2.4 Perpetuities and Annuities 2.5 Effective Annual Interest Rate Outline 2: Time Value of Money & Introduction to Discount Rates & Rate of Return 2.1 Future Values 2.2 Present Values 2.3 Multiple Cash Flows 2.4 Perpetuities and Annuities 2.5 Effective Annual Interest Rate 2.6 Loan Amortization Appendix on Time Value of Money 2

Future Values Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest. Simple Interest - Interest earned only on the original investment. 3

Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Interest Earned Per Year = 100 x .06 = $ 6 5

Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. 6

Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 1 2 3 4 5 Interest Earned Value 100 7

Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 1 2 3 4 5 Interest Earned 6 Value 100 106 8

Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 1 2 3 4 5 Interest Earned 6 6 Value 100 106 112 9

Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 1 2 3 4 5 Interest Earned 6 6 6 Value 100 106 112 118 10

Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 1 2 3 4 5 Interest Earned 6 6 6 6 Value 100 106 112 118 124 11

Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 1 2 3 4 5 Interest Earned 6 6 6 6 6 Value 100 106 112 118 124 130 Value at the end of Year 5 = $130 12

Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. 13

Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Interest Earned Per Year =Prior Year Balance x .06 14

Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 3 4 5 Interest Earned Value 100 15

Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 3 4 5 Interest Earned 6.00 Value 100 106.00 16

Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 3 4 5 Interest Earned 6.00 6.36 Value 100 106.00 112.36 17

Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 3 4 5 Interest Earned 6.00 6.36 6.74 Value 100 106.00 112.36 119.10 18

Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 3 4 5 Interest Earned 6.00 6.36 6.74 7.15 Value 100 106.00 112.36 119.10 126.25 19

Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 3 4 5 Interest Earned 6.00 6.36 6.74 7.15 7.57 Value 100 106.00 112.36 119.10 126.25 133.82 Value at the end of Year 5 = $133.82 20

Future Values Future Value of $100 = FV 21

Future Values Future Value of any Present Value = FV where t= number of time periods r=is the discount rate 21

Future Values if t=4: FV = PV(1+r)(1+r) (1+r)(1+r) = PV(1+r)4 if t=10: FV = PV(1+r)(1+r)(1+r)(1+r)(1+r)(1+r)(1+r) (1+r)(1+r)(1+r) = PV(1+r)10 21

Future Values if t=n: FV = PV(1+r)(1+r) (1+r)(1+r)…(1+r) = PV(1+r)n FV = PV(1+r) = PV(1+r)0 = PV 21

Future Values Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years? 22

Future Values Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years? 23

Future Values: FV with Compounding Interest Rates 24

Future Value: Manhattan Island Sale Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal? To answer, determine $24 is worth in the year 2006, compounded at 12.5% (long-term average annual return on S&P 500): FYI - The value of Manhattan Island land is a very small fraction of this number.

Present Values Present Value Value today of a future cash flow. Discount Factor Present value of a $1 future payment. Discount Rate Interest rate used to compute present values of future cash flows. 31

Present Values 32

Present Values Since FV = PV (1+r) then solve for PV by dividing both sides by (1+r): 34

Present Values Example You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years? 34

Present Values Example You are twenty years old and want to have $1 million in cash when you are 80 years old (you can expect to live to one-hundred or more). If you expect to earn the long-term average 12.4% in the stock market how much do you need to invest now? 34

Present Values Discount Factor = DF = PV of $1 Discount Factors can be used to compute the present value of any cash flow. r is the discount rate (of return) 36

Present Value The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable. 38

Present Value: PV of Multiple Cash Flows Example Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer? 42

Present Value: PV of Multiple Cash Flows PVs can be added together to evaluate multiple cash flows. 43

Present Value: Perpetuities & Annuities Perpetuity A stream of level cash payments that never ends. Annuity Equally spaced level stream of cash flows for a limited period of time. 44

Present Value: Perpetuities & Annuities PV of Perpetuity Formula C = constant cash payment r = interest rate or rate of return 45

Present Value: Perpetuities & Annuities Example - Perpetuity In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%? 47

Present Value: Perpetuities & Annuities Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today? 49

Present Value: Perpetuities & Annuities PV of Annuity Formula C = cash payment r = interest rate t = Number of years (periods) cash payment is received 50

Present Value: Perpetuities & Annuities If PV of Annuity Formula is: Then formula for annuity payment is: 50

Present Value: Perpetuities & Annuities Formula for annuity payment can be used to find loan payments. Just think of C as Payment, PV as loan amount, t as the number of months, and r must be the periodic loan r to coincide with the frequency of payments: 50

Present Value: Perpetuities & Annuities PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years. 51

Present Value: Perpetuities & Annuities Example - Annuity You are purchasing a car. You are scheduled to make 60 month installments of $500 for a $25,000 auto. Given an annual market rate of interest of 5% for a car loan, what is the price you are paying for the car (i.e. what is the PV)? 53

Present Value: Perpetuities & Annuities Example - Annuity You have just won the NJ lottery for $2 million over 25 years. How much is the “$2 million” NJ Lottery really worth at an opportunity cost rate of return of 12.4% - long-run annual stock market rate of return (ignoring income taxes)? 53

Present Value: Perpetuities & Annuities Example - Annuity Now what if you took the lump-sum based on a 5% discount rate by the State of New Jersey? 53

Perpetuities & Annuities Example - Future Value of annual payments You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account? 56

Perpetuities & Annuities Future Value of Ordinary Annuity: 56

Perpetuities & Annuities Present Value of Ordinary Annuity: 56

Effective Interest Rates Effective Annual Interest Rate - Interest rate that is annualized using compound interest. r = annual or nominal rate of interest or return m= number of compounding periods per year rnom/m=also known as the periodic interest rate 26

Effective Interest Rates example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? 27

Effective Interest Rates example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? 28

Amortization Amortization is the process by which a loan is paid off. During that process, the interest and contribution amounts change every month due to the mathematics of compounding. Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

Step 1: Find the required payments. Amortization Step 1: Find the required payments. 1 2 3 10% -1,000 PMT PMT PMT 3 10 -1000 0 INPUTS N I/YR PV PMT FV OUTPUT 402.11

Step 2: Find interest charge for Year 1. Amortization Step 2: Find interest charge for Year 1. INTt = Beg balt (i) INT1 = $1,000(0.10) = $100. Step 3: Find repayment of principal in Year 1. Repmt = PMT – INT = $402.11 – $100 = $302.11.

Step 4: Find ending balance after year 1. Amortization Step 4: Find ending balance after year 1. End bal = Beg bal – Repmt = $1,000 – $302.11 = $697.89. Repeat these steps for Years 2 and 3 to complete the amortization table.

Amortization BEG PRIN END YR BAL PMT INT PMT BAL 1 $1,000 $402 $100 $302 $698 2 698 402 70 332 366 3 366 402 37 366 0 TOT 1,206.34 206.34 1,000 Interest declines and contribution to principal grows. Tax implications from lower interest paid.

10% on loan outstanding, which is falling. $ 402.11 Interest 302.11 Principal Payments 1 2 3 Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling.

Amortization Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, etc. They are very important! Financial calculators (and spreadsheets) are great for setting up amortization tables.

Appendix on Time Value of Money Future value Present value Rates of return

Future Value 1 2 3 i% CF0 CF1 CF2 CF3 Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2. Time lines show timing of cash flows.

Time line for a $100 lump sum due at the end of Year 2. 1 2 Year i% 100

Time line for an ordinary annuity of $100 for 3 years. 1 2 3 i% 100 100 100

Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 to 3 1 2 3 r% -50 100 75 50

What’s the FV of an initial $100 after 3 years if r = 10%? 1 2 3 10% 100 FV = ? Finding FVs is compounding.

After 1 year: FV1 = PV + INT1 = PV + PV(r) = PV(1 + r) = $100(1.10) = $110.00 After 2 years: FV2 = PV(1 + r)2 = $100(1.10)2 = $121.00

After 3 years: FV3 = PV(1 + r)3 = 100(1.10)3 = $133.10 In general, FVn = PV(1 + r)n

Four Ways to Find FVs Solve the equation with a regular calculator. Use tables. Use a financial calculator. Use a spreadsheet.

Financial Calculator Solution Financial calculators solve this equation: FVn = PV(1 + r)n There are 4 variables. If 3 are known, the calculator will solve for the 4th.

Here’s the setup to find FV: INPUTS 3 10 -100 0 N r/YR PV PMT FV 133.10 OUTPUT Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set: P/YR = 1, END

What is the PV of $100 due in 3 years if r=10% What is the PV of $100 due in 3 years if r=10%? Finding PVs is discounting, and it’s the reverse of compounding. 1 2 3 10% PV = ? 100

What interest rate would cause $100 to grow to $125.97 in 3 years? Solve FVn = PV(1 + r )n for PV: . 3 1 æ ö ( ) PV = $100 ç ÷ = $100 PVIF è ø i, n 1.10 = $100 ( 0.7513 ) = $75.13.

Financial Calculator Solution 3 10 0 100 N r/YR PV PMT FV -75.13 INPUTS OUTPUT Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years.

If sales grow at 20% per year, how long before sales double? Solve for n: FVn = 1(1 + r)n; 2 = 1(1.20)n Use calculator to solve, see next slide.

Graphical Illustration: INPUTS 20 -1 0 2 N r/YR PV PMT FV 3.8 OUTPUT Graphical Illustration: FV 2 3.8 1 Year 1 2 3 4

What’s the difference between an ordinary annuity and an annuity due? 1 2 3 r% PMT PMT PMT Annuity Due 1 2 3 r% PMT PMT PMT

What’s the FV of a 3-year ordinary annuity of $100 at 10%? 1 2 3 10% 100 100 100 110 121 FV = 331

Financial Calculator Solution INPUTS 3 10 0 -100 331.00 N r/YR PV PMT FV OUTPUT Have payments but no lump sum PV, so enter 0 for present value.

What’s the PV of this ordinary annuity? 1 2 3 10% 100 100 100 90.91 82.64 75.13 248.68 = PV

Have payments but no lump sum FV, so enter 0 for future value. INPUTS 3 10 100 0 N r/YR PV PMT FV OUTPUT -248.69 Have payments but no lump sum FV, so enter 0 for future value.

Find the FV and PV if the annuity were an annuity due. 1 2 3 10% 100 100 100

Switch from “End” to “Begin.” Then enter variables to find PVA3 = $273.55. INPUTS 3 10 100 0 -273.55 N r/YR PV PMT FV OUTPUT Then enter PV = 0 and press FV to find FV = $364.10.

What is the PV of this uneven cash flow stream? 1 2 3 4 10% 100 300 300 -50 90.91 247.93 225.39 -34.15 530.08 = PV

Input in “CFLO” register: Enter r = 10, then press NPV button to get NPV = 530.09. (Here NPV = PV.)

Finding the interest rate or growth rate INPUTS 3 -100 0 125.97 N r/YR PV PMT FV OUTPUT 8%

LARGER! If compounding is more Will the FV of a lump sum be larger or smaller if we compound more often, holding interest rate constant? Why? LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

1 2 3 10% 100 133.10 Annually: FV3 = 100(1.10)3 = 133.10. 1 2 3 1 2 3 4 5 6 5% 100 134.01 Semiannually: FV6 = 100(1.05)6 = 134.01.

Rates of Return:We will deal with 3 different rates: rNom = nominal, or stated, or quoted, rate per year. rPer = periodic rate. EAR = EFF% = . effective annual rate

rNom is stated in contracts. Periods per year (m) must also be given. Examples: 8%; Quarterly 8%, Daily interest (365 days)

Periodic rate = rPer = rNom/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. Examples: 8% quarterly: rPer = 8%/4 = 2%. 8% daily (365): rPer = 8%/365 = 0.021918%.

Effective Annual Rate (EAR = EFF%): The annual rate that causes PV to grow to the same FV as under multi-period compounding. Example: EFF% for 10%, semiannual: FV = (1 + rNom/m)m = (1.05)2 = 1.1025. EFF% = 10.25% because (1.1025)1 = 1.1025. Any PV would grow to same FV at 10.25% annually or 10% semiannually.

An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons. Banks say “interest paid daily.” Same as compounded daily.

Find EFF% for a nominal rate of10%, compounded semi-annually Or use a financial calculator.

EAR = EFF% of 10% EARAnnual = 10%. EARQ = (1 + 0.10/4)4 – 1 = 10.38%. EARM = (1 + 0.10/12)12 – 1 = 10.47%. EARD(360) = (1 + 0.10/360)360 – 1 = 10.52%.

Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if m = 1. If m > 1, EFF% will always be greater than the nominal rate.

When is each rate used? iNom: Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

rPer: Used in calculations, shown on time lines. If rNom has annual compounding, then rPer = rNom/1 = rNom.

EAR = EFF%: Used to compare returns on investments with different payments per year. (Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.)

FV of $100 after 3 years under 10% semiannual compounding? Quarterly? mn i æ ö FV = PV ç 1 . + Nom ÷ n è ø m 2x3 0.10 æ ö FV = $100 ç 1 + ÷ 3S è ø 2 = $100(1.05)6 = $134.01. FV3Q = $100(1.025)12 = $134.49.

What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually? 1 2 3 4 5 6 6-mos. periods 5% 100 100 100

Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques.

1st Method: Compound Each CF 1 2 3 4 5 6 5% 100 100 100.00 110.25 121.55 331.80 FVA3 = 100(1.05)4 + 100(1.05)2 + 100 = 331.80.

Could you find FV with a financial calculator? 2nd Method: Treat as an Annuity Could you find FV with a financial calculator? Yes, by following these steps: a. Find the EAR for the quoted rate: EAR = (1 + ) – 1 = 10.25%. 0.10 2 2

Or, to find EAR with a calculator: NOM% = 10. P/YR = 2. EFF% = 10.25.

b. The cash flow stream is an annual annuity. Find rNom (annual) whose EFF% = 10.25%. In calculator, EFF% = 10.25 P/YR = 1 NOM% = 10.25 c. 3 10.25 0 -100 INPUTS N r/YR PV PMT FV OUTPUT 331.80

What’s the PV of this stream? 1 2 3 5% 100 100 100 90.70 82.27 74.62 247.59

On January 1 you deposit $100 in an account that pays a nominal interest rate of 10%, with daily compounding (365 days). How much will you have on October 1, or after 9 months (273 days)? (Days given.)

( ) ( ) iPer = 10.0% / 365 = 0.027397% per day. ... FV = $100 1 2 273 0.027397% ... -100 FV = ? FV = $100 ( 1.00027397 ) 273 273 = $100 ( 1.07765 ) = $107.77. Note: % in calculator, decimal in equation.

Leave data in calculator. rPer = rNom/m = 10.0/365 = 0.027397% per day. INPUTS 273 -100 0 107.77 N r/YR PV PMT FV OUTPUT Enter i in one step. Leave data in calculator.

Now suppose you leave your money in the bank for 21 months, which is 1 Now suppose you leave your money in the bank for 21 months, which is 1.75 years or 273 + 365 = 638 days. How much will be in your account at maturity? Answer: Override N = 273 with N = 638. FV = $119.10.

rPer = 0.027397% per day. ... ... FV = $100(1 + .10/365)638 365 638 days ... ... -100 FV = 119.10 FV = $100(1 + .10/365)638 = $100(1.00027397)638 = $100(1.1910) = $119.10.

You are offered a note that pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank that pays a 7.0% nominal rate, with 365 daily compounding, which is a daily rate of 0.019178% and an EAR of 7.25%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless. Should you buy it?

1. Greatest future wealth: FV 2. Greatest wealth today: PV rPer = 0.019178% per day. 365 456 days ... ... -850 1,000 3 Ways to Solve: 1. Greatest future wealth: FV 2. Greatest wealth today: PV 3. Highest rate of return: Highest EFF%

1. Greatest Future Wealth Find FV of $850 left in bank for 15 months and compare with note’s FV = $1,000. FVBank = $850(1.00019178)456 = $927.67 in bank. Buy the note: $1,000 > $927.67.

Calculator Solution to FV: rPer = rNom/m = 7.0/365 = 7.0/365 = 0.019178% per day. INPUTS 456 -850 0 927.67 N r/YR PV PMT FV OUTPUT Enter rPer in one step.

2. Greatest Present Wealth Find PV of note, and compare with its $850 cost: PV = $1,000/(1.00019178)456 = $916.27.

7/365 = INPUTS 456 .019178 0 1000 -916.27 N r/YR PV PMT FV OUTPUT PV of note is greater than its $850 cost, so buy the note. Raises your wealth.

3. Rate of Return Find the EFF% on note and compare with 7.25% bank pays, which is your opportunity cost of capital: FVn = PV(1 + r)n $1,000 = $850(1 + r)456 Now we must solve for r.

Convert % to decimal: Decimal = 0.035646/100 = 0.00035646. INPUTS 456 -850 0 1000 0.035646% per day N r/YR PV PMT FV OUTPUT Convert % to decimal: Decimal = 0.035646/100 = 0.00035646. EAR = EFF% = (1.00035646)365 – 1 = 13.89%.

Using interest conversion: P/YR = 365. NOM% = 0.035646(365) = 13.01. EFF% = 13.89. Since 13.89% > 7.25% opportunity cost, buy the note.