finance test 2

CathFoods will release a new range of candies which contain anti-oxidants. New equipment to manufacture the candy will cost $2 million, which will be depreciated by straight-line depreciation over five years. In addition, there will be $5 million spent on promoting the new candy line. It is expected that the range of candies will bring in revenues of $4 million per year for five years with production and support costs of $1.5 million per year . If CathFood's marginal tax rate is 35%, what are the incremental earnings in the second year of this project?

$1.365 million ============================== Δ Rev $ 4 million -Δ cost <$1.5 million> -Δdep <400,000>= 2mil/5years -------------------------------------------------------------- ΔT.I $2.1 million -tax rate <$735000> ----------------------------------------------------------- ΔNI $11.365 million

A small manufacturer that makes clothespins and other household products buys new injection molding equipment for a cost of $500,000. This will allow the manufacturer to make more clothespins in the same amount of time with an estimated increase in sales of 15%. If the manufacturer currently makes 75 tons of clothespins per year, which sell at $18,000 per ton, what will be the increase in revenue next year from the new equipment?

$202,500 ===================== current sales: 18,000 x 75= $1,350,000 increase 0.15 x 1,350,000 =$202,500

Use the information for the question below . The Sisyphean Corporation is considering investing in a new cane manufacturing machine that has an estimated life of three years. The cost of the machine is $30,000 and the machine will be depreciated straight line over its three-year life to a residual value of $0. The cane manufacturing machine will result in sales of 2000 canes in year 1. Sales are estimated to grow by 10% per year each year through year 3. The price per cane that Sisyphean will charge its customers is $18 each and is to remain constant. The canes have a cost per unit to manufacture of $9 each. Installation of the machine and the resulting increase in manufacturing capacity will require an increase in various net working capital accounts. It is estimated that the Sisyphean Corporation needs to hold 2% of its annual sales in cash, 4% of its annual sales in accounts receivable, 9% of its annual sales in inventory, and 5% of its annual sales in accounts payable. The firm is in the 35% tax bracket and has a cost of capital of 10%. 1. The depreciation tax shield for the Sisyphean Corporation's project in the first year is closest to....

$3500 ========= depreciation tax shield= tax rate x depreciation expense 30,000/3 years = 10,000 0.35 x 10,000 = 3500

A firm has an opportunity to invest $100,000 today that will yield $115,000 in one year. If interest rates are 6%, what is the net present value (NPV) of this investment?

$8491 $115,000 / 1.06 = $108,491; $108,491 - $100,000 = $8491

credit ratings

(investment grade bonds, relatively low prob. of defualt) AAA AA A BBB ------------------------------------ Speculative grade bonds/"junk bonds" High risk, High yield BB B C D

an investor holds a Ford bond with a face value of $5000, a coupon rate of 4%, and semi-annual payments that mature on 01/15/2012. How much will the investor receive on 01/12/2012?

* face value = par value 5000 x .04 = $200 (annual) 200/2 = 100 semi annual 100 + 5000 = 5100

Vernon-Nelson Chemicals is planning to release a new brand of insecticide, Bee-Safe, that will kill many insect pests but not harm useful pollinators. Buying new equipment to manufacture the product will cost $20 million, and there will be an additional $2 million cost to reconfigure the existing plant. The equipment is expected to have a lifetime of eight years and will be depreciated by the straight-line method over its lifetime. The firm expects that it should be able to sell 1,500,000 gallons per year at a price of $52 per gallon. It will take $38 per gallon to manufacture and support the product. If Vernon-Nelson's marginal tax rate is 40%, what are the incremental earnings in year 3 of this project?

*incremental earnings = NI $11.1 million ======================== Δ Rev $ 52 x 1.5 million -Δ cost <$38 x 1.5 million> -Δdep <2.5 million>= 20mil/8years -------------------------------------------------------------- ΔT.I $18.5 million -tax rate <$7.66 million> ----------------------------------------------------------- ΔNI $11.1 million

Cash Flow Investing (CFI

- Short-term: ΔNWC -Long-term: CAPEX

potential problems in capital budgeting

- inconsistent forecasts -incentive problems -behavioral biases -lack of economic reasoning

discount rate

- rate that companies and investors expect as a return on their investment. - is used to calculate the net present value (NPV) of an investment (what the value fo Future CF is worth today) - the required rate of return (discount rate) can be calculated according to risk related to that amount of return

bond valuation summary

- value of bond = PV of cash flows - bond prices fall when interest rates rise - bond prises rise when interest rates fall three kinds of info necessary to value a bond: 1.what are the CF (coupon rate + par value) 2. when and how freuently do CF arrive? frequency of coupon and maturity of bond 3. what is the appropriate discount rate? market interest rate

You want a 20 million budget, but you know that you will have to convince finance that spending that on advertising will add to the bottom line. You estimates that your product will sell for 3 years and your budget will allow your company to sell 10 million additional figures per year for $1.50 profit per figure. Discount rate of 11% per year given the risk involved. Should your company approve you?

-20,000 + (10 million x 1.5)/.11 [1- 1/(1.11)^3] = $16,655,720.73 ACCEPT

NPV example: Today 1 2 3 4 5 --------------------------------------------------------------------- 100,000 100,000 100,000 100000 100000 100000 1. If your discount rate is 8% per year, the value of your company is? You are considering a project that will cost $55000 today but will increase cash flow by $20,000 years 1-5. Projected cash flows from the new project: Today 1 2 3 4 5 --------------------------------------------------------- -55,000 20,000 20,000 20,000 20,000 20,000 projected cash flows for the company: Today 1 2 3 4 5 ------------------------------------------------------------------------ 45,000 120000 120000 120000 120000 120000 (100000-55000) 2. Value the company with the project

1. 100,000 + 100000/.08 [1- 1/(1.08)^5] = $499,271 2. 45,000 + 120000/.08 [1- 1/(1.08)^5]= $524,125 taking will increase the value of the firm by $24,854 Rather than revaluing the entire company based on new project, we can identify just the changes in the cash flows due to the project and value those directly. -55,000 +20,000/.08 [1- 1/(1.08)^5] = $24,854 (Δfirm value) ΔFCF

Cameron Industries is purchasing a new chemical vapor depositor in order to make silicon chips. It will cost $6 million to buy the machine and $10,000 to have it delivered and installed. Building a clean room in the plant for the machine will cost an additional $3 million. The machine is expected to raise gross profits by $4 million per year, starting at the end of the first year, with associated costs of $1 million for each of those years. The machine is expected to have a working life of six years and will be depreciated over those six years. The marginal tax rate is 40%. 1. What are the incremental free cash flows associated with the new machine in year 0? 2. What are the incremental free cash flows associated with the new machine in year 1?

1. -$6,000,000 - $10,000 -$3,000,000= -$9,010,00 2. Depreciation = (6,000,000) / 6 = 1,000,000, 4,000,000 - 1,000,000 - 1,000,000 = 2,000,000, 2,000,000 × (0.6) = 1,200,000; add back depreciation to get $2,200,000

Use the information for the question(s) below. Project A Project B Time 0 -10,000 -10,000 Time 1 5,000 4,000 Time 2 4,000 3,000 Time 3 3,000 10,000 1. If WiseGuy Inc. uses the payback period rule to choose projects, which of the projects (Project A or Project B) will rank highest? A) Project A B) Project B C) Project A and Project B have the same ranking. D) Cannot calculate a payback period without a discount rate. 2. If WiseGuy Inc. uses IRR rule to choose projects, which of the projects (Project A or Project B) will rank highest? A) Project A B) Project B C) Project A and Project B have the same ranking. D) Cannot calculate a payback period without a discount rate. 3. If WiseGuy Inc is choosing one of the above mutually exclusive projects (Project A or Project B), given a discount rate of 8%, which should the company choose?

1. A 2. B IRRA=10.7%, IRRB=26.4% 3. project B (higher NVP) Project A.) -10,000 + 5000/(1.08)^1 + 4000/(1.08)^2 + 3000/(1.08)^3= 440 Project B.) -10,000 + 4000/(1.08) + 3000/(1.08)^2 + 10,000/(1.08)^3 = 4212

value of a bond

1. Cash flow : coupon rate, par value 2. timing : frequency of coupon, maturity 3. discount rate : market interest rate (1 and 2 are set in contract/fixed)

important points about NPV

1. DOLLAR MEASURE OF THE EFFECTS OF AN INVESTMENT . reflects both near term and distant future 2. calculation does not depend on the source of financing (borrowing, selling stock, internal funds) 3. the challenges of implementation are estimating incremental free cash flows and discount rates (based on risk)

issues concerning IRR

1. It is generally inappropriate to use IRR to choose among projects b/c problems arise if: - initial investments are different - lives of the projects are different - multiple sign changes - different timing 2. does not measure the effect of an investment on the PV of equity (firms value) *choose NPV over IRR

which of the following is incremental cash flow: Market value of land and existing building cost of demolition cost of new road put in last year lost earnings from other projects due to managers' time being spent on the new plant portion of the cost of leasing the president plane future depreciation initial investment on inventories payments already made for engineering design of the new plant

1. Market value of land and existing building - Yes, O.C 2. cost of demolition - Yes 3. cost of new road put in last year - No, sunk cost 4. lost earnings from other projects due to managers' time being spent on the new plant - Yes, negative spillover effect 5. portion of the cost of leasing the president plane - No, existing overhead (if buying plane then yes) 6. future depreciation - Yes, depr. tax shield 7. initial investment on inventories - Yes, ΔNWC 8. payments already made for the engineering design of the new plant - No, sunk cost

Security: Treasury AAA corporate BBB corporate B corporate --------------------------------------------------------------------------- Yield: 5.2% 5.4% 6.2% 6.9% 1. The table shows the yields to maturity on a number of one-year coupon securities. What is the price per $100 of the face value of a one-year, zero-coupon corporate bond with a BBB rating? 2. What is the credit spread on a one-year, zero-coupon corporate bond with a B rating?

1. P= 100/(1.062)^1 = $94.16 2. 6.9 - 5.2 = 1.7% or 170 b.p

You're an investment banker specializing in helping corporations issue bonds. A client firm wants to issue a 5 year semi-annual coupon bond that has been rated Ba by moody's. Currently, the average Ba bond has a credit spread of 250 b.p. the yield to maturity of a 5-year treasury semi-annual coupon bond is 1.9%. if you price these bonds at par, - estimate the rate at which you would have to set the coupon - find the cash flow

1. at par so, r = coupon rate = 1.9% credit spread = risk - risk-free 2.5(given) = risk - 1.9 so, 4.4% or 440 b.p = rate 2. 1000 x .044 = $44 annual 44/2 = 22 semi-annual

risky bond pricing in the real world

1. find the yield on the treasury bond with a similar CF structures (similar timing of CF) 2. Find the credit spread on bonds with same level risk (credit rating) 3. add the two up to get required promised yield 4. discount the promised cash flows to obtain a price

why might interest rates vary among maturities?

1. interests rates are expected to rise in the future 2. lower prices paid for longer maturities b/c they are riskier 3. purchasers value short-term bonds more than long

The IRR and NPV criteria lead to the same investment decision when:

1. the initial net cash flow is negative and the following are all positive ( or vise versa) one sign change Investment: - + + + + Borrowing: + - - - - 2.the investment decision does not affect the decision to accept or reject another investment - NPV allows you to accurately choose between different projects, but cant use IRR to choose between projects ex.) 100% return on $1 vs 10% return on $1 million

Risky zero coupon bond example: Suppose I decide to start a corporation called Bailout Inc, which invests in politically-connected companies on the brink of bankruptcy, gambling in the hope that they'll get bailed out by the government. Suppose Bailout inc issues zero coupon bonds that promise to pay $1000 in 5 years' time. B/c this is very risky, there's a 50/50 chance bailout inc will go bankrupt. If it survives, investors get $1000.

1.) what is the expected cash flow 50 chance 0 50 chance 1000 so, (0.5)(1000) + (0.5)(0) = $500 2.) Assume 5 year treasury strips are currently yielding 3%(expressed as EAR). If everyone was risk neutral, bailout inc would have to provide an expected annual rate of return of 3%. What is their price(B) equal to? Bond price = expected cash flow/(1+r)^t = 500/(1.03)^5 = $431.30 3.) what is the yield (y) on this bond? Y=(FV/PV)^1/t - 1 = (1000/431.30)^1/5 -1= 0.1832 or 18.32% * as r goes up, p goes down *

finding the price of a bond

1.) zero coupon P= par/(1+r)^t 2.) annual coupon payments: P=CF/r [1- 1/(1+r)^t]+par/(1+r)^t

What is the bond price of a 30 year bond with a coupon rate of 5 % annual coupons and a market interest rate of 7%?

1000 x .05 = 50 P= 50/.07 [1- 1/(1.07)^30]+ 1000/(1.07)^30

A florist is buying a number of motorcycles to expand its delivery service. These will cost $87,000, but are expected to increase profits by $3000 per month over the next four years. What is the payback period in this case?

29 months Payback period = 87,000 / 3000 = 29 months

ΔEquity value or ΔFirm value

=ΔFCF0 + ΔFCF1/(1+R1)^1 + ΔFCF2/(1+R2)^2+....

A decrease in the sales of a current project because of the launching of a new project is A) cannibalization. B) a sunk cost. C) an overhead expense. D) irrelevant to the investment decision

A

Net Present Value (NPV)

A dollar measure of the impact of a project on the company's wealth. ex.) if a project has an npv of $100,000, doing it would increase the value of the fir by $100,000

Label True or False: A. bonds are securities sold by the government and corporations to raise moments from investors today in exchange for promised future payments B. By convention the coupon rate is expressed as an effective annual rate C. corporate bonds typically make two types of payments to their holders D. the time remaining until the repayment date is known as the term of the bond

A. True B. False C. True D. True

label true and false A. a fall in bond prices causes interest rates to fall B. a fall in interest rates causes a fall in bond prices C. A rise in interest rates causes bond prices to fall D. bond prices and interest rates are not correlated

A. false B. false C .true (inverse relationship) D. false

Decision rule for payback

Accept if the payback period is less than some benchmark length of payback time payback < benchmark

Which of the following costs would you consider when making a capital budgeting decision? A) sunk cost B) opportunity cost C) interest expense D) fixed overhead cost

B

Which of the following would you NOT consider when making a capital budgeting decision? A) The additional taxes a firm would have to pay in the next year. B) The cost of a marketing study completed last year. C) The opportunity to lease out a warehouse instead of using it to house a new production line. D) The change in direct labor expense due to the purchase of a new machine

B) The cost of a marketing study completed last year. - sunk cost

How are investors in zero-coupon bonds compensated for making such as investment? A. Purchased at face value and sold at a premium at a later date B. The bond makes regular interest payments C. Such bonds are purchases ar a discount to their face value D. The face value of these bonds is less than the value of the bond when the bond matures

C. Such bonds are purchased at a discount to their face values (par)

why are the interest rates of US treasury notes less than the interest rates of equivalent corporate bonds? A. The government has a high credit spread B. there is a significant risk that the US gov will default C. US treasury securities are widely regarded to be risk-free D. US treasury securities are generally used to determine interest rates

C. US treasury securities are widely regarded to be risk-free

Example: You are considering the purchase of a 3-year, 10% coupon bond. The bond pays coupons annually. The discount rate for a similar bond is 8%. What price are you willing to pay? discount rate does not equal the coupon rate discount rate = r

Coupon payment = 10% x 1000 = $100 V = (CF1)/(1 + r)^1 + (CF2)/(1 + r)^2 + (CF3)/(1 + r)^3 = 100/(1.08) + 100/(1.08)^2 + (100 + 1000)/(1.08)^3 V = $1051.54 > $1000 , so..... premium bond

A convenience store owner is contemplating putting a large neon sign over his store. It would cost $50,000, but is expected to bring an additional $24,000 of profit to the store every year for five years. Would this project be worthwhile if evaluated using a payback period of two years or less and if the cost of capital is 10%? A) Yes, since it will pay back its initial investment in two years. B) Yes, since the value of the cash flows into the store, in present dollars, are greater than the initial investment. C) Yes, since the cash flows after two years are greater than the initial investment. D) No, since the value of the cash flows over the first two years are less than the initial

D

Which of the following situations can lead to IRR giving a different decision than NPV? A) Different timing of cash flows B) Multiple IRRs C) Differences in project scale D) All of the above can lead to IRR giving a different decision than NPV

D

The owners of a chain of fast-food restaurants spend $28 million installing donut makers in all their restaurants. This is expected to increase cash flows by $10 million per year for the next five years. If the discount rate is 6.5%, were the owners correct in making the decision to install donut makers? A) No, as it has a net present value (NPV) of -$2.25 million. B) No, as it has a net present value (NPV) of-1.68 million. C) Yes, as it has a net present value (NPV) of $8.74 million. D) Yes, as it has a net present value (NPV) of $13.56 million

D) Yes, as it has a net present value (NPV) of $13.56 million -28 million + 10 mil/.065 [ 1- 1/(1.065)^5] = 13.56 million

Which of the following adjustments should NOT be made when computing free cash flow from incremental earnings? A) adding depreciation B) subtracting all non-cash expenses C) subtracting increases in Net Working Capital D) subtracting depreciation expenses from taxable earnings

D. subtracting depreciation expenses from taxable earnings

risk premium

Difference between the actual yield and the risk neutral yield.

Always ask the "with vs. without" questions!

Is this CF different with or without project? Yes, incremental (include) No, irrelevant (exclude)

Operating cash flow (CFO)

NI + Depreciation Project externality: 1. Positive spill over effect 2. Negative spill over effect (cannibalization)

A firm reports that in a certain year it had a net income of $4.5 million, depreciation expenses of $2.8 million, capital expenditures of $2.3 million, and Net Working Capital decreased by $1.5 million. What is the firm's free cash flow for that year?

NI 4.5 MIL +depr + 2.8 MIL -CAPEX - 2.3 MIL WC 1.5 MIL ---------------------- ----------- FCF = $6.5 MILLION

An auto-parts company is deciding whether to sponsor a racing team for a cost of $1 million. The sponsorship would last for three years and is expected to increase cash flows by $580,000 per year. If the discount rate is 7.5%, what will be the change in the value of the company if it chooses to go ahead with the sponsorship?

NPV = -1,000,000 + 580,000 / (1.075) + 580,000 / (1.075)^2 + 580,000 / 1.075)^3 = $508,305

A farmer sows a certain crop. It costs $250,000 to buy the seed, prepare the ground, and sow the crop. In one year's time it will cost $110,000 to harvest the crop. If the crop will be worth $380,000, and the interest rate is 8%, what is the net present value (NPV) of this investment

NPV= -250000 + 270000/(1.08)^1 = 0

changes in networking capital (NWC)

NWC ↑ , CF ↓ NWC ↓ , CF ↑

consider a zero coupon bond with a $1000 face value and ten year maturity. If the market interest rate of this bond is 10.4%, then the price of this bond is closest to:

P= 1000/(1.104)^10 = $372

in general, there is no single discount rate to discount a series of CF. To be precise in computing the PV of future CF, we use the interest rate for the date that applies to the timing

PV = CF1/(1 + r 1) + CF2/(1 + r2)^2 ....

Price of a bond

Price = PV of future CF

Priced at Par discount rate = r

Priced at par (p=1000) when the r = coupon rate

ΔFCF

Refers to incremental free cash flows represents all the changes in the firms free cash flows that result from the investment

After tax free FCF

Selling price - tax rate x profit . profit = selling price - book value . book value = purchase price - acc. depreciation

Internal Rate of Return (IRR)

THE DISCOUNT RATE THAT IMPLIES ZERO NET PRESENT VALUE FOR A SERIES OF CASH FLOWS NPV=0= CF0 + CF1/(1+ IRR)1 + CF2/(1+IRR)^2 .... determining IRR is like solving for the YTM or the return on investment depends only on the characteristics of the cash flows - IRR<benchmark (O.C.C) = ACCEPT - IRR>benchmark = REJECT IRR ≠ Firm value

Bond issuer

The borrower on one side of a bond issue.

Bonds with a high risk of default generally offer high yields?

True

computing returns on bond

We need to know: 1. CF or coupon payments 2. beginning price ( buying price) 3. end price (selling price) The YTM is not the return we actually earned from buying and selling the bond. To compute the return we we actually earned: r= (coupon payment + price change)/intial payment ex: Buy a bond for $950, sell for $1040. Received $100 coupon payment. r= 100+(1050-950)/950 = .2 or 20%

Evaluating a borrowing opportunity: You are offered a five year amortized loan that has a face value of $100,000 that will require 5 annual payments of $26,378 0 1 2 3 4 5 -------------------------------------------------------------------------------- +100000 -26378 -26378 -26378 -26378 -26378 The IRR turns out to be 10%, if the benchmark r is 12%, should the loan be undertaken?

Yes, want a lower rate as a borrowe If borrow: IRR< benchmark If invest: IRR>benchmark

credit spread

Yield - Risk-free rate = Expected loss + Risk premium - as credit risk goes up, risk yield goes up, credit spread goes up

you can decompose the yield into three components:

Yield= risk-free rate + expected loss + risk premium risk-free rate= given expected lost = premium - risk free rate

yield to maturity (conceptual, cannot be done with calculator)

a single interest rate that implies a PV of cash flows of a bond that equals its correct price - determined by the pattern of CF and the current value. = CF 1/(1 + YTM) + CF 2/(1 + YTM)^2 ...... A complicated average of interest rates in the term structure, it is not a market rate of interest *does NOT determine prices*

lender

an organization or person that lends money.

incremental cash flows

any and all changes in free cash flows caused by undertaking an investment

what happens to value when interest rate increases?

as r rises, bond price decresases b/c if r goes up, that means the other bond alternatives are better as they are offering a higher return. Our coupon rate is fixed and cannot be altered

Above Par value

at a premium or above par (p>1000) when r < coupon rate r < coupon rate

Below Par value

at discount or below par (p<1000) when the r > coupon rate

A company releases a 5 year bond with a face value of $1000 and coupons paid semiannual. If market interest rate is 6%, what should be the coupon rate offered if the bond is to trade at par?

at par is when r=coupon so, 6%

zero coupon bond

bond pays a single lump sum at maturity ex: treasury strips

definition of free cash flow

changes in operating CF (Δ rev - Δ expense) - changes in new working capital Δshortterm investments changes in capital spending (Δ long-term investments ------------------------------------ = changes in free cash flow

credit spreads are often expressed in terms of basis points (bp)

ex: 1% would be 100 basis points

evaluating proposal example 1

in course pack

evaluating proposal example 2

in course pack

evaluating proposal example 3

in course pack

The valuation principal

insures that the firm's assets equals the present value of all current and future free cash flows (FCF) Firms value= FCF 0 + FCF1/(1+R1)^1 + FCF2/(1+r2)^2 +FCF3/(1+r3)^3+.....

note

investor = want a higher rate borrower = want a lower rate

r = required return = discount rate=cost of capital

is the return that could be earned in the financial market for investments with the same risk (O.C)

Payback Period

is the time until the SUM of future cash flows equal the initial investment "time it takes to get money back" ex.) 0 1 2 3 -------------------------------------------- -30000 15000 15000 15000 It will take two years to get back initial investment

In order to attract investors into firms with high probability of default...

offer higher yield

what happens when interest rate declines?

our bond is more attractive so demand and price rise

interest only bond

pays regular coupon amounts and principal is repaid in full at maturity

coupon

percentage of par value that is paid every year (usually semi-annual = divide by two) ex: 3 yr, 10 % coupon =1000 x 10%= 100 annual or 50 every 6 months

why do all stockholder agree that you should take any positive NPV projects you come across?

positive NPV raises firms value so, stockholders' wealth increases

A company releases a 5 year bond with a face value of $1000 and coupons paid semiannually. If market interest rate is 6%, which coupon rate will cause the bond to be issued at s premium? A. 3% B. 4% C. 6% D. 8%

premium is when r>coupon, so.. r>coupon rate 6%>coupon rate

par value

principal amount to be paid at maturity. Usually $1000

A five year zero coupon bond with a $1000 face value is priced at $750. The yield to maturity of a 5 year treasury strip is 1.9%. what is the bond's credi spread? express answer in basis points.

r= (1000/750)^1/5 - 1 = 5.92% credit spread = Yield - Risk-free rate so, 5.92 - 1.9 = 4.02 % = 402 b.p

cannibalization

sales loss caused by a company's introduction of a new product

term structure of interest rates

set of interest rates for various maturities for a particular risk class of bonds

Current Yield current yield ≠ YTM ≠ Return

simply the total annual coupon payments divided by the current bond price Ex: 10% bond with a price of $1046.87 has a current yield of $100/$1046.87 = 9.55%

Market Value of Equity (MVE) also called equity value

stock price x number of stocks

tax shield

tax rate x Dep. expense

maturity

the amount of time that will pass until the par value is paid

The payback rule is based on the idea that an opportunity that pays back its initial investment quickly is a worthwhile opportunity.

true

You can evaluate alternative projects with different lives by calculating and comparing their equivalent annual annuity.

true