The Time Value of Money

The Time Value of Money FIN261 Personal Finance Spring 2006 Dana Smith

Outline • Time Value of Money • Future Value and Present Value • Compounding and Discounting • Applications of Time Value of Money • Lump sum • Evaluating multiple investments • Compounding more frequently than annually • Annuities • Evaluate a single investment using required rate of return FIN261 Spring 2006

Time Value of Money • The basic idea behind the concept of time value of money is: • $1 received today is worth more than $1 in the future OR • $1 received in the future is worth less than $1 today • Why? • because interest can be earned on the money • The connecting piece or link between present (today) and future is the interest rate, r or I/Y FIN261 Spring 2006

Time Value of Money • Economic decisions cannot be made without recognizing that cash flows occurring at different times have different values depending on • Time • Interest rates on similar alternatives • Adjustments allow comparisons FIN261 Spring 2006

Future Value and Present Value • Future Value (FV) is what money today will be worth at some point in the future • Present Value (PV) is what money at some point in the future is worth today FIN261 Spring 2006

The value of a lump sum or stream of cash payments at a future point in time: FV = PV x (1+r)n Future Value • Interest rate, r Future Value depends on: Number of periods, n FIN261 Spring 2006

FV4 = $146.41 FV3 = $133.10 FV2 = $121 FV1 = $110 Future Value of $100 (4 Years, 10% Interest ) PV = $100 0 1 2 3 4 End of Year FIN261 Spring 2006

Compounding Compounding: the process of earning interest in each successive year on 1.) the original balance or original principal 2.) past interest payments FIN261 Spring 2006

Year 1: FV1 = $110 • Earns 10% interest on initial $100 • FV1 = $100+$10 = $110 • Earn $10 interest again on $100 principal • Earns $1 on previous year’s interest of $10: $10 x 10% = 1 • FV2 = $110+$10+$1 = $121 Year 2: FV2 = $121 • Earn $10 interest again on $100 principal • Earns $2.10 on previous years’ interest of $21: $21 x 10% = $2.10 • FV3 = $121+$10+$2.10 = $133.10 Year 3: FV3 = $133.10 • Earn $10 interest again on $100 principal • Earns $3.15 on previous years’ interest of $33.10: $33.10 x 10% = $3.31 • FV4 = $133.10+$10+$3.31 = $146.41 Year 4: FV4 = $146.41 Compounding FIN261 Spring 2006

Present Value Today's value of a lump sum or stream of cash payments received at a future point in time: FIN261 Spring 2006

FV4 = $146.41 FV1 = $110 FV2 = $121 FV3 = $133.10 PV = $100 Present Value of $146.41 (4 Years, 10% Interest ) Discounting 0 1 2 3 4 End of Year FIN261 Spring 2006

Discounting Discounting: the process of converting a future cash flow into a present value Discounting: the reverse operation of compounding FIN261 Spring 2006

Lump Sum: Finding PV • Lump sum (single cash flow) decisions • Example: How much would you have to invest today at 10% interest to receive $150,000 in 10 years? • Formula FIN261 Spring 2006

Lump Sum: Business Calculator • TVM third row from top • N is number of periods • I/Y is annual rate of interest (r in formulas) • PV is present value • FV is future value FIN261 Spring 2006

Lump Sum: Finding PV • N = 10 • I/Y = 10 • FV = 150,000 • PV = ? • Computing the present value gives: $57,831.49 FIN261 Spring 2006

Lump Sum: Finding I/Y • Phil Goode wins $3,500 from a local radio station. He estimates that he will need $7,300 in 3 years to backpack across Europe. What must his rate of return (ROR) be on his investments if he wants to meet this financial goal? • N = 3 • PV = -3,500 • FV = 7,300 • I/Y = ? FIN261 Spring 2006

Lump Sum: Finding FV • Xiao wants to put a down payment on an apartment in Beijing within 4 years. How large will such a payment be if he has 22,000 yuan to invest at 17%? • N = 4 • PV = -22,000 • I/Y = 17% • FV = ? FIN261 Spring 2006

Evaluating Multiple Investments • Which one do you buy? • A single share of Axir Inc. is trading at $22.33. Analysts project share prices to be $60 in 5 years. • TackiWall Co. just announced its first IPO. Shares will begin selling at $17. The stock’s estimated share price is $25 in 2 years. • Your uncle wants to retire from his successful business and spend his days on the beach. He is selling ownership in his venture to the tune of $50 a share. In 10 years, shares will be worth $300. FIN261 Spring 2006

Compounding more frequently than annually • Interest is often compounded monthly, quarterly, or semiannually in the real world • Adjustments must be made to calculations and formulas • the number of years is multiplied by the number of compounding periods, m • the annual interest rate is divided by the number of compounding periods, m (or P/Y is adjusted on the business calculator) FIN261 Spring 2006

Compounding Intervals m compounding periods The more frequent the compounding period, the larger the FV! FIN261 Spring 2006

For semiannual compounding, m equals 2: Compounding More Frequently Than Annually FV at end of 2 years of $125,000 deposited at 5% interest FIN261 Spring 2006

For quarterly compounding, m equals 4: Compounding More Frequently Than Annually FV at end of 2 years of $125,000 deposited at 5% interest FIN261 Spring 2006

Compounding more frequently • EXAMPLE: Juan Garza invested $20,000 10 years ago at 12 percent, compounded quarterly. How much has he accumulated? • Using formula: • Using business calculator FIN261 Spring 2006

FV at end of 2 years of $125,000 at 5 % annual interest, compounded continuously: Continuous Compounding • In extreme case, interest compounded continuously: FV = PV x (e r x n) FIN261 Spring 2006

Effective annual rate: the annual rate actually paid or earned Stated Rate vs. the Effective Annual Rate Stated rate: the contractual annual rate charged by lender or promised by borrower FIN261 Spring 2006

Stated Rate vs. the Effective Annual Rate • FV of $100 at end of 1 year, invested at 5% stated annual interest, compounded: • Annually: FV = $100 (1.05)1 = $105 • Semiannually: FV = $100 (1.025)2 = $105.06 • Quarterly: FV = $100 (1.0125)4 = $105.09 Stated rate of 5% does not change.What about the effective rate? FIN261 Spring 2006

For annual compounding, effective = stated • For semiannual compounding • For quarterly compounding Effective Rates: Always Greater Than or Equal to Stated Rates FIN261 Spring 2006

Special considerations – Effective Annual Rate (EAR) • Effective Annual Rate (EAR) • I CONV • EXAMPLE: What rate does a depositor actually earn if a 6% annual interest rate is compounded monthly? • Using the business calculator FIN261 Spring 2006

Annuity • Annuity: • a stream or series of equal payments, equally spaced • The payments are assumed to be received at the end of each period • A good example of an annuity is an installment loan, where loan payments are paid over a number of years (ex. home mortgage loan) • PMT = is equal, periodic amounts (payments) FIN261 Spring 2006

Future Value and Present Value of an Ordinary Annuity Compounding FutureValue $1,000 $1,000 $1,000 $1,000 $1,000 0 1 2 3 4 5 End of Year Present Value FIN261 Spring 2006 Discounting

Annuity • EXAMPLE: What is the value of $1,000 to be received at the end of each of the next 5 years if an investor can earn only a 5.5% annual return? • Using calculator FIN261 Spring 2006

Present Value of Ordinary Annuity(5 Years, 5.5% Interest Rate) 0 1 2 3 4 5 $1,000 $1,000 $1,000 $1,000 $1,000 End of Year $947.87 $898.45 $851.61 $807.22 $765.13 Sum of present value payments: $4,270.28 FIN261 Spring 2006

Future Value of Ordinary Annuity(End of 5 Years, 5.5% Interest Rate) $1,238.82 $1,174.24 $1,113.02 $1,055.00 $1,000.00 $1,000 $1,000 $1,000 $1,000 $1,000 0 1 2 3 4 5 End of Year Sum of future value payments: 5,581.08 FIN261 Spring 2006

Annuity • Installment Loans • EXAMPLE: Jim and Kathy want to buy a house costing $100,000. They plan to put $20,000 down and finance the rest. What will their monthly loan payments be if they can get a loan at 7% for 30 years? FIN261 Spring 2006

Annuity • Using the business calculator • EXAMPLE: What is the outstanding balance on an installment loan with 3 years to maturity, annual loan payments of $1,200 and an interest rate of 10%? FIN261 Spring 2006

Special considerations – Annuities Due • Annuities received at the beginning rather than the end of the period. • Annuities due • EXAMPLE: How much will a landlord have accumulated if he deposits his monthly $1000 rent payments and earns 5.5% for 5 years? Rent payments are received and deposited at the beginning of each month. FIN261 Spring 2006

Special considerations – Annuities Due • Annuities received at the beginning rather than the end of the period. • Annuities due • EXAMPLE: How much will a landlord be willing to pay for a potential investment property that has $1000 monthly rent payments and earns 5.5% for 5 years? Rent payments are received and deposited at the beginning of each month. FIN261 Spring 2006

Required Rate of Return • What is your required rate of return? • Businesses and investors use their required ROR to analyze investment opportunities • Their required rate of return is used in the I/Y input of the calculator in TVM calculations • Can be thought of as opportunity cost • This is how to analyze an investment by itself FIN261 Spring 2006

2 Questions to Ask in Time Value of Money Problems • Future Value or Present Value? • Finding Future Value: Present (Now)  Future • Finding Present Value: Present  Future • Single amount or Annuity? • Single amount: one-time (or lump) sum • Annuity: same amount per period for a number of periods FIN261 Spring 2006

For Next Week • Read chapter 11 in the textbook • Quiz on the reading • Homework assignment 3: construct a personal income statement for the year ending 1/31/2006; refer to worksheet 2.3 FIN261 Spring 2006

The Time Value of Money