Fin Ch. 4

A lottery winner will receive $6 million at the end of each of the next twelve years. What is the future value (FV) of her winnings at the time of her final payment, given that the interest rate is 8.6% per year?

$118.00 million

Suppose you invest $1000 into a mutual fund that is expected to earn a rate of return of 11%. The amount of money will you have in ten years is closest to which of the following? The amount you will have in 50 years is closest to which of the following?

$2839 ; $184,565

If $8000 is invested in a certain business at the start of the year, the investor will receive $2400 at the end of each of the next four years. What is the present value of this business opportunity if the interest rate is 6% per year?

$316.25

A business promises to pay the investor of $6000 today for a payment of $1500 in one yearʹs time, $3000 in two yearsʹ time, and $3000 in three yearsʹ time. What is the present value of this business opportunity if the interest rate is 6% per year?

$603.94

An investment pays you $30,000 at the end of this year, and $10,000 at the end of each of the four following years. What is the present value (PV) of this investment, given that the interest rate is 5% per year?

$79,228

Define the following terms: (a) perpetuity (b) annuity (c) growing perpetuity (d) growing annuity

(a) A perpetuity is a stream of equal cash flows that occur at regular intervals and lasts forever. (b) An annuity is a stream of N equal cash flows paid at regular intervals. (c) A growing perpetuity is a cash flow stream that occurs at regular intervals and grows at a constant rate forever. (d) A growing annuity is a stream of N growing cash flows, paid at regular intervals.

4) Ally wishes to leave a provision in her will that $7000 will be paid annually in perpetuity to a local charity. How much must she provide in her will for this perpetuity if the interest rate is 6%?

) PV perpetuity = $7000 / 0.06 = $116,666.67

14) Joe just inherited the family business, and having no desire to run the family business, he has decided to sell it to an entrepreneur. In exchange for the family business, Joe has been offered an immediate payment of $100,000. Joe will also receive payments of $50,000 in one year, $50,000 in two years, and $75,000 in three years. The current market rate of interest for Joe is 6%. In terms of present value (PV), how much will Joe receive for selling the family business?

254,641

Salvatore has the opportunity to invest in a scheme which will pay $5000 at the end of each of the next 5 years. He must invest $10,000 at the start of the first year and an additional $10,000 at the end of the first year. What is the present value of this investment if the interest rate is 3%?

3189.80

Which of the following is true about perpetuities?

All of the above are true statements.

Faisal has $12,000 in his savings account and can save an additional $3600 per year. If interest rates are 12%, how long will it take his savings to grow to $47,000 ?

Calculate N using TVM keys: input PV = $12,000 , interest rate = 12% , PMT = $3600 ; N = 5.3 years.

How long will it take $50,000 placed in a savings account at 10% interest to grow into $75,000 ?

Calculate N using TVM keys: input PV = 50,000 , interest rate = 10%, and FV = 75,000 ; N = 4.25 years.

A bank offers a home buyer a 20-year loan at 8% per year. If the home buyer borrows $130,000 from the bank, how much must be repaid every year?

Calculate PMT using TVM keys: input PV = 130,000 , N = 20, and interest rate = 8%; PMT = $13,240.787 .

Dan buys a property for $210,000 . He is offered a 30-year loan by the bank, at an interest rate of 8% per year. What is the annual loan payment Dan must make?

Calculate PMT using TVM keys: input PV = 210,000 , N = 30, and interest rate = 8%; PMT = $18,653.76 .

A bank is negotiating a loan. The loan can either be paid off as a lump sum of $80,000 at the end of four years, or as equal annual payments at the end of each of the next four years. If the interest rate on the loan is 6%, what annual payments should be made so that both forms of payment are equivalent?

Calculate PMT with FV = $80,000 , interest = 6% and N = 4, which gives PMT = $18,287.32 .

An annuity pays $10 per year for 98 years. What is the present value (PV) of this annuity given that the discount rate is 7%?

Calculate PV annuity using TVM keys input PMT = $10, number of years = 98, and interest rate = 7%; computing PV = $142.67 .

An annuity is set up that will pay $1500 per year for ten years. What is the present value (PV) of this annuity given that the discount rate is 9%?

Calculate PV annuity using TVM keys: input PMT = $1500 , number of years = 10, and interest rate = 9%; PV = $9626.49 .

Matthew wants to take out a loan to buy a car. He calculates that he can make repayments of $5000 per year. If he can get a four-year loan with an interest rate of 7.9%, what is the maximum price he can pay for the car?

Calculate PV using TVM keys: input PMT = $5000 , N= 4, and interest rate = 7.9%; PV = $16,597.5634 .

A businessman wants to buy a truck. The dealer offers to sell the truck for either $120,000 now, or six yearly payments of $25,000 . Which of the following is closest to the interest rate being offered by the dealer?

Calculate interest rate using TVM keys: input PV = $120,000 , N = 6, and PMT = $25,000 ; interest rate = 6.8%.

What is the internal rate of return (IRR) of an investment that requires an initial investment of $11,000 today and pays $15,400 in one yearʹs time?

Calculate interest rate using TVM keys: input PV = 11,000 , N = 1, and FV = -15,400 ; interest rate = 40%.

A homeowner in a sunny climate has the opportunity to install a solar water heater in his home for a cost of $2900 . After installation the solar water heater will produce a small amount of hot water every day, forever, and will require no maintenance. How much must the homeowner save on water heating costs every year if this is to be a sound investment? (The interest rate is 5% per year.)

Calculate the cash flow as the perpetuity whose PV = $2900 ; hence, annual heating cost = 2900 × 0.05 = $145 .

If a few intermediate cash flows in valuing a stream of cash flows are zero can we delete those points on the timeline and squeeze the timeline to show only nonzero cash flows?

Every cash flow contains two pieces of informationthe nominal value and the time stamp. If we decide to eliminate the zero cash flows from the timeline and concentrate only on the nonzero ones, we will be distorting the time stamp of some nonzero cash flows. Hence, we need to show the timeline in full, including all cash flows zero as well as nonzero.

Cash flows from an annuity occur every year in the future.

False

Investment X and Investment Y are both growing perpetuities with initial cash flow of $100. Both investments have the same interest rate (r) and cash flows. The present value of Investment X is $5,000, while the present value of Investment Y is $4,000. Which of the following is true?

Investment X has a higher growth rate than Investment Y.

If the current rate of interest is 7%, then the future value (FV) of an investment that pays $1200 per year and lasts 18 years is closest to ________.

N = 18 I=7 PMT = $1200 PV = 0 Compute FV = $40,799

Since your first birthday, your grandparents have been depositing $1200 into a savings account on every one of your birthdays. The account pays 6% interest annually. Immediately after your grandparents make the deposit on your 18th birthday, the amount of money in your savings account will be closest to ________.

N = 18 PMT = $1200 I=6 PV = 0 Compute FV = $37,086.78 .

Since your first birthday, your grandparents have been depositing $100 into a savings account every month. The account pays 9% interest annually. Immediately after your grandparents make the deposit on your 18th birthday, the amount of money in your savings account will be closest to ________.

N = 216 PMT = $100 I = 9/12 PV = 0 Compute FV = $53,635.167

If the current rate of interest is 8%, then the present value (PV) of an investment that pays $1200 per year and lasts 24 years is closest to ________.

N = 24 I=8 PMT = $1200 FV = 0 Compute PV = $12,635 .

You are borrowing money to buy a car. If you can make payments of $320 per month starting one month from now at an interest rate of 12%, how much will you be able to borrow for the car today if you finance the amount over 4 years?

N = 48 I = 12 /12 PMT = $320 FV = 0 PV = $12,151.67

You are saving money to buy a car. If you save $310 per month starting one month from now at an interest rate of 6%, how much will you be able to spend on the car after saving for 4 years?

N = 48 I = 6/12 PMT = $310 PV = 0 Compute FV = $16,770.33

You are considering investing in a zero-coupon bond that will pay you its face value of $1000 in twelve years. If the bond is currently selling for $496.97 , then the internal rate of return (IRR) for investing in this bond is closest to ________.

PV = -496.97 FV = 1000 PMT = 0 N = 12 Compute I = 6.0%.

You are considering purchasing a new home. You will need to borrow $290,000 to purchase the home. A mortgage company offers you a 20-year fixed rate mortgage (240 months) at 12% APR (1% month). If you borrow the money from this mortgage company, your monthly mortgage payment will be closest to ________.

PV = 290,000 I=1 N = 240 FV = 0 Compute payment = $3193.15 .

You are interested in purchasing a new automobile that costs $33,000 . The dealership offers you a special financing rate of 9% APR (0.75% per month) for 60 months. Assuming that you do not make a down payment on the auto and you take the dealerʹs financing deal, then your monthly car payments would be closest to ________.

PV = 33,000 I = 0.75 N = 60 FV = 0 Compute payment = $685.03 .

What is the present value (PV) of an investment that will pay $500 in one yearʹs time, and $500 every year after that, when the interest rate is 10%?

PV Perpetuity = 500 /0.1 = $5000

Clarissa wants to fund a growing perpetuity that will pay $10,000 per year to a local museum, starting next year. She wants the annual amount paid to the museum to grow by 5% per year. Given that the interest rate is 9%, how much does she need to fund this perpetuity?

PV growth perpetuity = $10,000 / (0.09 - 0.05) = $250,000.00

Martin wants to provide money in his will for an annual bequest to whichever of his living relatives is oldest. That bequest will provide $4000 in the first year, and will grow by 7% per year, forever. If the interest rate is 9%, how much must Martin provide to fund this bequest?

PV growth perpetuity = $4000 / (0.09 - 0.07) = $200,000.00

Which of the following statements regarding perpetuities is FALSE?

PV of a perpetuity = r/C

You are thinking about investing in a mine that will produce $10,000 worth of ore in the first year. As the ore closest to the surface is removed it will become more difficult to extract the ore. Therefore, the value of the ore that you mine will decline at a rate of 7% per year forever. If the appropriate interest rate is 3%, then the value of this mining operation is closest to ________.

PVP = C / (r - g) = $10,000 /(0.03 - -0.07 ) = $10,000 / 0.1 = $100,000

A perpetuity has a PV of $20,000 . If the interest rate is 6%, how much will the perpetuity pay every year?

Payment = 20,000 × 0.06 = $1200

A rich donor gives a hospital $1,040,000 one year from today. Each year after that, the hospital will receive a payment 6% larger than the previous payment, with the last payment occurring in ten yearsʹ time. What is the present value (PV) of this donation, given that the interest rate is 11 %?

Payment in 10th year = $1,040,000 × (1 + 0.06)^10 = $1,862,481.6044 ; PV of growth perpetuity in year 10 = $1,862,481.6044 /(0.11 - 0.06) = $37,249,632.088 ; PV of this CF at time zero = $13,118,742.261 ; PV of entire CF = $1,040,000 / (0.11 - 0.06) = $20,800,000.000 ; Differenceofthetwocashflows=$20,800,000.000 -$13,118,742.261 = $7,681,257.739

Which of the following is true about perpetuities?

Since a perpetuity generates cash flows every period infinitely, the cash flow generated equals the PV times the interest rate.

How do the growth perpetuity results differ with negative and positive growths of similar magnitude assuming everything else remains unchanged?

The denominator in the formula for growth perpetuity plays in important role on the results for negative and positive growths of similar magnitude. A positive growth results in a smaller denominator thereby increasing the present value (PV). Contrarily, a negative growth results in a larger denominator giving a smaller present value (PV).

Which of the following statements regarding annuities is FALSE?

The difference between an annuity and a perpetuity is that a perpetuity ends after some fixed number of payments.

A perpetuity will pay $900 per year, starting five years after the perpetuity is purchased. What is the present value (PV) of this perpetuity on the date that it is purchased, given that the interest rate is 11%?

The first step is to calculate the PV perpetuity = $900 / 0.11 = $8181.82 ; the next step is to calculate its PV using TVM keys: input FV = $8181.82 , number of years = 4, and interest rate = 11%; PV = $5389.6171 .

A perpetuity will pay $1000 per year, starting five years after the perpetuity is purchased. What is the future value (FV) of this perpetuity, given that the interest rate is 3%?

The future value of a perpetuity cannot be calculated, since there is no ending date.

You are given two choices of investments, Investment A and Investment B. Both investments have the same future cash flows. Investment A has a discount rate of 4%, and Investment B has a discount rate of 5%. Which of the following is true?

The present value of cash flows in Investment A is higher than the present value of cash flows in Investment B.

2) Trial and error is the only way to compute the internal rate of return (IRR) when interest is calculated over five or more periods

True

A growing perpetuity, where the rate of growth is greater than the discount rate, will have an infinitely large present value (PV).

True

The internal rate of return (IRR) is the interest rate that sets the net present value (NPV) of the cash flows equal to zero.

True

The present value (PV) of a stream of cash flows is just the sum of the present values of each individual cash flow.

True

An annuity pays $13 per year for 53 years. What is the future value (FV) of this annuity at the end of that 53 years given that the discount rate is 9%?

Using TVM keys input PMT = $13, number of years = 53, and interest rate = 9%; computing PV = $13,764.85 .

An annuity pays $47 per year for 22 years. What is the future value (FV) of this annuity at the end of those 22 years, given that the discount rate is 8%?

Using TVM keys input PMT = $47, number of years = 22, and interest rate = 8%; computing FV = $2606.47 .

Which of the following statements regarding growing perpetuities is FALSE?

We assume that r < g for a growing perpetuity.

Can we apply the growing perpetuity equation for negative growth as well?

Yes, it is perfectly in order to apply the growth perpetuity for negative growth. A negative growth gives two negatives in the denominator making it larger than a positive growth thus reducing the valuation compared to a positive growth of similar magnitude.