FIN305W E1 Questions

■ Consider a cash flow of $70 for the next four years, with first payment one year form now. You also get $1,000 four years from now. ■ What is the present value of this cash flow if you use a discount rate of 6%? What is the present value of this cash flow if you use a discount rate of 10%?

$1,035 $904.9

A project will generate a cash flow of $20,000 next year, and the cash flow will be increasing by 3% every year over the following 14 years. The project will pay for 15 years. What is the present value of this project if the discount rate is 10%?

$179,153

Your grandma put $10,000 in a bank account 23 years ago. She withdrew $10,000 two years ago and she believes the account has zero balance. How much money does she have if the bank was paying a constant annual interest rate of 5% on that account?

$19,690

A friend has to collect some money from a customer. In particular, he will collect $200 in a year, and $900 in three years. If the discount rate is 7%, how much are you willing to pay to your friend to take over the collection?

$200/(1+0.07)^1 + $900/(1+0.07)^3

What's the amount I am borrowing if I agree to pay $1,130 each month for the next 360 months (30-year mortgage) with a flat 30-year APR or 3.75% (Hint: my monthly interest rate is 0.3125%.)

$244,000

How much money would you have nine years from now if you invest $200 two years from now at an effective annual interest rate of 7%?

$321

What is the present value of receiving a $20 annual dividend forever starting in five years? Assume an interest rate of 4%.

$427

Suppose you begin saving for your retirement by depositing $2,000 per year in an IRA starting in a year. If the interest rate is 7.5%, how much will you have in 40 years?

$454,513

Suppose you had a relative deposit $10 at 8% two hundred years ago. How much will you have today?

$48,389,496

What is the present value of receiving a $20 annual dividend forever starting next year? Assume an interest rate of 4%.

$500

What is the present value of receiving a $20 annual dividend forever starting today? Assume an interest rate of 4%.

$520

What's the present value of a constant cash flow of $50 forever assuming a discount rate of 7%? Assume that you receive the first payment five years from now.

$544.93

Suppose you need $7,000 in two years for the down payment on a new car. If you can earn 4% annually, how much do you need to invest today?

$6,471.9

What's the present value of an annuity that will pay ten payments of $10,000 each year. The first payment is today, and the next nine payments will be paid over the next nine years. The discount rate is 7%?

$75,152

You plan to invest $750 in a bank account one year from now. How much money will you have in your account five years from now if the bank gives you an interest rate of 105 basis points (1.05%)?

$782

Suppose you begin saving for your retirement by depositing $2,000 per year in an IRA starting in a year. If the interest rate is 7.5%, how much money will you have in 50 years if you stop making payments in 40 years?

$936,766

Last year you bought a bond for $1,050. It was a 20 year 7% coupon rate bond with yield-to-maturity of 6.54%. It's face value is $1,000. This year you want to sell the bond. Bonds with similar maturity and risk profile now trade at a 7.5% yield-to-maturity. What is your one-year return from investing in this bond?

-2.8%

Your grandma put $1,250 in a bank account ten years ago. Suddenly she remembers about the money and gives them to you as a gift. How much money do you get if the bank was paying a constant annual interest rate of 3.5% on that account?

1,250*1.03510

Suppose you invest in a one year $1,000 face value bond that has an annual coupon rate of 7% and YTM of 20%? Let's assume that this bond has a one in five chance of default. In case of default you will not collect the coupon and will recover $400 of the face value. 1. What is the current price of the bond? 2. Your return over the year if you do not sell the bond and it does not default? 3. Your return over the year if you do not sell the bond and it defaults? 4. What is your expected return?

1. 891.66 2. 20% 3. -55.14% 4. 4.97%

Suppose you have $1,000 now in a savings account that is earning 6%. You will add $500 one year from now and $700 two years from now. How much will you have three years from now in your savings account?

1000 × 1.06 3 + 500 × 1.06 2 + 700 × 1.06

Suppose you invest $1,000 at 5% per year. Now (time=0) Next Year (time=1) Two years from Now (time=2) ■ What is the future value in TWO years? ■ What is the future value in FIVE years?

1102.5 1276.28

You are planning your retirement. You want to receive twenty-five annual payments of $7,000 each, beginning in 40 years. How much would you need to invest today if you can invest at an interest rate of 5%?

14,715

If I sell my house, I have to pay the bank the remaining balance on my 30-year mortgage. My starting loan balance was $244,000, with monthly payments of $1,130. Assume that I made the first payment of $1,130 in July 2015 and the last payment was made in August 2022. My monthly interest rate is 0.3125%. What is my remaining balance as of August 2022 (after I made the August payment)?

207,804.51

What will be the 6-month YTM on the following bond with semi-annual coupon payments: 5 year bond $960 price 6% annual coupon interest rate $1,000 maturity value

3.48%

What is the amount of money one needs to donate today to endow a permanent scholarship that pays equal annual payment of $30,000 if the discount factor is 7.5% and you want the scholarship payments to grow at a 3.5% rate to compensate for future tuition and cost of living increases? The scholarship will be awarded next year and will start paying two years from now. A new student will be selected each year. The scholarship should be running forever. What will be the first annual payment of the same scholarship if the donor can donate only $250,000 right now?

30,000/(1.075x(0.075-0.035) $10,750

In 2015, I got a $244,000, 30-year flat APR (3.75%) mortgage. My monthly payment is $1,130. (N=360, I/Y=0.3125, PV=244,000, FV=0) Over the next 30 years, I will pay Northwest Savings Bank Still, the amount (price) of my mortgage is $244,000. If I paid all cash, I would have paid

360*1,130=$406,800 $244,000

You invested in an asset that had a nominal rate of return of 13% over the year, and the inflation rate was 7%. Your real rate of return was:

5.6%

An Ultima Ratio Corp. bond carries a 12 percent coupon, paid semiannually. The face value is $1,000, and the bond matures in eight years. The bond currently sells for $1,250. What is its annual yield to maturity? (Hint: Annual YTMs on bonds are quoted as twice the effective six-month YTM.)

7.75%

Macrohard Corp. bond carries an 8 percent coupon, paid annually. The bond trades at face value, $1,000, and the bond matures in six years. What is the effective annual yield?

8%. If price=face value, then the yield on an annual coupon bond is equal to its coupon rate

The inflation rate over the last 12 months (Aug 2021 to August 2022) was roughly:

8.2%

A one year $1,000 face value bond that has an annual coupon rate of 6% and YTM of 15%? Let's assume that this bond has a one in ten (10%) chance of default. In case of default, you will not collect the coupon and will recover $500 of the face value. What is your expected return?

8.9%

Suppose you begin saving for your retirement by depositing $7,500 next year in an IRA account. You will increase your payments by 3% each year thereafter (your deposit two years from now will be $7,725=$7,500*1.03). If the effective annual rate is 9%, how much money will you have 40 years from now if you stop making payments in 25 years? (Hint: be careful with the timing here, the first payment is in year 1 and the last payment is in year 25. You withdraw the money in year 40)

Answer: 7,500/(0.09-0.03)*(1-(1.03/1.09)^25)=$94,648.64 FV=$94,648.64*1.09^40=$2,972,859

A project will generate a cash payment of $30,000 next year, and the cash flow will be increasing by 6% every year over the following seven years. Then, in year nine, the project will require a one-time investment (clean-up cost) of $500,000. The project will exist for nine years, paying out cash for eight years and requiring one cash outflow in year nine. What is the present value of this project if the discount rate is 7%?

Answer: First apply the growing annuity formula to get the present value of the growing annuity in period one to eight: 30,000/(0.07-0.06)*(1-(1.06/1.07)^8)= 217,098 Add in the negative present value of the one-time cash outflow in period nine: -500,000/(1.07^9)= -271,967 Final answer: 217,098-271,967=-54,869

You're prepared to make monthly payments of $400, beginning in a month, into an account that pays 0.5% monthly interest rate. How many payments will you have made when your account balance exceeds $20,000?

Answer: I/Y=0.5 PV=0 PMT=-400 FV=20,000 CPT N=44.7 You'll need 45 payments to exceed 20,000

You ran a little short on your spring break vacation, so you put $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month, starting next month. The interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000?

Answer: I/Y=1.5 PV=1,000 PMT=-20 FV=0 CPT N=93.11

You're trying to buy a new $230,000 Bentley. You have $80,000 today that can be invested at your bank. The bank pays 6 percent annual interest on its accounts. How long will it be before you have enough to buy the car?

Answer: I/Y=6, PV=-80,000, PMT=0, FV=230,000 CPT N gets you 18.12

Last year a bond delivered nominal return of 11%. The inflation rate over the same period was 4%. That means that the real return on your investment was:

Answer: Less than 7.00%. Use formula 1.11/1.04-1=0.0673=6.73%

You are looking into an investment that will pay you $12,000 per year for the next 10 years (starting in one year). If you require a 15 percent annual return, what is the most you would pay for this investment?

Answer: N=10 I/Y=15 PMT=12,000 FV=0 CPT PV=60,225

Assume the total cost of a college education will be $200,000 when your child enters college in 15 years. You presently have $60,000 to invest. What annual rate of interest must you earn on your investment to cover the cost of your child's college education?

Answer: N=15, PV=-60,000, PMT=0, FV=200,000 CPT I/Y gets you 8.36%

An Ultima Ratio Corp. bond carries a 12 percent coupon, paid semiannually. The face value is $1,000, and the bond matures in eight years. The bond currently sells for $1,250. What is its annual yield to maturity? (Hint: Annual YTMs on bonds are quoted as twice the effective six-month YTM.)

Answer: N=16, PMT=60, FV=1000, PV=-1250, CPT I/y=3.874 (annual YTM=7.75)

Suppose a German company issues a bond with a par value of €1,000, 25 years to maturity, and a coupon rate of 6.4 percent paid annually. If the yield to maturity is 7.5 percent, what is the current price of the bond?

Answer: N=25 I/Y=7.5 PMT=64 FV=1,000 CPT PV=-877.38

You are scheduled to receive $15,000 in three years. When you receive it, you will invest it for five more years at 7 percent per year. How much will you have in eight years?

Answer: N=5, I/Y=7, PV=-15,000, PMT=0, CPT FV gets you 21,038.27

Suppose you borrow $25,000 from your parents to buy a car. You agree to pay them $500 per month for 60 months, starting in a month. What is the monthly interest rate?

Answer: N=60 PV=25000 PMT=-500 FV=0 CPT I/Y=0.618%

You receive a credit card application from Crooks United Bank offering an introductory rate of 1.2 percent per year, compounded monthly for the first six months, increasing thereafter to 24 percent compounded monthly. Assuming you transfer the $24,000 balance from your existing credit card and make no subsequent payments, how much will you owe at the end of the first year?

Answer: The first six months you will be charged a monthly rate of 1.2%/12=0.1%=0.001, the next six months this rate will change to 24%/12=2%. FV= 24000*1.001^6*1.02^6= 27,190

You plan to make a series of deposits in an individual retirement account. You will deposit $1,000 today, $2,000 in two years, and $2,000 in five years. If you withdraw $1,500 in three years and $1,000 in seven years, assuming no withdrawal penalties, how much will you have after eight years if the interest rate is 7 percent?

Answer: There are two "easy" approaches. One is to use the CF function in the calculator like this: CF0=1,000, C02=2,000, C03=-1,500, C05=2,000 , C07=-1,000, I=7, CPT NPV=2,325.65 And then move this into future value like that: N=8, I/Y=7, PV=-2,325.65, PMT=0, CPT FV gets you 3,995.90 Or do this one by one and sum (again can be done with calculator financial functions or just normal calculator and formulas): FV8=1,000*1.07^8+2,000*1.07^6-1,500*1.07^5+2,000*1.07^3-1000*1.07=3,995.90 Note that in here each cash flow is pushed to the eight period depending on when it occurs. For example, the $2,000 that comes in period two is invested for six years (to get to period eight), and hence using the formula, its FV=PV*(1+r)6=2,000*1.07^6. Same for all the other cash flows.

Other things equal, a bond buyer will ask for a higher yield if a bond A. is first to collect in case of bankruptcy (is senior bond). B. has protective covenants. C. has a low credit rating. D. has a low default risk. E. offers collateral.

C. has a low credit rating.

A first-round draft choice quarterback has been signed to a two-year $18 million contract. The details provide for an immediate cash bonus of $6 million. The player is to receive $5 million in salary at the end of the first year, and $7 million at the end of the second year. Assuming a 15 percent discount rate, how much is it worth right now?

Either using calculator or formulas, calculate each cash flow and sum. Or do it in one take with the CF function like that: CF0=6, C01=5, C02=7, I=15, CPT NPV=15.65 The alternative is to just compute 6+5/1.15+7/(1.152)= 15.65

Assume that you are promised $100 three years from now. What happens to the future value of this promise six years from now (FV6) if you increase the discount rate r? What happens to the Present Value (PV0) of this promise if you increase the discount rate r?

FV6 increases, PV0 decreases, If unsure just use an example. Let's say we use r=10% and then increase it to r=12%. N=3, I/Y=10, PV=-100, PMT=0 CPT FV gets you 133.10 N=3, I/Y=12, PV=-100, PMT=0 CPT FV gets you 140.49 So FV increases as r increases. Same result can be derived from the formula FV=PV*(1+r)t . N=3, I/Y=10, PMT=0, FV=-100 CPT PV gets you 75.13 N=3, I/Y=12, PMT=0, FV=-100 CPT PV gets you 71.18 So, PV decreases as r increases. Same result can be derived from the formula PV=FV/(1+r)t .

Suppose you bought a 7 percent annual coupon, a 20-year bond last year when it was first issued. You paid 1,000 for a 1,000 face value bond. The yield-to-maturity of your bond was therefore 7%. If interest rates suddenly rise to 15 percent this year, and hence your bond now has to deliver a 15% yield to maturity. What is your one-year rate of return if you sell the bond this year?

First, calculate the new price: N=19 I/Y=15 PMT=70 FV=1000 CPT PV=-504.14 Return=(New Price+ Coupon -Old Price)/Old Price=(504.14+70-1000)/1000=-0.4259=-42.59%

What happens to the future value of an annuity if you increase the discount rate r?

Higher discount rate will increase the future value of each payment and thus increase the overall future value of the annuity. To see this come up with an example, N=10 I/Y=5 PV=0 PMT=-100 CPT FV=1,258 and compare to N=10 I/Y=10 PV=0 PMT=100 CPT FV=1,594

You want to have $2 million in real dollars in an account when you retire in 40 years. The nominal return on your investment is 10 percent and the inflation rate is 3.8 percent. What nominal amount must you deposit each year (starting next year) to achieve your goal?

If you want to have $2 million in real dollars in 40 years, then you need to have $2,000,000*(1.038)^40=$8,890,458 dollars in nominal terms. Hence, you need to save: N=40, I/Y=10, PV=0, FV=8,890,458 CPT PMT=-20,087.23

An investment will double your money in 5 years. What annual rate of return are you being offered?

N=5, PV=-1, PMT=0, FV=2 CPT I/Y gets you 14.87%

Suppose you buy a 4 percent annual coupon, a 10-year bond when it's first issued. You pay $731.60 for a 1,000 face value bond. The yield-to-maturity of your bond was therefore 8%. If the yield-to-maturity of your bond stays at 8%, you collect your coupon, what will be the price of this bond next year?

N=9 (one year is gone) I/Y=8 PMT=40 FV=1000 CPT PV=-750.12 The price goes up to make sure you get an 8% return on a 4% coupon bond. Note that if nothing changes, the price will increase to $1,000 at the end of the bond life.

How much money do you have to set aside every year for the next 40 years to guarantee yourself a 25 year annuity of $100,000 starting in 41 years? You make the first payment next year. Assume an interest rate of 5%.

Start with the 25 years annuity in the future. How much money do we need to fund this annuity? N=25 I/Y=5 PMT=100,000 FV=0 CPT PV=-1,409,394.45 The answer here is not in present value money. We get the total value as of period 40 because the first payment will be in period 41. Next, we need to make this amount equal to the future value of your savings. We are looking for the payment on a 40 period annuity starting in a year that can exactly fund the future annuity. In a calculator: N=40 I/Y=5 PV=0FV=1,409,394.45 CPT PMT=-11,667

What is the present value of a stream of cash that starts in nine years with a payment of $23,000 and pays the same payment forever? Assume an annual interest rate of 1%.

Step one, use the perpetuity formula but keep in mind you are getting the future value as of period eight (a period before first payment in nine): FV8=C/r=23,000/0.01=2,300,000 Step two, bring this to present value using the PV/FV formula: PV= FV8/(1+r)8=2,300,000/(1.01)8=2,124,011.41

You just bought a 10 year corporate bond. Which of these things makes your bond less valuable (decreases the bond price)?

The FED announces an unexpected increase in the interest rates in the economy by 75 basis points. This raises the required YTM on your bond and lowers the price.

Corporate ownership varies around the world. Historically individuals have owned the majority of shares in public corporations in the United States. In Germany and Japan, however, banks, other large financial institutions, and other companies own most of the stock in public corporations. Do you think agency problems are likely to be more or less severe in Germany and Japan than in the United States? Why?

The agency costs refer to the conflict of interest between management and shareholders and to the costs that dispersed shareholders have to incur to monitor and motivate management. We would expect agency problems to be less severe in countries with more concentrated ownership and a relatively small percentage of individual ownership. Fewer individual owners should reduce the number of diverse opinions concerning corporate goals. It is also much cheaper (per investor) to implement efficient monitoring and to incentivize managers. For example, it is very costly for a small investor to read and analyze all company filings and information. However, a concentrated bank-centered system like the one in Germany can create another agency problem where large owners do not care about the interest of small minority shareholders.

Suppose that a seven-year coupon bond trades for less than the face value. The annual coupon rate is 4%. Then,

The coupon rate is smaller than the YTM. If the Price is lower than the face value, then an investor will realize capital appreciation on the bond, and the total return will be higher than the coupon yield.

Suppose you own stock in a company. The current price per share is $25. Another company has just announced that it wants to buy your company and will pay $35 per share to acquire all the outstanding stock. Your company's management immediately begins fighting off this hostile bid. Is management acting in the shareholders' best interests? Why or why not?

The goal of management should be to maximize the share price for the current shareholders. If management believes that it can improve the profitability of the firm so that the share price will exceed $35, then they should fight the offer from the outside company. If management believes that this bidder or other unidentified bidders will pay more than $35 per share to acquire the company, then they should still fight the offer. However, if the current management cannot increase the value of the firm beyond the bid price, and no other higher bids come in, then management is not acting in the interests of the shareholders by fighting the offer. Since current managers often lose their jobs when the corporation is acquired, poorly monitored managers have an incentive to fight corporate takeovers in situations such as this.

In 21 years you would like to receive $25,000 income annually to perpetuity. You will receive the first payment in exactly 21 years. What equal annual deposits are required to guarantee that income if you make your first deposit in one year and you make your last deposit in 20 years. The annual interest rate is 9%.

The value of the perpetuity 20 years from now is 25,000/0.09=277,777.78. This is the future value of the annuity deposits to be made. Hence, N=20, I/Y=9 PV=0 FV=277,777.78 CPT PMT=-5,429.58

Can our goal of maximizing the value of the stock conflict with other goals, such as avoiding unethical or illegal behavior? In particular, do you think subjects like customer and employee safety, the environment, and the general good of society fit in this framework, or are they essentially ignored? Think of some specific scenarios to illustrate your answer.

There is no correct answer here. An argument can be made either way. At one extreme, we could argue that in a market economy, all of these things are priced. There is thus an optimal level of, for example, ethical and/or illegal behavior, and the framework of stock valuation explicitly includes these. At the other extreme, we could argue many ESG practices have an immediate negative impact on firm profits. For example, dumping pollution in the rivers (up to the EPA allowable limits) probably reduces the firm's clean-up costs and drives profits up. However, it is hard to see how this behavior will not backfire in the long run.

The Maybe Pay Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $30,000 per year forever. The first payment is right now. A sales associate told you the policy costs $475,000. At what interest rate would this be a fair deal?

This is an annuity due now. So, we will use the formula PV=C/r+C. We need to solve for r: 475,000=30,000/r+30,000 445,000=30,000/r R=30,000/445,000=0.0674=6.74%

What is the present value of a growing cash flow that starts at $50 five years from now and grows at a 6% rate thereafter? Assume a discount rate of 7%. The first payment of $50 comes five years from now.

tep one, use the growing perpetuity formula but keep in mind you are getting the future value as of period four (a period before first payment in five): FV4=C/(r-g)=50/(0.07-0.06)=5,000 Step two, bring this to present value using the PV/FV formula: PV= FV4/(1+r)4=5,000/(1.07)4=3,814,48