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Chapter 3 Time Value of Money: An IntroductionProblem 4Suppose Bank One offers a risk-free interest rate of 5.5% on both savings and loans, and Bank Enn offers a risk-free interest rate of 6% on both savings and loans.a.What arbitrage opportunity is available?b.Which bank would experience a surge in the demand for loans? Which bankwould receive a surge in deposits?c.What would you expect to happen to the interest rates the two banks areoffering?a.Take a loan from Bank One at 5.5% and save the money in Bank Enn at 6%.b.Bank One would experience a surge in the demand for loans, while Bank Ennwould receive a surge in deposits.c.Bank One would increase the interest rate, and/or Bank Enn would decrease itsrate.Problem 7Bubba is a shrimp farmer. In an ironic twist, Bubba is allergic to shellfish, so he cannot eat any shrimp. Each day he has one-ton supply of shrimp. The market price of shrimp is $10,000 per ton.a.What is the value of a ton of shrimp to him?b.Would this value change if he were not allergic to shrimp? Why or why not?a.The value of one ton of shrimp to Bubba is $10,000 because that is the marketprice.b.No. As long as he can buy or sell shrimp at $10,000 per ton, his personalpreference or use for shrimp is irrelevant to the value of the shrimp.Problem 11A friend asks to borrow $55 from you and in return will pay you $58 in one year. If your bank is offering a 6% interest rate on deposits and loans:a. How much would you have in one year if you deposited the $55 instead?b. How much money could you borrow today is you pay the bank $58 in one year?c. Should you loan the money to your friend or deposit it in the bank?a. I f you deposit the money in the bank today you will have:$1.06 in one year FV $55 today $58.30 in one year $ today ⎛⎫=⨯= ⎪⎝⎭b.If you lend the money to your friend for one year and borrow against the promised $58 repayment, then you could borrow:$1.06 in one year PV $58 in one year $54.72 today $ today ⎛⎫=÷= ⎪⎝⎭c. F rom a financial perspective, you should deposit the money in the bank, as it will result in more money for you at the end of the year.Problem 16Calculate the future value of $2000 ina. Five years at an interest rate of 5% per year.b. Ten years at an interest rate of 5% per year.c. Five years at an interest rate of 10% per year.d. Why is the amount of interest earned in part (a) less than half the amount of interest earned in part (b)?a. Timeline:0 1 2 555FV 2,000 1.052,552.56=⨯=b. Timeline:0 1 2 101010FV 2,000 1.053,257.79=⨯=c. Timeline:0 1 2 555FV 2,000 1.13,221.02=⨯=d. Because in the last 5 years you get interest on the interest earned in the first 5 years as well as interest on the original $2,000.Problem 22Your grandfather put some money in an account for you on the day you were born. You are now 18 years old and are allowed to withdraw the money for the first time. The account currently has $3996 in it and pays an 8% interest rate.a. How much money would be in the account if you left the money there until your 25th birthday?b. What if you left the money until your 65th birthday?c. How much money did your grandfather originally put in the account?a. Timeline:18 19 20 21 25 0 1 2 3 77FV 3,996(1.08)6,848.44==b. Timeline:18 19 20 21 65 0 1 2 3 4747FV 3,996(1.08)148,779==c. Timeline:0 1 2 3 4 18183,996PV 1,0001.08==Chapter 4 Time Value of Money: Valuing Cash Flow StreamsProblem 1You have just taken out a five-year loan from a bank to buy an engagement ring. The ring costs $5000. You plan to put down $1000 and borrow $4000. You will need to make annual payments of $1000 at the end of each year. Show the timeline of the loan from your perspective. How would the timeline differ if you created it from the bank’s perspective?0 1 2 3 4 5From the bank’s perspective, the timeline is the same except all the signs are reversed.Problem 9The British government has a consol bond outstanding paying £100 per year forever. Assume the current interest rate is 4% per year.a. What is the value of the bond immediately after a payment is made?b. What is the value of the bond immediately before a payment is made? Timeline:0 12 3a. The value of the bond is equal to the present value of the cash flows. By the perpetuity formula:100PV 2,500.0.04£==b. The value of the bond is equal to the present value of the cash flows. The cash flows are the perpetuity plus the payment that will be received immediately.PV =100+100=£2,600 Problem 30You are saving for retirement. To live comfortably, you decide you will need to save $2 million by the time you are 65. Today is your 30th birthday, and you decide, starting today and continuing on every birthday up to and including your 65th birthday, that you will put the same amount into a savings account. If the interest rate is 5%, how much must you set aside each year to make sure that you will have $2 million in the account on your 65th birthday? Timeline:30 31 32 33 65 0 12 3 35FV = $2 millionThe PV of the cash flows must equal the PV of $2 million in 35 years. The cash flows consist of a 35-year annuity, plus the contribution today, so the PV is:()35C 1PV 1 C.0.05 1.05=-+⎛⎫ ⎪⎝⎭The PV of $2 million in 35 years is()352,000,000$362,580.57.1.05=Setting these equal gives:()()3535C 11C 362,580.570.05 1.05362,580.57C $20,868.91.11110.05 1.05-+=⇒==-+⎛⎫⎪⎝⎭⎛⎫⎪⎝⎭。
Chapter 3 Time Value of Money: An IntroductionProblem 4Suppose Bank One offers a risk-free interest rate of 5.5% on both savings and loans, and Bank Enn offers a risk-free interest rate of 6% on both savings and loans.a.What arbitrage opportunity is available?b.Which bank would experience a surge in the demand for loans? Which bankwould receive a surge in deposits?c.What would you expect to happen to the interest rates the two banks areoffering?a.Take a loan from Bank One at 5.5% and save the money in Bank Enn at 6%.b.Bank One would experience a surge in the demand for loans, while Bank Ennwould receive a surge in deposits.c.Bank One would increase the interest rate, and/or Bank Enn would decrease itsrate.Problem 7Bubba is a shrimp farmer. In an ironic twist, Bubba is allergic to shellfish, so he cannot eat any shrimp. Each day he has one-ton supply of shrimp. The market price of shrimp is $10,000 per ton.a.What is the value of a ton of shrimp to him?b.Would this value change if he were not allergic to shrimp? Why or why not?a.The value of one ton of shrimp to Bubba is $10,000 because that is the marketprice.b.No. As long as he can buy or sell shrimp at $10,000 per ton, his personalpreference or use for shrimp is irrelevant to the value of the shrimp.Problem 11A friend asks to borrow $55 from you and in return will pay you $58 in one year. If your bank is offering a 6% interest rate on deposits and loans:a. How much would you have in one year if you deposited the $55 instead?b. How much money could you borrow today is you pay the bank $58 in one year?c. Should you loan the money to your friend or deposit it in the bank?a. I f you deposit the money in the bank today you will have:$1.06 in one year FV $55 today $58.30 in one year $ today ⎛⎫=⨯= ⎪⎝⎭b.If you lend the money to your friend for one year and borrow against the promised $58 repayment, then you could borrow:$1.06 in one year PV $58 in one year $54.72 today $ today ⎛⎫=÷= ⎪⎝⎭c. F rom a financial perspective, you should deposit the money in the bank, as it will result in more money for you at the end of the year.Problem 16Calculate the future value of $2000 ina. Five years at an interest rate of 5% per year.b. Ten years at an interest rate of 5% per year.c. Five years at an interest rate of 10% per year.d. Why is the amount of interest earned in part (a) less than half the amount of interest earned in part (b)?a. Timeline:0 1 2 555FV 2,000 1.052,552.56=⨯=b. Timeline:0 1 2 101010FV 2,000 1.053,257.79=⨯=c. Timeline:0 1 2 555FV 2,000 1.13,221.02=⨯=d. Because in the last 5 years you get interest on the interest earned in the first 5 years as well as interest on the original $2,000.Problem 22Your grandfather put some money in an account for you on the day you were born. You are now 18 years old and are allowed to withdraw the money for the first time. The account currently has $3996 in it and pays an 8% interest rate.a. How much money would be in the account if you left the money there until your 25th birthday?b. What if you left the money until your 65th birthday?c. How much money did your grandfather originally put in the account?a. Timeline:18 19 20 21 25 0 1 2 3 77FV 3,996(1.08)6,848.44==b. Timeline:18 19 20 21 65 0 1 2 3 4747FV 3,996(1.08)148,779==c. Timeline:0 1 2 3 4 18183,996PV 1,0001.08==Chapter 4 Time Value of Money: Valuing Cash Flow StreamsProblem 1You have just taken out a five-year loan from a bank to buy an engagement ring. The ring costs $5000. You plan to put down $1000 and borrow $4000. You will need to make annual payments of $1000 at the end of each year. Show the timeline of the loan from your perspective. How would the timeline differ if you created it from the bank’s perspective?0 1 2 3 4 5From the bank’s perspective, the timeline is the same except all the signs are reversed.Problem 9The British government has a consol bond outstanding paying £100 per year forever. Assume the current interest rate is 4% per year.a. What is the value of the bond immediately after a payment is made?b. What is the value of the bond immediately before a payment is made? Timeline:0 12 3a. The value of the bond is equal to the present value of the cash flows. By the perpetuity formula:100PV 2,500.0.04£==b. The value of the bond is equal to the present value of the cash flows. The cash flows are the perpetuity plus the payment that will be received immediately.PV =100+100=£2,600 Problem 30You are saving for retirement. To live comfortably, you decide you will need to save $2 million by the time you are 65. Today is your 30th birthday, and you decide, starting today and continuing on every birthday up to and including your 65th birthday, that you will put the same amount into a savings account. If the interest rate is 5%, how much must you set aside each year to make sure that you will have $2 million in the account on your 65th birthday? Timeline:30 31 32 33 65 0 12 3 35FV = $2 millionThe PV of the cash flows must equal the PV of $2 million in 35 years. The cash flows consist of a 35-year annuity, plus the contribution today, so the PV is:()35C 1PV 1 C.0.05 1.05=-+⎛⎫ ⎪⎝⎭The PV of $2 million in 35 years is()352,000,000$362,580.57.1.05=Setting these equal gives:()()3535C 11C 362,580.570.05 1.05362,580.57C $20,868.91.11110.05 1.05-+=⇒==-+⎛⎫⎪⎝⎭⎛⎫⎪⎝⎭。
Time Value of Money ( 貨幣的時間價值 )0、今年的100元與明年的100元相比，何者的財富價值較高?答：今年的100元的財富價值較高，因為我如果將100元存放在台銀，一年後，以定存年利率1.5%計算，到期時我將擁有100*(1+1.5%)=101.5元>100元(明年)；所以明年的100元與今年的100元在作財富價值比較時，須將明年的100元以適當的折現率( Appropriate Discounting Rate )折現成現值( Present Value )，這就是折現的觀念。
反之如果明年我欲擁有100元，華銀的一年定存利率同為1.5%，則我只須存入100/(1+1.5%)=98.522元，此98.522元就是明年100元的折現值(在折現率1.5%條件下)。
1、Compound Interest ( 複利 ) and Future Value( 未年值或終值 ))1(r C + 2)1(r C +nr C )1(+n r C FV )1(+= Future Value ( 終值 ) C=現金流量，r =利率(or 報酬率2、Future Value ( 終值，未來值 ))(,n r n FVIF PV FV =FVIF =Future Value Interest Factor( 終期利率因子 )，r = 利率，n=期數範例 1、小明存入東和銀行30,000元，年利率=6%，每年複利一次，5年後會變成多少錢?)(,n r n FVIF PV FV =78.146,40338226.1000,30%)61(000,3055=⨯=+⨯=FV而338226.15%,6=FVIF 0 1 2 n3、Present Value ( 現值 )在某一時點之金錢價值折現( Discounting )成目前的金錢價值。
)1(1r C+ )1(2r C+ )1(r nC+)1(r nCPV +=PV=Present Value( 現值 )，C=現金流量，r =折現率，n =期數)()1(,n r n nnPVIF FV r FV PV =+=PVIF r,n =Present Value Interest Factor(現值利率因子 )範例：小明希望存一筆錢在東和銀行，3年後能有100,000元，定存年利率為1.7%，則小明此時應存入多少錢才能達成目標?)1(r nCPV +=61.068,959506861.0000,100000,100%)7.11(3=⨯==+PV而 9506861.03%,7.1=PVIF0 1 2n4、Annuity ( 年金 )是指在某固定時間點的等額金額支付。
