FINAN 450 Exam 2

True/ false: Using the Excel formula to convert a quoted rate (or an APR) to an effective rate, use the formula EFFECT(nominal_rate, npery).

true

True/ false: When calculating the present value of an annuity using the financial calculator, you enter the cash flows of the annuity in the PMT key.

true

True/ false: When using a financial calculator to find the number of payments, the PMT value should be entered as a negative.

true

True/ false: You need the future value factor and the discount rate to find the annuity future value factor.

true

If you invest at a rate of r for ___________ periods, under compounding, your investment will grow to (1+r)^2 per dollar invested.

two

The cash flows of an annuity due are the same as those of an ordinary annuity except that there is an extra cash flow at Time

zero

Present Values

• The current value of future cash flows discounted at the appropriate discount rate • Value at t=0 on a time line - Present Value = the current value of an amount to be received in the future - Why is it worth less than face value? - Opportunity cost - Risk & Uncertainty Discount Rate = ƒ (time, risk)

Finding the Number of Payments

$1,000 due on credit card • Payment = $20 month minimum • Rate = 1.5% per month • The sign convention matters!!! 1.5 I/Y 1000 PV -20 PMT 0 FV CPT N = 93.111 months = 7.75 years

To find the present value of an annuity of $100 per year for 10 years at 10% per year using the tables, find a present value factor of 6.1446 and multiply it by ______.

$100

Larry's Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $25,000 per year forever. A representative for Larry's tells you the policy costs $645,000. At what interest rate would this be a fair deal?

$25,000/ $645,000 = .0388 = 3.90%

Which of the following is the formula for the future value of an annuity factor?

((1+r)t−1/r)

EAR formula

(1+APR/m)^m-1 APR= the quoted rate M= number of compounds per year

The formula for the present value of an annuity due is:

(1+r)×(PV of an ordinary annuity)

Pure Discount Loans: If a T-bill promises to repay $10,000 in 12 months and the market interest rate is 7 percent, how much will the bill sell for in the market?

- 1 N; 10,000 FV; 7 I/Y; CPT PV = -9345.79 - =PV(.07,1,0,10000)

First account:

- EAR = (1 + .0525/365)^365 - 1 = 5.39% • ICONV: NOM=5.25; C/Y=365 EFF=5.3899

True/ false: The annuity due calculation assumes cash flows occur at the beginning of the period.

true

True/ false: The annuity present value factor equals one minus the present value factor all divided by the discount rate.

true

True/ false: The annuity present value of an amount C is calculated as C multiplied by {1-[1/(1+r)t]}r.

true

True/ false: The interest rate charged per period on a loan multiplied by the number of periods per year equals the annual percentage rate.

true

Wells, Inc., has identified an investment project with the following cash flows. YearCash Flow 1 $865 2 1,040 3 1,290 4 1,385 a.If the discount rate is 8 percent, what is the future value of these cash flows in Year 4? b.What is the future value at an interest rate of 11 percent? c.What is the future value at an interest rate of 24 percent?

- Future value is the sum of Future values of all the cash flows - Future value = C1*(1+r)3 + C2*(1+r)2 + C3*(1+r)1 + C4 a. When discount rate = r = 8% - Future value = C1*(1+r)3 + C2*(1+r)2 + C3*(1+r)1 + C4 = 865*(1.08)3 + 1040*(1.08)2 + 1290*(1.08)1 + 1385 - Future value = 1089.65088 + 1213.056 + 1393.2 + 1385 = 5080.90688 b. When Discount rate = r = 11% - Future value = C1*(1+r)3 + C2*(1+r)2 + C3*(1+r)1 + C4 = 865*(1.11)3 + 1040*(1.11)2 + 1290*(1.11)1 + 1385 = 1183.000815 + 1281.384 + 1431.9 + 1385 = 5281.284815 c. When discount rate = r = 24% Future value = C1*(1+r)3 + C2*(1+r)2 + C3*(1+r)1 + C4 = 865*(1.24)3 + 1040*(1.24)2 + 1290*(1.24)1 + 1385 Future value = 1649.22976 + 1599.104 + 1599.6 + 1385 = 6232.93376

Interest Rates: Annual Percentage Rate (APR) "Nominal"

- The annual rate quoted by law- APR = periodic rate X number of periods per year - Periodic rate = APR / periods per year

Interest Rates: Effective Annual Rate (EAR)

- The interest rate expressed as if it were compounded once per year. - Used to compare two alternative investments with different compounding periods

We Pay Insurance Co. will pay you $1,025 each quarter for 21 years. You want to earn a minimum interest rate of .82 percent per quarter. What is the most you are willing to pay today for these payments?

-$1,025 PMT 21 X 4 = 84 N .82 I/Y 0 FV PV = $62051.15

Suppose you have $200,000 to deposit and can earn .75% per month. - How much could you receive every month for 5 years?

-200000 PV .75 I/Y 5 N 0 FV PMT= 4,151.67

You're prepared to make monthly payments of $250, beginning at the end of this month, into an account that pays 8 percent interest compounded monthly. How many payments will you have made when your account balance reaches $50,000?

-250 PMT 8/12= 0.6667 I/Y 50,000 FV 0 PV CPT N=127.52 payments

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%

Suppose you have $200,000 to deposit and can earn .75% per month. - How many months could you receive the $5,000 payment?

.75 I/Y -200000 PV 5000 PMT 0 FV N= 47.73 months / 12 = 3.98 years

You have your choice of two investment accounts. Investment A is a 10-year annuity that features end-of-month $1,525 payments and has an interest rate of 7 percent compounded monthly. Investment B is an annually compounded lump-sum investment with an interest rate of 9 percent, also good for 10 years. How much money would you need to invest in B today for it to be worth as much as Investment A 10 years from now?

10 X 12 = 120 N 7/12 = .58333 I/Y 0 PV -1525 PMT CPT FV: $263,896.24 So, Investment in B = 263,896.24 = FV N = 10 I/Y = 9 PMT=0 COMPUTE PV= 111,497.162

What is the monthly rate if the APR is 12% with monthly compounding?

12%/12 = 1% Can you divide the above APR by 2 to get the semiannual rate? NO. You need an APR based on semiannual compounding to find the semiannual rate.

Computing Payments with APRs: Suppose you want to buy a new computer. 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. The interest rate is 16.9% with monthly compounding. What is your monthly payment?

2 (12) = 24 N 16.9 / 12 = 1.40833 I/Y 3500 PV 0 FV CPT PMT = -172.88

Suppose you borrow $2,000 at 5% and you are going to make annual payments of $734.42. How long before you pay off the loan?

2000 PV 5 I/Y -734.42 PMT 0 FV N= 2.998 = 3 years

One of your customers has just made a purchase in the amount of $24,800. You have agreed to payments of $515 per month and will charge a monthly interest rate of 1.32 percent. How many months will it take for the account to be paid off?

24,800 PV -515 PMT 1.32 I/Y 0 FV CPT N= 76.99 months.

Larry's Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $25,000 per year forever. Assume the required return on this investment is 4 percent. How much will you pay for the policy?

25000/.04 = 625,000

Your parents are giving you $265 a month for 4 years while you are in college. At an interest rate of .33 percent per month, what are these payments worth to you when you first start college?

265x(1-(1/1.0033^48)))/.0033 = 11745.85

Annuity Due: 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 3 N 8 I/Y 0 PV -10000 PMT CPT FV = 35061.12 2nd BGN 2nd SET

EAR and APR in TI BA II+

2nd ICONV2nd CE/C (to clear the memory) • 3 fields in worksheet: - NOM (Nominal rate-APR) - EFF (Effective annual rate) - C/Y (Compounding periods/yr) - To compute EFF, enter the NOM and C/Y values, move to EFF and press CPT - To compute NOM, enter the EFF and C/Y values, move to NOM and press CPT

You want to receive $5,000 per month in retirement. If you can earn .75% per month and you expect to need the income for 25 years, how much do you need to have in your account at retirement?

300 N .75 I/Y 5000 PMT 0 FV PV = -595808.11 (neg because we deposit)

Bob has been investing $3,750 in stock at the end of every year for the past 14 years. If the account is currently worth $105,700, what was his annual return on this investment?

3750 PMT 14 N 105,700 FV 0 PV CPT I/Y= 10.10

Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 = .66667% per month). Ifyoutakea4yearloan, what is your monthly payment?

4(12)=48 N 0.66667 I/Y 20,000 PV 0 FV PMT= -488.26 =PMT(0.006667,48,20000,0)

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

40 N 7.5 I/Y 0 PV -2000 PMT CPT FV = 454513.04 =FV(0.075,40,-2000,0)

Although you may know William Shakespeare from his classic literature, what is not well-known is that he was an astute investor. In 1604, when he was 40 and writing King Lear, Shakespeare grew worried about his eventual retirement. Afraid that he would become like King Lear in his retirement and beg hospitality from his children, he purchased grain "tithes," or shares in farm output, for 440 pounds. The tithes paid him 60 pounds per year for 31 years. Even though he died at the age of 52, his children received the remaining payments. What interest rate did the Bard of Avon receive on this investment?

440=60((1-(1/1+r)^31)/r)= 13.36 ????

Annuity example 1 : You can afford $632 per month. Going rate = 1%/month for 48 months. How much can you borrow? You borrow money TODAY so you need to compute the present value.

48 N 1 I/Y 632 PMT 0 FV CPT PV = 23,999.54 ($24,000) =PV(0.01,48,-632,0)

You want to receive $5,000 per month for the next 5 years. What monthly rate would you need to earn if you only have $200,000 to deposit?

5 x 12 = 60 N 5000PMT -200000 PV 0 FV I/Y= 1.44 % per month

Amortized Loan with Fixed Payment: Each payment covers the interest expense plus reduces principal Consider a 4-year loan with annual payments. The interest rate is 8% and the principal amount is $5000.- What is the annual payment?

5,000 = PMT[1 - 1 / 1.08^4] / .08 PMT = 1,509.60 • =PMT(0.08,4,5000,0) = 1509.60 • 4 N; 8 I/Y; 5000 PV, 0 FV, CPT PMT = 1509.60

Ken just purchased new furniture for his house at a cost of $15,400. The loan calls for weekly payments for the next 5 years at an annual interest rate of 10.39 percent. How much are his weekly payments?

52 weeks in a year pv= -15400 i/y = 10.39/52 = .20 n= 52x5 = 260 cpt pmt = 76

You want to receive $5,000 per month for the next 5 years. How much would you need to deposit today if you can earn .75% per month?

5x12= 60 N .75 I/Y 5000 PMT 0 FV N= -240866.87

Which one of the following has the highest effective annual rate?

6 percent compounded daily or monthly

Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the monthly interest rate?

60 N 10000 PV -207.58 PMT 0 FV CPT I/Y = 0.75%. per mont =RATE(60,-207.58,10000,0)

Marko, Inc., is considering the purchase of ABC Co. Marko believes that ABC Co. can generate cash flows of $6,100, $11,100, and $17,300 over the next three years, respectively. After that time, they feel the business will be worthless. Marko has determined that a rate of return of 15 percent is applicable to this potential purchase. What is Marko willing to pay today to buy ABC Co.?

=6100/1.15+11100/1.15^2+17300/1.15^3 =$25072.57(Approx).

One of your customers is delinquent on his accounts payable balance. You've mutually agreed to a repayment schedule of $400 per month. You will charge 1.4 percent per month interest on the overdue balance. If the current balance is $17,320, how long will it take for the account to be paid off?

=NPER(rate,pmt,pv,FV) =NPER(1.4%,-400,17320,0) =67.03 months

Which of the following is a perpetuity?

A constant stream of cash flows forever

True/ false: To find the future value of multiple cash flows, calculate the future value of each cash flow first and then sum them.

true

Which of the following is the simplest form of loan?

A pure discount loan

You expect to receive $30,000 at graduation in two years. You plan on investing it at 7 percent until you have $125,000. How long will you wait from now?

A=P(1+r/100)^n PV= -30,000 FV= 125,000 I/Y = 7 PMT = 0 CPT N=21.09 years(Approx). Hence time to wait from now=21.09+2 years =23.09 years(Approx).

What is the definition of an APR?

ANNUAL PERCENTAGE RATE: LEGALY QUOTED RATE

Computing APRs from EARs

APR = m {(1 + EAR)^1/m-1} M = number of compounding periods per year

APR - ExamplE: Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay?

APR=12[(1+.12)1/12 −1]=.1138655 or11.39% ICONV: EFF = 12 C/Y = 12 NOM = 11.3866

You want to have $1 million to use for retirement in 35 years. If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made today?

Annuity due: 2nd BGN 35 x 12 = 420 1 I/Y 0 pV 1,000,000 FV PMT= -153.96

You want to have $1 million to use for retirement in 35 years. If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made in one month?

Annuity payment stream 35 x 12 = 420 N 1 I/Y 0 pV 1,000,000 FV PMT= -155.50

To calculate the future value of $100 invested for t years at r interest rate, you enter the present value in your calculator as a negative number. Why?

Because the $100 is an outflow from you which should be negative.

Which of the following processes can be used to calculate the future value of multiple cash flows?

Calculate the future value of each cash flow first and then sum them Compound the accumulated balance forward one year at a time

Perpetuity: Suppose that Fellini Co. wants to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend is Fellini going to have to offer if the preferred stock is going to sell?

Current required return: - 40 = 1 / r - r = .025 or 2.5% per quarter Dividend for new preferred: - 100=PMT/.025 - PMT = 2.50 per quarter

Lucas expects to receive a sales bonus of $7,500 one year from now. The process of determining how much that bonus is worth today is called:

DISCOUNTING

Second account::

EAR=(1+.053/2)2 -1. =5.37% ICONV: NOM=5.3; C/Y=2 EFF=5.3702

Which of the following is the appropriate spreadsheet function to convert a quoted rate of 12% compounded quarterly to an EAR?

EFFECT(0.12,4)

When calculating annuity present values using a financial calculator, the ________ , amount is left blank. (Enter the abbreviation only.)

FV

Future Values: General Formula

FV = PV(1 + r)^t FV = future value PV = present value r = period interest rate, expressed as a decimal t = number of periods • Future value interest factor = (1 + r)^t Note: "y^x" key on your calculator

You have just made your first $5,000 contribution to your individual retirement account. Assume you earn a 10.2 percent rate of return and make no additional contributions. a.What will your account be worth when you retire in 45 years? b.What if you wait 10 years before contributing?

FV = PV(1 + r)t a. Account value if you start now = PV= $5,000 I/Y = 10.2 N= 45 PMT = 0 CPT FV= $395,495.69 b. Account value if you wait 10 years = PV= $5,000 I/Y = 10.2 N= 35 PMT = 0 CPT FV= $149,735.86

"interest on Interest"

FV/CI - FV/SI

More frequent compounding leads to:

Higher EARs

Future Value: Important Relationship I

For a given interest rate: - The longer the time period, - The higher the future value FV = PV(1 + r)t - For a given r, as t increases, FV increases

Present Value: Important Relationship I

For a given interest rate: - The longer the time period, - The lower the present value PV = FV / (1+r)^t - For a given r, as t increases, PV decreases

Present Value Important Relationship II

For a given time period: - The higher the interest rate, - The smaller the present value PV = FV / (1+r)t For a given t, as r increases, PV decreases

Number of Periods - Example

Formula Solution: - FV/PV = 20,000/15,000 = 1.333 - ln(1.333) = 0.2877 - ln(1.10) = 0.0953 - t = 0.2877/0.0953 = 3.0189

You have just started a new job and plan to save $4,850 per year for 43 years until you retire. You will make your first deposit in one year. How much will you have when you retire if you earn an annual interest rate of 10.03 percent?

I/Y 10.03 N 43 PMT -4850 PV 0 FV= 2898906.51

You have just won the lottery and will receive a lump sum payment of $22.85 million after taxes. Instead of immediately spending your money, you plan to deposit all of the money into an account that will earn 5.05 percent. If you make equal annual withdrawals for the next 25 years, how much can you withdraw each year starting exactly one year from now?

I/Y 5.05 N 25 PV 22850000 FV 0 P 1629397.85

You are to make monthly deposits of $500 into a retirement account that earns an APR of 9.5 percent compounded monthly (9.5/12 % per month). If your first deposit will be made one month from now, how large will your retirement account be in 35 years?

I/Y = 9.5% / 12 = 0.79167% N = 35 * 12 = 420 Retirement account value = Annuity * [(1 + r)n - 1] / r Retirement account value = 500 * [(1 + 0.0079167)420 - 1] / 0.0079167 Retirement account value = 500 * 3,339.497034 Retirement account value = $1,669,731.48 ?????

You plan to save $150 per month starting today for the next 30 years "just to start the month off right." You feel that you can earn an interest rate of 9.1 percent compounded monthly. How much will there be in the account 30 years from today?

INT: 9.1 # YEARS: 30 # COMPONDED 12 R= 9.1/ 12= .76 N= 30X12= 360 PMT= 150 (1+R)XPX[(1+R)^N-1]/R (1+.76)X150X((1+.76)^360-1)/.76 = 264(1.42)/.76 ????????????

Kendall is investing $3,333 today at 3 percent annual interest for three years. Which one of the following will increase the future value of that amount?

Increasing the interest rate

Effects of Compounding: Simple interest

Interest earned only on the original principal

You take out a loan for $100,000 at an annual interest rate of 5.9% that is to be paid with three equal annual payments of $37,341.79. How much principal will be paid in the second year?

Interest in year 1 = 100,000 * 0.059 = $5900 Principle paid in 1st year = 37,341.79 - 5900 = $31441.79 Outstanding loan in year 2 = 100,000 - 31441.79 = $68558.21 Interest in year 2 = 68558.21 * 0.059 = $4044.9344 Principle paid in 2nd year = 37,341.79 - 4044.9344= $33296.86

Amortized Loan with Fixed Payment - Example

LOOK AT TABLE 5000 LOAN @8 = 5000 X .08 = 400 INTEREST PAID PAYED 1509.60 = -400 + 1509.60 = 1109.6 PRINCIPLE PAID -1109.6+5000 = 3890.40 ENDING BALANCE (BEG BAL FOR NEXT YEAR)

Which of the following could not be evaluated as annuities or annuities due?

Monthly electric bills Tips to a waiter

You are solving a present value equation using a financial calculator and are given the number of years for compounding. This should be entered as the _____ value on the financial calculator.

N

What is the present value of $2,625 per year, at a discount rate of 6.9 percent, if the first payment is received six years from now and the last payment is received 20 years from now?

N 15 I/Y 6.9 PMT -2625 FV 0 CPT PV = $24,060.02 PV = $24,060.02/(1 + .069)^5PV = $17,234.85

IF IT SAYS PAY IN 17 YEARS WHEN HE IS ONE NOW

N WOULD BE 16

To calculate FV: 10% 5 years PV=$100

N: 5 I/Y: 10 PV: -100 PMT: 0 CPT FV: 161.05

Present Value with Daily Compounding: 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?

ORDINARY ANNUITY 3 X 365 = 1095 N 5.5/365 = .02 I/Y 0 PMT 15,000 FV CPT PV= -12718.56

Future Values with Monthly Compounding: 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?

ORDINARY ANNUITY 35 X 12 = 420 N 9 / 12 = .75 I/Y 0 PV -50 PMT CPT FV= 147089.22

Perpetuity: You want to fund a College of Business scholarship that will pay $1,000 each year. If such funds earn 8.5% annual interest, how much do you need to deposit into the fund?

Ordinary Annuity: end of period PMT PVP: PMT/r (interest as decimal) PMT= rX PVP r = PMT/PVP PMT: 1000 r= 8.6 PVP: 1000/.085= 11,764.71 0 1 2 3 --------------------- ^ 1000 1000 1000

Which rate should you use to compare alternative investments or loans?

START ANNUAL PERCENTAGE RATE BUT USE PERIODIC RATE

Prepare an amortization schedule for a three-year loan of $57,000. The interest rate is 8 percent per year, and the loan calls for equal annual payments. How much total interest is paid over the life of the loan?

P= r(PV) / 1- (1+R)^-N P = (.08) X 57,000/ 1-((1/1.08)^3)) = 22117.91 = TP INT: a x 8% = 57,000 X .08 = 4560 PRINCIPAL: TP - INT = 22117.91 - 4560 = 17557.91 EB: BB - PRINCIPAL= 39,442.09 LOOK AT TABLE

You plan to save $7,400 per year for the next 12 years. After the last deposit, you will keep the money in the account for 3 more years. The account will earn an interest rate of 7.7 percent. How much will there be in the account 15 years from today?

PMT = 7400 N = 12 PV = 0 I/Y = 7.7 Compute FV = 137,957.307 PV = 137,957.307 N = 3 PMT = 0 I/Y = 7.7 Compute FV = 172,342.2743

Annuity - Sweepstakes Example Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?

PMT: 333,333.33 N: 30 I/Y: 5 FV= 0 PV= -5124150.24 PV = $333,333.33 [1 - (1/1.05)^30)] / .05 = $5,124,150.29

You know the payment amount for a loan and you want to know how much was borrowed. - Do you compute a present value or a future value?

PV

You have $13,500 and will invest the money at an interest rate of .38 percent per month until the account is worth $20,200. How many years do you have to wait until you reach your target account value?

PV 13500 I/Y .38 FV 20200 0 PMT CPT I/Y = 106.25/12=8.85

Given an interest rate of 6.35 percent per year, what is the value at Year 7 of a perpetual stream of $7,000 payments that begin at Year 20?

PV = C/r PV = $7,000/.0635 = $110,236.22 PV = FV/(1 + r)^t PV = $110,236.22/(1 + .0635)^12PV = $52,659.22

Fox Co. has identified an investment project with the following cash flows. Year Cash Flow 1 $570 2 430 3 840 4 1,230 a.If the discount rate is 10 percent, what is the present value of these cash flows? b.What is the present value at 18 percent? c.What is the present value at 24 percent?

PV = FV / (1 + r)t - PV@10% = $570 / 1.10 + $430 / 1.102 + $840 / 1.103 + $1,230 / 1.104 = $2,344.76 - PV@18% = $570 / 1.18 + $430 / 1.182 + $840 / 1.183 + $1,230 / 1.184 = $1,937.54 - PV@24% = $570 / 1.24 + $430 / 1.242 + $840 / 1.243 + $1,230 / 1.244 = $1,700.16

You have just received an offer in the mail from Friendly Loans. The company is offering to loan you $3,250 with low monthly payments of $75 per month. If the interest rate on the loan is an APR of 14.9 percent compounded monthly, how long will it take for you to pay off the loan?

PV= -3250 fv= 0 pmt= 75 i/y = 14.9/12= 1.24 cpt n = 62.54

You are considering preferred stock that pays a quarterly dividend of $1.50. If your desired return is 3% per quarter, how much would you be willing to pay?

PVP: PMT/r = 1.50/.03= $50 for investment

Which of the following are ways to amortize a loan?

Pay the interest each period plus some fixed amount of the principal. Pay principal and interest every period in a fixed payment.

Which of the following are real-world examples of annuities?

Pensions Mortgages Leases

What is the effective annual rate?

REAL RATE OF RETURN, TP COMPARE INVESTMENT ALTERNATIVES

Set Annuity Time Value Parameters

Set END for an ordinary annuity or BGN for an annuity due - Press 2nd BGN (above PMT) - This is a toggle switch. The default is END. - To change to BEGIN, press 2nd SET (above ENTER) to go back and forth.

Ordinary annuity versus Annuity due

Switch your calculator between the two types (next slide) - If you see "BGN" or "Begin" in the display of your calculator, you have it set for an annuity due - Most problems are ordinary annuities

True/ false: All else equal, an increase in the discount rate decreases the present value and increase the future value of an annuity

TRUE

Which one of the following statements is correct?

The EAR, rather than the APR, should be used to compare both investment and loan options.

What is the primary difference between time value of money data entries in your calculator and in a spreadsheet function?

The interest rate in your calculator is entered as a whole number while in the spreadsheet function it is entered as a decimal.

You are comparing three investments, all of which pay $100 a month and have an interest rate of 8 percent. One is ordinary annuity, one is an annuity due, and the third investment is a perpetuity. Which one of the following statements is correct given these three investment options?

The present value of the perpetuity has to be higher than the present value of either the ordinaryannuity or the annuity due.

Time Line of Cash Flows

Tick marks at ends of periods Time 0 is today; Time 1 is the end of Period 1 +CF = Cash INFLOW -CF = Cash OUTFLOW PMT = Constant CF

Which one of the following features distinguishes an ordinary annuity from an annuity due?

Timing of the annuity payment

In the Excel setup of a loan amortization problem, which of the following occurs?

To find the principal payment each month, you subtract the dollar interest payment from the fixed payment. The payment is found with = PMT(rate, nper, -pv, fv).

Pure Discount Loans:

Treasury bills are excellent examples of pure discount loans. - Principal amount is repaid at some future date - No periodic interest payments

Suppose you can earn 1% per month on $1 invested today.

What is the APR? 1 (12) = 12% How much are you effectively earning? EAR= 1(1.01)^12= 1.1268-1 = 12.68 % NOM = 12 C/Y =12 EFF =12.68

Suppose you put it in another account, earning 3% per quarter.

What is the APR? 3 (4) = 12% How much are you effectively earning? EAR= 1(1.03)^4= 12.55 % NOM = 12 C/Y =4 EFF =12.55 %

The present value of a lump-sum future amount:

increases as the interest rate decreases.

Perpetuity

infinite series of equal payments.

The most common way to repay a loan is to pay ____.

a single fixed payment every period

Fox Co. has identified an investment project with the following cash flows. Year Cash Flow 1 $570 2 430 3 840 4 1,230 a.If the discount rate is 10 percent, what is the present value of these cash flows? b.What is the present value at 18 percent? c.What is the present value at 24 percent?

a) 1/1.1 =.9091 PV= 570 x .9091 = 518.18 .9091/1.1 = .8264 PV= 430 x .8264 = 355.38 .8264/ 1.1 = .7513 PV= 840 x 7513 = 631.07 .7513/1.1= .6830 PV= 1230 x .6830 = 840.09 PV= 2344.72 b) 1/1.18 =.8475 PV= 570 x .8475 = 483.05 .8475 /1.18 = .7182 PV= 430 x .7182 = 308.82 .7182/ 1.18 = .6086 PV= 840 x .6086 = 511.25 .6086 /1.18= .5158 PV= 1230 x .5158 = 634.42 PV= 1937.54 C) 1/1.24 =.8065 PV= 570 x .8065 = 459.68 .8065 /1.24 = .6504 PV= 430 x .6504 = 279.66 .6504/ 1.24 = .5245 PV= 840 x .5245 = 440.57 .5245 /1.24= .4230 PV= 1230 x .4230 = 520.26 PV= 1700.16

Suppose you just bought a 10-year annuity of $5,200 per year at the current interest rate of 10 percent per year. a.What is the value of the investment at the current interest rate of 10 percent? b.What happens to the value of your investment if interest rates suddenly drop to 5 percent? c.What happens to the value of your investment if interest rates suddenly rise to 15 percent?

a) N = 10, PMT=5,200, I/Y=10, FV=0, Compute PV pv= 31951.75 b) Same as a.) just change I/Y to 5 pv= 40153.02 c) Same as a.) just change I/Y to 15 pv= 26097.60

What type of payment stream do the following have: a) Monthly payments on a car loan, paid at the end of each month b) Five years of quarterly deposits into a savings account starting today c) $1000 per month forever d) Weekly grocery bill

a) Ordinary Annuity b) Regular payment Annuity is due C) Perpetuity d) uneven cash flow stream

Investment X offers to pay you $3,100 per year for 9 years, whereas Investment Y offers to pay you $4,800 per year for 5 years. a.If the discount rate is 6 percent, what is the present value of these cash flows? b.If the discount rate is 22 percent, what is the present value of these cash flows?

a. Investment X CF: COF= 0 CO1= 3100 FO1= 9 NVP=6 CPT= 21085.25 Investment Y CF: COF= 0 CO1= 4800 FO1= 5 NVP=6 CPT= 20219.35 B. Investment X CF: COF= 0 CO1= 3100 FO1= 9 NVP=22 CPT= $11,737.48. Investment Y CF: COF= 0 CO1= 4800 FO1= 5 NVP=22 CPT= $13,745.47.

The interest rate charged per period multiplied by the number of periods per year is equal to ________ on a loan.

annual percentage rate

An annuity with payments beginning immediately rather than at the end of the period is called an _________.

annuity due

An annuity due is a series of payments that are made ____.

at the beginning of each period

Which one of the following qualifies as an annuity payment?

auto loan payment

The effective annual rate (EAR) takes into account the ______ of interest that occurs within a year.

compounding

One step in calculating an EAR is to _____________ the quoted rate by the number of times that the interest is compounded.

divide

Assume interest is compounded monthly. The ______ annual rate will express this rate as though it were compounded annually.

effective

The _______________ annual rate is the interest rate expressed as if it were compounded once per year.

effective

In almost all multiple cash flow calculations, it is implicitly assumed that the cash flows occur at the _____ of each period.

end

True/ false: The payment for an annuity can be calculated using the annuity present value, the present value factor, and the interest rate.

false: The payment for an annuity can be calculated using the annuity present value, the present value factor, and the discount rate.

True/ false: The effective annual rate is the interest rate expressed in terms of the interest payment made each period.

false: The stated interest rate is the interest rate expressed in terms of the interest payment made each period.

When entering variables in a spreadsheet function (or in a financial calculator) the "sign convention" can be critical to achieving a correct answer. The sign convention says that outflows are negative values; inflows are positive values. For which variables is this a consideration?

future value payment present value

If the interest rate is greater than zero, the value of an annuity due is always ______ an ordinary annuity.

greater than

Travis borrowed $10,000 four years ago at an annual interest rate of 7 percent. The loan term is six years. Since he borrowed the money, Travis has been making annual payments of $700 to the bank. Which type of loan does he have?

interest only

When finding the present or future value of an annuity using a financial calculator, the ______ ______ should be entered as a percentage.

interest rate

An ordinary annuity consists of a(n) ________ stream of cash flows for a fixed period of time.

level

You have just purchased a new warehouse. To finance the purchase, you've arranged for a 30-year mortgage loan for 80 percent of the $3,500,000 purchase price. The monthly payment on this loan will be $15,100. a.What is the APR on this loan? b.What is the EAR on this loan?

loan amount=80%*3,500,000 =2800000 Monthly rate=RATE(nper,pmt,pv,fv) n= 30*12= 360 pmt= -15,100 pv= 2800000 fv= 0 i/y=0.4200% What is the APR on this loan=0.42%*12=5.04% What is the EAR on this loan=(1+0.42%)^12-1=5.16%

A simple way to amortize a loan is to have the borrower pay the interest each period plus some fixed amount. This approach is common with ___________ -term business loans.

medium

The annuity present value factor equals one ______ the present value factor all divided by the discount rate.

minus

An investment will pay you $100,000 in 9 years. Assume the appropriate discount rate is 5.5 percent compounded daily. What is the present value?

n = 9 * 365 days = 3285 days r = 5.5%/365 = 0.0151% (daily) 100,000=PVX(1+0.000151)^3285 100,000 = PV * 1.640437 PV = $60,959.36

You have arranged for a loan on your new car that will require the first payment today. The loan is for $28,500, and the monthly payments are $525. If the loan will be paid off over the next 60 months, what is the APR of the loan?

n= 60 pv= -28500 pmt = 525 fv= 0 i/y = 0.3342 APR= .33x12= 4.01 4.01 x (1.0334) = 4.15

Using an Excel spreadsheet to solve for the payment in an amortized loan, enter the number of periods as the ________value.

nper

Ordinary Annuity vs. Annuity Due

ord: 0 1 2 3 --------------------- # x x x ann: 0 1 2 3 --------------------- x x x # PVAD= PVA (1+r) FVAD= FVA (1+r)

The present value of an annuity due is equal to the present value of a(an) ______ annuity multiplied by (1+ r).

ordinary

C/r is the formula for the present value of a(n) ____.

perpetuity

The present value formula for a(n) ______ is PV = C/r, where C is the constant and regularly timed cash flow to infinity, and r is the interest rate.

perpetuity

The formula for the ______ value interest factor of an annuity is: [1- 1/(1+r)τ]/r.

present

The original amount of a loan is termed the loan ___________.

principal

If you borrow $15,000 today at 5% annual interest to be repaid in one year as a lump sum, this is termed a _______________ .

pure discount loan

Compounding during the year can lead to a difference between the ___________ rate and the effective rate.

quoted

With typical interest-only loans, the entire principal is:

repaid at some point in the future

The general formula for ______ is (1+quoted rate/m)m - 1.

the EAR

_________ present value can be found using the perpetual cash flow and the discount rate.

the perpetuity

One year ago, the Jenkins Family Fun Center deposited $5,300 into an investment account for the purpose of buying new equipment four years from today. Today, they are adding another $7,100 to this account. They plan on making a final deposit of $9,300 to the account next year. How much will be available when they are ready to buy the equipment, assuming they earn a rate of return of 8 percent?

total avilable return= $ 5300*(1+0.08)5+$ 7100(1+0.08)4+$ 9300(1+0.08)3 = $ 5300(1.4693)+$ 7100*(1.3605)+$ 9300(1.2597) =$ 9659.55+$ 7787.29+$ 11715.21 =$ 29162.23

Because of __________ and _________, interest rates are often quoted in many different ways.

tradition; legislation

True/ false: A simple way to amortize a loan is to have the borrower pay the interest each period plus a fixed amount.

true

Future Value: Important Relationship II

For a given time period: - The higher the interest rate, - The larger the future value FV = PV(1 + r)t - For a given t, as r increases, FV increases

First City Bank pays 7 percent simple interest on its savings account balances, whereas Second City Bank pays 7 percent interest compounded annually. If you made a deposit of $7,900 in each bank, how much more money would you earn from your Second City Bank account at the end of 10 years?

first city bank: - SI= p x r x t - SI= 7900 x 7% x 10 = 5530 - value @ the end of 10 years = 5530 +7900= 13430 second city bank: - FV=PV(1+i)^r - FV= 7900 (1+7%)^10= 15540.50 Diff: 2110.50

Given an investment amount and a set rate of interest, the _____ the time horizon the _____ the future value.

longer; greater

With discounting, the resulting value is called the _____ value; while with compounding the result is called the ____ value.

present; future

interest rate risk

say 8% annual compound int will perfectly compensate for three risk (10x1.08x1.08x1.08=12.60)

with ____________ interest, the interest is not reinvested.

simple

interest rate risk: uncertainty risk

there is some doubt about whether the $10 promised after 3 years will actually be received

The real world has moved away from using ____________ for calculating future and present values.

time value of money tables

True/ false: Given the PV, FV, and life of the investment, you can determine the discount rate.

true

True/ false: The formula for a present value factor is 1/(1+r)t.

true

True/ false: The process of leaving your money and any accumulated interest in an investment for more than one period is called compounding.

true

True/ false: If you invest for two periods at an interest rate of r, then your money will grow to (1 + r) per dollar invested.

false

True/false: Future value refers to the amount of money an investment is worth today.

false

Suppose you have $500 to invest and you believe that you can earn 8% per year over the next 15 years

500(1.08)^15 = 1586.08 N: 15 I/Y: 8 PV: -500 PMT: 0 = 1586.08 - INTEREST ON INTEREST 1586.08-1100 = 486.08

Decisions, Decisions: Yourbrokercallsyouandtells you that he has this great investment opportunity. Ifyouinvest$100today,youwill receive $40 in one year and $75 in two years. Ifyourequirea15%returnon investments of this risk, should you take the investment?

0 1 2 ---------------------- -100 40 75 PV= 40/1.15 + 75/1.15^2 = 91.49 cf 100 we dont want to invest

Future Values - Example 3: Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today?

Formula Solution: FV = PV(1+r)t = 10(1.055)^200 =10(44718.984) =447,189.84 CAlC: N: 200 I/Y: 5.5 PV:10 PMT: 0 CPT FV: -447,189.84

Present Values: Example 2 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?

Formula Solution: PV =FV(1+r)-t =150,000(1.08)-17 =150,000/(1.08)17 =40,540.34 N 17 I/Y 8 PMT 0 FV 150,000 CPT PV: -40,540.34

Present Values: Example 3 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?

Formula Solution: PV =FV(1+r)-t =19,671.51(1.07)-10 = -10,000

Present Value Important Relationship II EXAMPLE What is the present value of $500 received in 5 years if the interest rate is 10%? 15%?

Rate = 10% Calculator Solution: 5N10 I/Y0 PMT 500 FVCPT PV = -310.46 Rate = 15% Calculator Solution: 5N15 I/Y0 PMT 500 FVCPT PV = -248.59

What's the PV of $100 due in 3 Years if r = 10%?

- Finding PVs is discounting, and it's the reverse of compounding. 0 10% 1 2 3 ------------------------------------------------------- PV=? 100 Formula: PV = FV(1+r)-t = 100(1.10)-3 = $75.13 Calculator: 3 N 10 I/Y 0 PMT 100 FV CPT PV = -75.13

Discount Rate - Example 1 You are looking at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest?

- Formula: r = (1200 / 1000)^1/5 - 1 = .03714 = 3.714% - Calculator - the sign convention matters!!! 5N -1000 PV (you pay $1,000 today) 0 PMT 1200 FV (you receive $1,200 in 5 years) CPT I/Y = 3.714%

interest rate risk: risk of inflation

the $10 received after 3 years would have less purchasing power over goods and services than the $10 today

Discount Rate

-To find the implied interest rate, rearrange the basic PV equation and solve for r: FV = PV(1 + r)^t r=(FV/PV)^1/t -1 - If using formulas with a calculator, make use of both the yx and the 1/x keys

Which of the following are the primary as well as easy ways used to perform financial calculations today?

Spreadsheet functions Financial calculator

Finding the Number of Periods

Start with basic equation and solve for t: FV = PV(1 + r)^t T= IN (FV/PV)/IN (1+R) CALC: CPT N

Interest rate (r)

- Discount rate - Cost of capital - Opportunity cost of capital - Required return Terminology depends on usage

Using a time value of money table, what is the future value interest factor for 10 percent for 2 years?

(1+.10)^2= 1.21

If you invest for a single period at an interest rate of r, your money will grow to ______ per dollar invested.

(1+r)

Interest rate and time period must match!

- Annual periods ⇒ annual rate - Monthly periods ⇒ monthly rate

The Sign Convention

- Cash inflows are positive - Cash outflows are negative

Saving For Retirement: You are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years. - How much would you be willing to invest today if you desire an interest rate of 12%?

39 40 41 42 43 44 ----------------------------------------- 25k 25k 25k 25k 25k CO1= 0 = Fo= 39, CO2 = 25000 = FO= 5 NPV= 12 CPT = 1084.71

Future Value: General Growth Formula: 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?

Formula Solution: FV = PV(1+r)t = 3(1.15)^5= 6.0341 calc: N: 5 I/Y: 15 PV: 3 PMT: 0 CPT FV: -6.0341

Future Values - Example 2: Suppose you invest the $100 from the previous example for 5 years. How much would you have?

Formula Solution: FV =PV(1+r)^t =100(1.10)^5 =100(1.6105)= 161.05

Present Value: Example 1: 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?

Formula Solution: PV =FV(1+r)-t =10,000(1.07)-1 =10,000/1.07 =9,345.79 N: 1 I/Y 7 PMT 0 FV 10000 CPT PV: -9,345.79

a.At 5.3 percent interest, how long does it take to double your money? b.At 5.3 percent interest, how long does it take to quadruple your money?

a) Time to double the money: =NPER(5.3%,0,-100,200) =13.42 years b) Time to quadruple the money: =NPER(5.3%,0,-100,400)=26.84

Future value is the ____________ value of an investment at some time in the future.

cash

The idea behind ______ is that interest is earned on interest.

compounding

The process of accumulating interest in an investment over time to earn more interest is called

compounding

Calculating the present value of a future cash flow to determine its worth today is commonly called ___________ valuation.

discounted cash flow (DCF)

Which of the following is the correct Excel function to calculate the present value of $300 due in 5 years at a discount rate of 10%?

=PV(0.10,5,0,-300)

Beatrice invests $1,440 in an account that pays 3 percent simple interest. How much more could she have earned over a 4-year period if the interest had been compounded annually?

Interest earned in simple interest = 1440 *0.03*4 = 172.80 Interest earned in compounded interest = 1440( 1.03^4-1)= 180.73 Extra interest she could have earned = 180.73 - 172.80 = 7.93

If FV= PV x (1+r) is the single period formula for future value, which of the following is the single period present value formula?

PV = FV/(1+r)

Present Value Equation

PV = FV/(1+r)^t "Discounting" = finding the present value of one or more future amounts.

Imprudential, Inc., has an unfunded pension liability of $645 million that must be paid in 25 years. To assess the value of the firm's stock, financial analysts want to discount this liability back to the present. If the relevant discount rate is 5.5 percent, what is the present value of this liability?

PV = FVn / (1 + r )n PV = FV= $ 645,000,000 I/Y =5.5 N= 25 PMT=0 PV= 169140739.02

Perpetuity formula

PV = PMT / r

Annuities Formula

PV = PMT [ (1- (1/(1+r)^t))/r} FV = PMT { ((1+r)^t-1) / r }

An investment offers to triple your money in 24 months (don't believe it). What rate per three months are you being offered?

PV= -1 FV= 3 N= 24/3 = 8 PMT = 0 CPT I/Y = 14.72%

You made an investment of $13,000 into an account that paid you an annual interest rate of 3.6 percent for the first 6 years and 8 percent for the next 7 years. What was your annual rate of return over the entire 13 years?

Part I: PV = -13,000 I/Y=3.6 N=6 PMT=0 Compute FV = 16,073.1828 Part II: PV = -16,073.1828 I/Y = 8 N = 7 PMT = 0 Compute FV = 27,546.61 Part III: PV: 13,000 N = 13 PMT = 0 FV = 27,546.61 Compute I/Y = 5.95

Effects of Compounding: Compound interest

- Interest earned on principal and on interest received - "Interest on interest" - interest earned on reinvestment of previous interest payments

Future Value: Multiple Cash Flows 5: If you deposit $100 in one year, $200 in two years and $300 in three years. How much will you have in three years at 7 percent interest? How much in five years if you don't add additional amounts?

0 1 2 3 4 5 ------------------------------------------------- 100 200 300 1. FV3= 100(1.07)^2+200(1.07)+300 = 628.49 2. FV5= 100(1.07)^4+200(1.07)^3+300(1.07)^2 = 719.56 or FV5= FV3 (1.07)^2= 628.49 (1.07)^2

Future Value (FV)

- The amount an investment is worth after one or more periods. - "Later" money on a time line

Present Value (PV)

- The current value of future cash flows discounted at the appropriate discount rate - "Earlier" (often at t=0) on a time line

Annuity

- finite series of equal payments that occur at regular intervals - If the first payment occurs at the end of the period, it is called an ordinary annuity - If the first payment occurs at the beginning of the period, it is called an annuity due

Time value of money tables are not as common as they once were because:

- it is easier to use inexpensive financial calculators instead - they are available for only a relatively small number of interest rates.

Rick deposited $2,950 into an account 6 years ago for an emergency fund. Today, that account is worth $3,970. What annual rate of return did Rick earn on this account assuming no other deposits and no withdrawals?

-2950 PV 6 N 3970 FV 0 PMT CPT I/Y= 5.07%

Which of the following methods are used to calculate present value?

A financial calculator An algebraic formula A time value of money table

Future Value: Multiple Cash Flows Example 4: You think you will be able to deposit $4,000 at the end of each of the next three years in a bank account paying 8 percent interest. You currently have $7,000 in the account. How much will you have in 3 years? How much in 4 years?

0 1 2 3 -------------------------------- 7000. 4000. 4000. 4000 1. FV2: 7000(1.08)^3 +4000(1.08)^2+4000(1.08)+4000 = 8817.98+4665.6+4320+4000 =21803.58 @ the end of 3 years 2. FV4= 7000(1.08)^4+4000(1.08)^3+4000(1.08)^@ +4000(1.08) = 23547.87 @ year 4 0r FV4=FV3(1+r)= 21803.58(1.08) = 23547.86

Present Value: Multiple Cash Flows 2: You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3,000 in three years. If you want to earn 10% on your money, how much would you be willing to pay?

0 1 2 3 ------------------------------ X 1000 2000 3000 PV: 1000/(1.10)+ 2000/(1.10)^2 +3000/(1.10)^3 = 4815.93

Present Value: Multiple Cash Flows 1: You are offered an investment that will pay • $200 in year 1, • $400 the next year, • $600 the following year, and • $800 at the end of the 4th year. • You can earn 12% on similar investments. • What is the most you should pay for this one?

0 1 2 3 4 ------------------------------------ X 200 400 600 800 PV= 200/1.12 + 400/(1.12)^2+600/(1.12)^3+800/(1.12)^4 = 1432.93 (calculator: press 2nd clear, enter all Co, clear, NPV 12, arrow down, cpt)

Future Value: Multiple Cash Flows 6: Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years? How much will you have in 5 years if you make no further deposits?

0 1 2 3 4 5 ----------------------------------------- 500 600 X X 1. FV2= 500(1.09)^2 + 600(1.09) = 1248.05 2. FV5= FV2 (1.09) ^3 = 1248.05 (1.09) ^3 = 1616.26

Suppose you are looking at the following possible cash flows: - Year 1 CF = $100;- Years 2 and 3 CFs = $200;- Years 4 and 5 CFs = $300.- The required discount rate is 7% • What is the value of the CFs at year 5? • What is the value of the CFs today?

0 1 2 3 4 5 ------------------------------------------------- 100 200 200. 300 300 FV5= 100(1.07)^4 +200(1.07) ^3 +200(1.07) ^2 +300(1.07) +300 = 1226.07 or FV5= PV (1+r)^5 = 1226.07 PV= 100/(1.07) +200/(1.07) ^2 + 200/(1.07) ^3 + 300/(1.07)^4 +300/(1.07) ^5= 874.17

Effects of Compounding example

0 10% 1. 2 -I-------------------I------------------------- 100 100 110 10/110 and 11/121 FV1: 100 (1.10) = 110 FV2: 110 ( 1.10) = 121 = 100 (1.10)^2 Compound int: FV = PV(1 +r)^t Simple interest: FV = PV(1+rt) "interest on Interest": FV/CI - FV/SI

Present Value: Important Relationship I EXAMPLE What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10%

0 10% 5 10 ------------------------------ 310.46 PV? 500 -192.77 PV? 500 5 YR: PV = 500/(1.10)5 = -310.46. (1.10)5 =1.6105 10 yrs: PV = 500/(1.10)10= -192.77. (1.10)10 = 2.5937

For a given time period (t) and interest rate (r), the present value factor is _______ the future value factor. (Select all that apply.)

1 divided by the reciprocal of

interest rate risk: risk free rate of return

1: risk free rate of return - $10 invested today would grow to more than $10 after 3 years

Retirement Investment Advisors, Inc., has just offered you an annual interest rate of 4.6 percent until you retire in 35 years. You believe that interest rates will increase over the next year and you would be offered 5.2 percent per year one year from today. If you plan to deposit $14,000 into the account either this year or next year, how much more will you have when you retire if you wait one year to make your deposit?

35: 4.6 I/Y 35 N -14000 PV O PMT CPT FV= $67,567.37 34: 5.2 I/Y 34 N -14000 PV 0 PMT CPT FV= $78,462.69 78,462.6893066413 - 67,567.3668177835 = $10,895.3224888578 You will have $10,895.3224888578 more in your account if you wait for one year.

Sign Convention Example

5 N 10 I/Y -100 PV 20 PMT CPT FV = $38.95 Implies you deposited $100 today and plan to WITHDRAW $20 a year for 5 years (cash inflow to you) 5 N 10 I/Y -100 PV -20 PMT CPT FV = $283.15 Implies you deposited $100 today and plan to ADD $20 a year for 5 years (cash outflow from you)

Your bank will pay you an interest rate of .137 percent per week. You want to have $26,000 in 8 years. How much will you have to deposit today? Assume 52 weeks per year.

52*8 = 416 26000/(1+0.00137)^(416) =$14710.57(Approx).

Discount Rate - Example 2 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?

6N -10000 PV 0 PMT 20000 FV CPT I/Y = 12.25%

South Central Bank pays 2.5 percent interest, compounded annually, on its savings accounts. Northern Bank pays 2.5 percent simple interest on its savings accounts. You want to deposit sufficient funds today so that you will have $1,500 in your account 2 years from today. The amount you must deposit today:

CI = 1500 / (1 + 2.5%)^2 = 1427.72 SI = 1500 = x + x * 2.5% * 2 1.05x = 1500 x = 1428.57 Greater if you invest with Northern Bank

Number of Periods - Example 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?

Calculator Solution: 10 I/Y; -15000 PV; 20000 FV; CPT N = 3.02 years

Discount Rate - Example 3 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?

Calculator: 17 N, -5000 PV, 0 PMT, 75000 FV, CPT I/Y = 17.27%

Which of the following is the multi-period formula for compounding a present value into a future value?

FV = PV×(1 + r)^t

Although appealing to more refined tastes, art as a collectible has not always performed so profitably. During 2010, Deutscher-Menzies sold Arkie under the Shower, a painting by renowned Australian painter Brett Whiteley, at auction for a price of $1,100,000. Unfortunately for the previous owner, he had purchased it 3 years earlier at a price of $1,680,000. What was his annual rate of return on this painting?

FV=PV(1 + r )n FV = $1,100,000 PV= $1,680,000 N= 3 PMT = 0 CPT I/Y= -.1317 OR 13.17%