FIN 301 Exam II
Future Value
the value of a current asset at a specified date in the future based on an assumed rate of growth. amount an investment is worth after one or more periods. Simplest case is the single investment period. ---depend critically on the assumed interest rate, particularly for long-lived investments
A perpetuity is defined as:
unending equal payments paid at equal time intervals.
After-tax salvage
• If the salvage value is different from the book value of the asset, then there is a tax effect. • Book value = initial cost - accumulated depreciation • After-tax salvage = salvage - T × (salvage - book value)
Mutually exclusive projects
• Initial investments are substantially different (issue of scale). • Timing of cash flows is substantially different. If you choose one, you can't choose the other. Example: You can choose to attend graduate school at either Harvard or Stanford, but not both. NPV - choose the project with the higher NPVIRR - choose the project with the higher IRR
Internal Rate of Return Advantages
• Knowing a return is intuitively appealing • It is a simple way to communicate the value of a project to someone who doesn't know all the estimation details. • If the IRR is high enough, you may not need to estimate a required return, which is often a difficult task.
Which one of the following is an example of a sunk cost?
$2,000 paid last year to rent equipment
Interest Rate
"exchange rate" between earlier money and later money
Annuity due example
$1 million 35 years 1 % month. First payment today? BGN 1,000,000 FV 1 I/Y 35 * 12 = 420 N CPT PMT
Simple Interest ex
$10,000 5 years 5% simple interest A: $2500
Discount rate example
$100 end each year 10 years. PV annually discount rate 8%. 100 PMT 10 N 8 I/Y CPT PV
You just paid $480,000 for an annuity that will pay you and your heirs $15,000 a year forever. What rate of return are you earning on this policy?
Payment/ Rate 15000/480,000 3.13%
Implied rate example
$1200 in 5 years invest $1000. Implied? 5 N -1000 PV 1200 FV =3.71 CPT I/Y
Number example
$20,000 invest 10% and have currently $15,000. How long? 20,000 FV -15,000 PV 10 I/Y = 3.0184 CPT N
Ordinary Annuity Example
$200,000 deposit 0.75%. Receive every month for 5 years? 5 * 12 = 60.00 -200,000 PV .75 I/Y 60 N CPT PMT
Pham & Davis currently sells 9,820 motor homes per year at $45,500 each, and 3,680 luxury motor coaches per year at $89,700 each. The company wants to introduce a new portable camper to fill out its product line. It hopes to sell 4,000 of these campers per year at $14,750 each. An independent consultant has determined that if the new campers are introduced, sales of its existing motor homes will most likely increase by 250 units per year while the sales of its motor coaches will probably decline by 368 units per year. What is the amount that should be used as the annual sales figure when evaluating the portable camper project?
$37,365,400Sales = 4,000($14,750) + 250($45,500) - 368($89,700)Sales = $37,365,400
Perpetuity Rate of Return
Payment/investment
Number of Periods
't' or 'n'-number of periods.
On your tenth birthday, you received $300 which you invested at 4.5 percent interest, compounded annually. Your investment is now worth $756. How old are you today?
(756/300) = (1.045)^n n = log(1.045)(756/300) = 20.9978 approx 21 year's So the person is 31 years old today
Compound interest formula
(Principal + Interest) X Interest
Taxes
- cash flows should be evaluated on an after-tax basis
Your credit card company charges you 1.15 percent interest per month. What is the APR?
1.15 * 12 = 13.8%
Cash Flow Benefits
Positive side effects, benefits to other projects
Assume the total cost of a college education will be $245,000 when your child enters college in 15 years. You presently have $108,000 to invest for this purpose. What annually compounded rate of interest must you earn to cover the cost of your child's college education?
245,000 FV 15 N -10800 PV CPT I/Y I/Y = 5.61
Jonathan invested $6,220 in an account that pays 11 percent simple interest. How much money will he have at the end of 40 years?
Principle* rate * time = 6220 * 11% * 40 = 27368 Total amount = principal + interest = 6220 + 27368 = 33588
Aidan can afford $240 a month for five years for a car loan. If the interest rate is 8.5 percent, what is the most he can afford to borrow?
8.5/12 = 0.7083% I/Y 12 * 5 = 60 n 240 A: $116988
Integrity Materials is considering expanding on some land that it currently owns. The initial cost of the land was $364,500 and it is currently valued at $357,900. The company has some unused equipment that it currently owns valued at $29,000 that could be used for this project if $8,200 is spent for equipment modifications. Other equipment costing $157,900 will also be required. What is the amount of the initial cash flow for this expansion project?
= - current value of land - value of unused equipment - equipment modification - other equipment costing = - $ 357,900 - $ 29,000 - $ 8,200 - $ 157,900 = - $ 553,000
Fifteen years ago, you invested $5,000. Today, it is worth $18,250. What annually compounded rate of interest did you earn?
=($18250 / $5000)^(1/15) -1 =(3.65) ^ 0.0666666667 - 1 =1.09014983 -1 =0.09014983 or 9.01%
Sixty years ago, your grandmother invested $4,500. Today, that investment is worth $430,065.11. What is the average annually compounded rate of return she earned on this investment?
=(430,065.11/4500)^(1/50)-1 =95.57 ^ 0.0167 - 1 = 0.0791 = 7.90%
Your parents have made you two offers. The first offer includes annual gifts of $4,000, $4,500, and $5,200 at the end of each of the next three years, respectively. The other offer is the payment of one lump sum amount today. You are trying to decide which offer to accept given the fact that your discount rate is 9.7 percent. What is the minimum amount that you will accept today if you are to select the lump sum offer?
A: $11324.66 **Practice, refer to chegg
The interest rate that is most commonly quoted by a lender is referred to as the:
APR Annual percentage rate
Isaac has analyzed two mutually exclusive projects that have 3-year lives. Project A has an NPV of $81,406, a payback period of 2.48 years, and an AAR of 9.31 percent. Project B has an NPV of $82,909, a payback period of 2.57 years, and an AAR of 9.22 percent. The required return for Project A is 11.5 percent while it is 12 percent for Project B. Both projects have a required AAR of 9.25 percent. Isaac must make a recommendation and justify it in 15 words or less. What should his recommendation be?
Accept Project B and reject Project A based on the NPVs
Payback
Accept if the payback period is less than some preset limit.
Twenty years from now, you hope to have $175,000 to buy a parcel of land. How much must you deposit as a lump sum today to achieve this goal at an interest rate of 6.6 percent, compounded annually?
Amount required after 20 years (FV) = $175,000Rate (r ) = 6.60%Time (n) = 20 yearsAmount to be deposited today (PV) = ?? Future Value = Present Value * (1 + r)^ n$175,000 = PV * (1 + 0.0660) ^ 20$175,000 = PV * 1.0660^ 20$175,000 = PV * 3.59041PV = $48,740.95
annual rate example
Annual rate is $1000 grows into $4000 in 20 years? -1000 PV 4000 FV 20 N CPT I/Y
Theresa adds $1,500 to her savings account on the first day of each year. Marcus adds $1,500 to his savings account on the last day of each year. They both earn 6.5 percent annual interest. What is the difference in their savings account balances at the end of 35 years?
Annuity/rate*((1+rate)^n-1)*rate=1500/6.5%*(1.065^35-1)*6.5%=12093.3823
4th Variable Example
Assume we are offered an investment that costs us $100 and will double our money in eight years. To compare this to other investments, we would like to know what discount rate (i.e., rate of return, or return) is implicit in these numbers. (8th root both sides or use a future value table) • Here, PV = $100, FV = $200 (double our money), and an 8-year life • To calculate the return, we can write the basic present value equation as: Use a financial calculator.• PV = -100, FV = 200, N = 8; Solve for I/Y = 9%
Payback Example I
Assume we will accept the project if it pays back within two years. Year 1: 165,000 - 63,120 = 101,880 still to recover Year 2: 101,880 - 70,800 = 31,080 still to recover Year 3: 31,080 - 91,080 = -60,000 project pays back in year 3
accumulated depreciation
Book value = initial cost - accumulated depreciation
Which cash flow is not included as incremental evaluating new project?
Cash flow generated from existing products
Net working capital
Changes in net working capital (increases in inventory or receivables). GAAP requires that sales be recorded on the income statement when made, not when cash is received. GAAP also requires that we record cost of goods sold when the corresponding sales are made, whether we have actually paid our suppliers yet. Finally, we have to buy inventory to support sales, although we haven't collected cash yet.
Simple vs. compound
Compound interest will generate a higher future value
Cullen invested $5,000 five years ago and earns 6 percent annual interest. By leaving his interest earnings in her account, he increases the amount of interest he earns each year. His investment is best described as benefitting from:
Compounding
Compounding interest
Compounding the interest means earning interest on interest. The effect of compounding is small for a small number of periods but increases as the number of periods increases. -process of accumulating interest on an investment over time to earn more interest
Meek's is considering a five-year project that will require $738,000 for new fixed assets that will be depreciated straight-line to a zero book value over five years. No bonus depreciation will be taken. At the end of the project, the fixed assets can be sold for 18 percent of their original cost. The project is expected to generate annual sales of $679,000 with costs of $321,000. The tax rate is 22 percent and the required rate of return is 15.2 percent. What is the amount of the aftertax salvage value?
Cost of Fixed Assets = $738,000 Salvage Value = 18% * Cost of Fixed AssetsSalvage Value = 18% * $738,000Salvage Value = $132,840 After-tax Salvage Value = Salvage Value * (1 - tax)After-tax Salvage Value = $132,840 * (1 - 0.22)After-tax Salvage Value = $103,615.20
opportunity cost
Cost of lost options (existing land or building)
Nirav just opened a savings account paying 2 percent interest, compounded annually. After four years, the savings account will be worth $5,000. Assume there are no additional deposits or withdrawals. Given this information, Nirav:
Could have deposited less money today and still had 5k in four years if the account paid higher interest
straight-line depreciation
D = (Initial cost - salvage) / number of years Very few assets are depreciated straight-line for tax purposes.
Newson Minerals is considering a project that will require the purchase of $479,000 of equipment. The equipment will be depreciated straight-line to a zero book value over the five-year life of the project after which it will be worthless. The required return is 12 percent and the tax rate is 25 percent. What is the value of the depreciation tax shield in Year 4 of the project assuming no bonus depreciation is taken?
Depreciation tax shield =( $479,000 / 5 ) * 25% Depreciation tax shield = $95,800 * 25% Depreciation tax shield = $23,950
Depreciation tax shield
Depreciation × Tax rate (D X T) D = depreciation expense T = marginal tax rate
NPV
Difference between market value and cost Take the project if the NPV is positive. Has no serious problems Preferred decision criterion
IRR
Discount rate that makes NPV = 0 Take the project if the IRR is greater than the required return. Same decision as NPV with conventional cash flows IRR is unreliable with nonconventional cash flows or mutually exclusive projects.
The internal rate of return is defined as the:
Discount rate which causes the net present value of a project to equal zero.
Time value of money
Double 10k 5% annually A: 72/5 = 14.4
True
EAR always higher than APR
Payback Advanages
Easy to understand Adjusts for the uncertainty of later cash flows Biased toward liquidity
Internal Rate of Return Example I
Enter the cash flows as you did with NPV. Press IRR and then CPT. IRR = 16.13% > 12% required return
Sunk Cost
Example: 10k of maintenance costs incurred in the prior year to repair an existing piece of equipment that will be used in an upcoming investment project
Negative incremental Value
Example: 10k reduction in sales of an existing product due to the introduction of a new project
Incremental project cost
Example: 12,500 of material and labor costs to associated with sales related to a new project investment
investment in working capital
Example: 50k in raw material inventory that will result due to the launch of a new project
Positive incremental value
Example: 50k in revenue from a new product investment
Opportunity cost
Example: A building you own valued at 1 million which will be used in an investment project
Tax Shield
Example: Tax rate 21%, annual equipment $750k and ongoing depreciation $250K? A: 21% * 250,000 = 52,500 F: Tax rate * depreciation
Sunk costs
Excluded; costs that have accrued in the past.
Present Value Example IV
For a given interest rate - the longer the time period, the lower the present value What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10% 5 years: PV = 500 / (1.1)5 = 310.46 10 years: PV = 500 / (1.1)10 = 192.77
Interest-only loan example I
For example, with a three-year, 10 percent, interest-only loan of $1,000, the borrower would pay $1,000 × .10 = $100 in interest at the end of the first and second years• At the end of the third year, the borrower would return the $1,000 along with another $100 in interest for that year • Likewise, a 50-year interest-only loan would call for the borrower to pay interest every year for the next 50 years and then repay the principal
What is the future value of $8,500 invested at the end of each year for 40 years, at 10.8 percent interest compounded annually?
Future Value of Annuity = Periodic Payments [(1+r)n-1]/r Future Value of Annuity = $8,500[(1+0.108)40-1]/0.108 Future Value of Annuity = $8,500[550.713190053] Future Value of Annuity =$4,681,062.12
Assume you are investing $100 today in a savings account. Which one of the following terms refers to the total value of this investment one year from now?
Future value
What is the future value in 60 years of $7,440 invested today at 9 percent interest, compounded annually?
Future value = Present value(1 + Rate)^Time Future value = $7,440(1 + 0.09)^60 Future value = $1,309,673
Interest rate Example II
Going back to our $100 investment, what will you have after two years, assuming the interest rate doesn't change? • $110 x .10 = $11 in interest during the second year • Total = $110 + 11 = $121• $121 is the future value of $100 in two years at 10%
Number of Periods Example III
How much do you need to have in the future? Down payment = .1(150,000) = 15,000 Closing costs = .05(150,000 - 15,000) = 6,750 Total needed = 15,000 + 6,750 = 21,750• Compute the number of periods• Using the formula: t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years• Using a financial calculator: PV = -15,000, FV = 21,750, I/Y = 7.5, CPT N = 5.14 years
A relative will support your education by paying you $500 a month for 50 months. If you can earn 7 percent on your money, what is this gift worth to you today?
I/Y = 7%/12= 0.5833333333% Monthly payment, PMT = 500 Number of months, N = 50 FV = 0 (No future value) Input the below into a financial calculator: N = 50 I/Y = 0.5833333333 PMT = 500 FV = 0 CPT PV PV = $21,629.93 (Ignore negative sign) The gift is worth $21,629.93 today
Which one of the following statements related to the internal rate of return (IRR) is correct?
IRR is equal to Required Rate of Return when NPV is 0.
payback Disadvantages
Ignores the time value of money Requires an arbitrary cutoff point Ignores cash flows beyond the cutoff date Biased against long-term projects, such as research and development, and new projects
Opportunity costs
Included; costs of lost options (existing land or buildings)
Assume your mother invested a lump sum 28 years ago at 4.05 percent interest, compounded annually. Today, she gave you the proceeds of that investment, totaling $48,613.24. How much did your mother originally invest?
Interest Rate = r = 4.05% Number of years = n = 28 Value of investment now = FV = $48613.24
Rule of 72
Invest 8% interest rate A: 72/8 = 9 years
Payback Period
Length of time until initial investment is recovered Take the project if it pays back within some specified period. Doesn't account for time value of money, and there is an arbitrary cutoff period
Salazar's Salads is considering two projects. Project X consists of creating an outdoor eating area on the unused portion of the restaurant's property. Project Z would instead use that outdoor space for creating a drive-thru service window. When trying to decide which project to accept, the firm should rely most heavily on which one of the following analytical methods?
NPV
NPV Example I
NPV = -165,000 + 63,120/(1.12) + 70,800/(1.12)2 + 91,080/(1.12)3 = 12,627.41 • Using the calculator: CF0 = -165,000; C01 = 63,120; F01 = 1; C02 = 70,800; F02 = 1; C03 = 91,080; F03 = 1; NPV; I = 12; CPT NPV = 12,627.41
Non-conventional Cash Flows
NPV and IRR will generally give us the same decision. cash flow signs change more than once.
MACRS
Need to know which asset class is appropriate for tax purposes Multiply percentage given in table by the initial cost. Depreciate to zero Mid-year convention - assumes all assets are purchased mid year for depreciation purposes
Operating Cash flow
Net income + Depreciation
Discount Rate
Often we will want to know what the implied interest rate is on an investment.
Bui Bakery has a required payback period of two years for all of its projects. Currently, the firm is analyzing two independent projects. Project X has an expected payback period of 1.4 years and a net present value of $6,100. Project Z has an expected payback period of 2.6 years with a net present value of $18,600. Which project(s) should be accepted based on the payback decision rule?
Project X will be selected as Actual payback period ( 1.4 years ) < Expected Payback period ( 2 Years ) Project Z will be Rejectedas Actual payback period ( 2.6 years ) > Expected Payback period ( 2 Years ) Answer - Project X only
What is the EAR if a bank charges you an APR of 7.65 percent compounded quarterly?
Quarterly Interest Rate = 7.65%/4Quarterly Interest Rate = 1.9125% Effective Annual Rate = (1 + Quarterly Interest Rate)^4 - 1Effective Annual Rate = (1 + 0.019125)^4 - 1Effective Annual Rate = 1.019125^4 - 1Effective Annual Rate = 1.0787 - 1Effective Annual Rate = 0.0787 or 7.87%
Net Income
Sales X profit margin
Eunchae invested $2,000 six years ago at 4.5 percent interest. She spends all of her interest earnings immediately so she only receives interest on her initial $2,000 investment. Which type of interest is she earning?
Simple Interest
A project will require $512,000 for fixed assets and $47,000 for net working capital. The fixed assets will be depreciated straight-line to a zero book value over the six-year life of the project. No bonus depreciation will be taken. At the end of the project, the fixed assets will be worthless. The net working capital returns to its original level at the end of the project. The project is expected to generate annual sales of $965,000 and costs of $508,000. The tax rate is 21 percent and the required rate of return is 14.7 percent. What is the amount of the annual operating cash flow?
Straight line Depreciation = Fixed Assets Cost / Project Life = $ 512,000 / 6 = $ 85,333.3333 ; Thus applying the above information in the formula we have = [ ( $ 965,000 - $ 508,000 - $ 85,333.3333 ) * ( 1 - 0.21 ) ] + $ 85,333.3333 = [ $ 371,666.6667 * ( 1 - 0.21 ) ] + $ 85,333.3333 = ( $ 371,666.6667 * 0.79 ) + $ 85,333.3333 = $ 293,616.6667 + $ 85,333.3333 = $ 378,950.00
Which one of the following types of costs was incurred in the past and cannot be recouped?
Sunk
Nonconventional Cash Flows Example I
Suppose an investment will cost $90,000 initially and will generate the following cash flows: Year 1: 132,000 Year 2: 100,000 Year 3: -150,000 • The required return is 15%. • The NPV is positive at a required return of 15%, so you should Accept. • If you use the financial calculator, you would get an IRR of 10.11% which would tell you to Reject.
Number of Periods Example I
Suppose we are interested in purchasing an asset that costs $50,000. We currently have $25,000. If we can earn 12 percent on this $25,000, how long until we have the $50,000? • We can again manipulate the basic present value equation to solve for the number of periods: • I/Y = 12, PV = -25,000, FV = 50,000; Solve for N = 6.1163
Discount Rate Example II
Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest? r = (20,000 / 10,000)1/6 - 1 = .122462 = 12.25%
Effective Annual Rate Example I
Suppose you can earn 1% per month on $1 invested today. What is the APR? 1(12) = 12% How much are you effectively earning? • FV = 1(1.01)12 = 1.1268• Rate = (1.1268 - 1) / 1 = .1268 = 12.68% Suppose you put it in another account and earn 3% per quarter. What is the APR? 3(4) = 12% How much are you effectively earning? • FV = 1(1.03)4 = 1.1255• Rate = (1.1255 - 1) / 1 = .1255 = 12.55%
Simple Interest Example II
Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today? FV = 10(1.055)200 = 447,189.84 What is the effect of compounding? Simple interest = 10 + 200(10)(.055) = 120.00 Compounding added $447,069.84 to the value of the investment.
Discount Rate Example III
Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. • You currently have $5,000 to invest. • What interest rate must you earn to have the $75,000 when you need it? r = (75,000 / 5,000)1/17 - 1 = .172688 = 17.27%
Interest Rate Example I
Suppose you invest $100 in a savings account that pays 10% interest per year. How much will you have in one year? In general, if you invest for one period at an interest rate of r, your investment will grow to (1 + r) per dollar invested • r = 10%• 1 + .10 = 1.1 dollars per dollar invested • $100 x 1.1 = $110
Simple Vs. Compounding Example I
Suppose you invest the $1,000 from the previous example for 5 years. How much would you have with simple interest? FV = 1,000(.05)+1,000(.05)+1,000(.05)+1,000(.05)+1,000(.05)+1000 = 1,250.00 How much would you have with compound interest? FV = 1,000(1.05)5 = 1,276.28
Present Value Multiple Periods Example
Suppose you need $1,000 in three years. You can earn 15% on your money. How much do you have to invest today? (In other words, what is the present value of $1,000 in three years at 15%?) • We need to discount $1,000 back three periods at 15% • Discount factor = 1/(1 + .15)3 = 1/1.5209 = .6575• PV = $1,000 x 0.6575 = $657.52 • N = 3 • I/Y = 15 • FV = 1,000 • Solve for PV = -657.52
Present Value Example I
Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today? • PV = 10,000 / (1.07)1 = 9,345.79 Answer: Calculator 1 N 7 I/Y 10,000 FV CPT PV = -9,345.79
APR Payment Example I
Suppose you want to buy a new computer system and the store is willing to allow you to make monthly payments. The entire computer system costs $3,500. • The loan period is for 2 years, and the interest rate is 16.9% with monthly compounding. • What is your monthly payment? 2(12) = 24 N; 16.9 / 12 = 1.408333333 I/Y; 3,500 PV; CPT PMT = -172.88
Future Value Example I
Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years? FV = 3,000,000(1.15)5 = 6,034,072
Incremental Cash Flows
The cash flows that should be included in a capital budgeting analysis are those that will only occur (or not occur) if the project is accepted.
Net Present Value
The difference between the market value of a project and its cost. If the NPV is positive, accept the project. A positive NPV means that the project is expected to add value to the firm and will therefore increase the wealth of the owners.
Which one of the following statements concerning interest rates is correct?
The effective annual rate equals the annual percentage rate when interest is compounded annually.
Effective Annual Rate
This is the actual rate paid (or received) after accounting for compounding that occurs during the year. If you want to compare two alternative investments with different compounding periods, you need to compute the EAR and use that for comparison.
Annual Percentage rate
This is the annual rate that is quoted by law, period rate times the number of periods per year. Period rate = APR / number of periods per year.
Claire's coin collection contains fifty 1948 silver dollars. Her grandparents purchased them at their face value in 1948. These coins have appreciated by 7.6 percent annually. How much is the collection expected to be worth in 2025?
Value in 1948 = P = 50*1 = $50 Number of years = n = 2025 - 1948 = 77 Annual rate = r = 7.6% Hence, Value now = P(1+r)n = 50(1+0.076)77 = $14077.16
Which one of the following statements related to loan interest rates is correct?
When comparing loans you should compare the effective annual rates.
Discount Rate Example I
You are looking at an investment that will pay $1,200 in 5 years if you invest $1,000 today. What is the implied rate of interest? r = (1,200 / 1,000)1/5 - 1 = .03714 = 3.714% Calculator - the sign convention matters! • N = 5 • PV = -1,000 (you pay 1,000 today) • FV = 1,200 (you receive 1,200 in 5 years) • CPT I/Y = 3.714%
Effective Annual Rate Example II
You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use? First account:• EAR = (1 + .0525/365)365 - 1 = 5.39% Second account:• EAR = (1 + .053/2)2 - 1 = 5.37% Suppose you invest $100 in each account. How much will you have in each account in one year? First Account: • 365 N; 5.25 / 365 = .014383562 I/Y; 100 PV; CPT FV = 105.39 Second Account: • 2 N; 5.3 / 2 = 2.65 I/Y; 100 PV; CPT FV = 105.37 • You have more money in the first account.
Annuity Due Example
You are saving for a new house and you put $10,000 per year in an account paying 8%. The first payment is made today. • How much will you have at the end of 3 years? 2nd BGN 2nd Set (you should see BGN in the display) 3 N -10,000 PMT 8 I/Y CPT FV = 35,061.12 2nd BGN 2nd Set (be sure to change it back to an ordinary annuity)
Present Value Example II
You want to begin saving for your daughter's college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today? PV = 150,000 / (1.08)17 = 40,540.34
Number of Periods Example II
You want to purchase a new car, and you are willing to pay $20,000. • If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car? t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years
Present Value Example III
Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest? PV = 19,671.51 / (1.07)10 = 10,000
EAR
[1 + (APR/m)]]^m -1
Annuity Due
a repeating payment that is made at the beginning of each period. It has the following characteristics: • All payments are in the same amount• All payments are made at the same intervals of time (such as once a quarter or year).
Ordinary Annuity
a series of equal payments made at the end of consecutive periods over a fixed length of time. While the payments in an annuity can be made as frequently as every week. In practice, ordinary annuity payments are made monthly, quarterly, semi-annually or annually.
perpetuity
a type of annuity that lasts forever, into perpetuity. The stream of cash flows continues for an infinite amount of time.
Stand-alone principle
allows us to analyze each project in isolation from the firm simply by focusing on incremental cash flows
Positive side effects
benefits to other projects
Interest-only loan
call for the borrower to pay interest each period and to repay the entire principal (the original loan amount) at some point in the future • Most corporate bonds have a general form of an interest-only loan
negative side effects
costs to other projects (erosion)
Financing costs
included in the required return
Changes in net working capital
increases in inventory or recievables
The stand-alone principle advocates that project analysis should be based solely on which one of the following costs?
incremental
Simple Interest
interest is earned only on the original principal amount invested and results in earning interest only on the original present value amount
Net present value:
is the best method of analyzing mutually exclusive projects.
If a firm accepts Project X it will not be feasible to also accept Project Z because both projects would require the simultaneous and exclusive use of the same piece of machinery. These projects are considered to be:
mutually exclusive
Operating cash flow
net income + depreciation
Simple Interest Formula
principal X interest
The fact that a proposed project is analyzed based on the project's incremental cash flows is the assumption behind which one of the following principles?
standalone principle
Perpetuity Example I
suppose your grandparents invested $100,000 in an annuity that pays a guaranteed $4,000 per year forever. What would the guaranteed return on the investment be? Payment/Investment or $4,000 / $100,000 = 4.0%
Present Value
the current value of a future sum of money or stream of cash flows given a specified rate of return. discounted at the appropriate discount rate. -is the reverse of future value -Instead of compounding the money forward into the future, we discount it back to the present -To discount is to calculate the present value of some future amount -For a given interest rate - the longer the time period, the lower the present value -For a given time period - the higher the interest rate, the smaller the present value
NPV vs. IRR
• NPV directly measures the increase in value to the firm. • Whenever there is a conflict between NPV and another decision rule, you should always use NPV. • IRR is unreliable in the following situations: Nonconventional cash flows Mutually exclusive projects
Internal Rate of Return
• This is the most important alternative to NPV. • It is often used in practice and is intuitively appealing.• It is based entirely on the estimated cash flows and is independent of interest rates found elsewhere. RR is the return that makes the NPV = 0. Accept the project if the IRR is greater than the required return!!!
Future Value APR Example I
•Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. • How much will you have in the account in 35 years?35(12) = 420 N 9 / 12 = .75 I/Y 50 PMT CPT FV = 147,089.22
Annual Percentage Rate Examples
•What is the APR if the monthly rate is .5%? .5(12) = 6% •What is the APR if the semiannual rate is .5%? .5(2) = 1% •What is the monthly rate if the APR is 12% with monthly compounding? 12 / 12 = 1%
Present Value APR Example I
•You need $15,000 in 3 years for a new car. • If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit? 3(365) = 1,095 N 5.5 / 365 = .015068493 I/Y 15,000 FV CPT PV = -12,718.56
