doing so, you would retain the ability to use the money for your own benefit by either soending it or investing it to earn interest. By oaying immediately, you would incur an opportunty cost, which is the calue you would be forgoing.

Nominal rate of return

Its realized on most investments should be adjusted downward to an after-tax return

Present value of a single sum (PVSS)

Moving back in time by discounting future value to present value. We will ise an interest ir discount rate to calculate PVSS

PVA formula

PVA = annual payment x {1-[1/(1+i)^n)}/i

PVA factor

PVA factor = {1-[1/(1+i)^n)}

PVAD formula

PVAD = PVA formula x (1+i)

PVSS factor

PVSS factor= 1/(1+i)n

PVSS Formula

PVSS=FVSSx[1/(1+i)n] PVSS=FVSS x PVSS factor Solve is present value is $100,000 single sum due 4 yrs bad 10% ⏏️ C ALL; 100000, FV; 4 and N; 10 i/yr; PV Answer is -68,301.35

24. What is the future value of annuity due of $1,500 per year for 17 years and 11.5% interest?

The FVAD of $1,500 per year for 17 years at 11.5 percent interest is $77,998.98. HP calculator= ⏏️ C ALL; ⏏️ Begin/End (If BEG display); 1500 +/- PMT; 11.5 I/yr; 17 N; FV

Sinking fund

The amount of the annual payment or deposit is the unknown value; expected to the funds be deposited at the end of each year.

33. The account in which you deposited to your inheritance as a present balance of $48,000. If the account is credited with 10% compound annual interest if you plan to withdraw $7500 from it per year beginning one year from now how, long would it be before the balance is zero?

The balance in the account will reach zero in 10.72 years HP calculator= ⏏️ C ALL; ⏏️ BEG/END (If BEG display), 48000, +/-, PV; 7500 PMT; 10 i/yr; N

37. A bond with a $1000 face value matures in 15 years and pays $80 in annual interest. If the bond's yield to maturity (discount rate) is 9 percent, what is the current value of the bond? What will be the size of the annual payment?

The current value of the bond is $919.39 HP calculator= ⏏️ C ALL; ⏏️ BEG/END (If Beg Display), 1000 FV; 80 PMT; 9 i/yr; 15 N; PV

14. The factor for calculating the present value of a single sum for a particular number of years in discount rate can be represented as [1/(1.11)7]. What discount rate is being used and how many years of discounting are involved?

The discount rate is 11 percent per year. The number of years of discounting is 7

11. A real estate appraiser has advise you that the value of the homes in your neighborhood has been rising a compound annual rate of about 6% in recent years. On the basis of this information, what is the value today of the home you bought 7 years ago for 119,500?

The house should be worth about $179,700 today. A PV of $119,500 compounded at 6 percent for 7 years produces a FV=$179,683.82 HP calculator = C ALL; 119500, +/-, and PV; 7 and N, 6 and i/yr; FV

4. The FVSS formula for calculating the future value of a $5,000 Single sum for particular number of years and a particular interest rates can be represented as $5,000 x 1.076. What compound annual interest rate is being used, and how many years of compounding are involved?

The interest rate is 7 percent per year. The number of years of compounding is 6.

17. Assume that you owe $10,000 and that is to be repaid in a lump sum at the end of five years. If the lender is willing to accept $6000 today in full settlement of the loan, what annual rate of return (discount) it is the lender affectively offering you?

The lender is offering you a discount rate of 10.76 percent HP calculator=⏏️ C ALL; 10000 and FV; 6000 +/- PV; 5 N; i/yr

30. A bank is willing to lend $20,000 for a home improvement loan to be repaid annually over 5 years based on 7 percent interest. What is the amount of each level payment required to repay the loan?

The level annual payment would be $4,877.81 HP calculator= ⏏️ C ALL; ⏏️ BEG/END (If BEG display), 20000 PV; 5 N; 7 i/yr; PMT

19. There is an attractive piece of undeveloped land that you are considered purchasing. You think that in 5 years it would sell for $30,000. What would you pay for it today if you want to earn a compound annual rate of 12% on your investment?

You should be willing to pay $17,022.81 for the property today. HP calculator= ⏏️ CALL; 30000 FV; 12 i/yr; 5 N; PV

10. Your personal net worth has risen in the past four years from 110,000 to 260,000 due to your shrewd investing. What has been the compound annual rate of growth of your net worth doing this period?

Your personal net worth has grown at a compound annual rate of 23.99 percent. HP calculator= ⏏️ and C ALL; 110000 +/-PV; 260000 and FV; 4 and N;i/yr

FVA factor

[(1+i)n-1]

Rule of 72

a quick formula for computing how long it will take to double money invested at a given interest rate 72/i=n