Understanding and Appreciating the Time Value of Money Chapter 3 Understanding and Appreciating the Time Value of Money

Introduction Always comparing money from different time periods – a dollar received today is worth more than a dollar received in the future. Why is that? Everything in personal finance involves time value of money

Compound Interest and Future Values Interest paid on interest. Reinvestment of interest paid on an investment’s principal Principal is the face value of the deposit or debt instrument.

How Compound Interest Works Future value—the value of an investment at some point in the future Present value—the current value in today’s dollars of a future sum of money Facts of Life page 67

Figure 3.1 Compound Interest at 6 Percent Over Time

Ways to Calculate TVM Formulas Financial calculators* Spreadsheets (ex: Excel)

The Rule of 72 How long will it take to double your money? Numbers of years for a given sum to double by dividing the investment’s annual growth or interest rate into 72.

The Rule of 72 Example: If an investment grows at an annual rate of 9% per year, then it should take 72/9 = 8 years to double.

Calculator Clues Before solving problem: Working a problem: Set to one payment per year Set to display at least four decimal places Set to “end” mode Working a problem: Positive and negative numbers – 1 negative number for each equation Enter zero for variables not in the problem Enter interest rate as a %, 10 not 0.10

Calculating the Future Value of a Single Sum Example: What will $5000 grow to become if invested at 10% for 6 years?

Calculating the Future Value of a Single Sum Chapter 2 Calculating the Future Value of a Single Sum Calculator N = 6 I/Y = 10 PV = -5000 PMT = 0 CPT FV

Compound Interest with Nonannual Periods Compounding may be quarterly, monthly, daily, or even a continuous basis. Money grows faster as the compounding period becomes shorter. Interest earned on interest more frequently grows money faster.

Compound Interest with Nonannual Periods Interest is often compounded monthly, quarterly, or semiannually in the real world Adjustments must be made to calculations and formulas the number of years, N, is multiplied by the number of compounding periods, m Divided I/Y by m Make m = number of compounding periods per year.

Calculating the Future Value of a Single Sum Chapter 2 Calculating the Future Value of a Single Sum Calculator N = 2 x m I/Y = 8/m PV = -100 PMT = 0 CPT FV

Chapter 2 Calculating the Future Value of a Single Sum of $100 for 2 years compounded quarterly Calculator N = 2 x 4 I/Y = 8 / 4 PV = -100 PMT = 0 CPT FV

The Importance of the Interest Rate The interest rate plays a critical role in how much an investment grows. Time is important also. Stop & Think page 72 Daily compounding – 1 cent compounded daily for 31 days at 100% interest.

Growth of $1,000 at 8 % interest Years

Growth of $1000 at 10% interest 45,259 21,725 Years

Present Value What’s it worth in today’s dollars? Strip away inflation to see what future cash flows are worth today. Inverse of compounding. Discount rate is the interest rate used to bring future money back to present.

Present Value Example You’re on vacation in Florida and you see an advertisement stating that you’ll receive $100 simply for taking a tour of a model condominium. You discover that the $100 is in the form of a savings bond that will not pay you the $100 for 10 years. What is the PV of the $100 to be received 10 years from today if your discount rate is 6%?

Calculating the Present Value of a Single Sum Chapter 2 Calculating the Present Value of a Single Sum Calculator N = 10 I/Y = 6 FV = 100 PMT = 0 CPT PV

Annuities An annuity is a series of equal dollar payments coming at the end of each time period for a specific number of time period. Depositing an equal amount of money at the end of a specific time period and allowing it to grow – investing in a college education

Calculating the Future Value of an Annuity Example: What would you accumulate if you could invest $5000 every year for the next 6 years at 10%? Assume the first investment occurs a year from today.

Calculating the Future Value of a Single Sum Chapter 2 Calculating the Future Value of a Single Sum Calculator N = 6 I/Y = 10 PV = 0 PMT = 5000 CPT FV

Annuities Example You’ll need $10,000 for education in 8 years. How much must you put away at the end of each year at 6% interest to have the college money ready?

Calculating the Future Value of an annuity Chapter 2 Calculating the Future Value of an annuity Calculator N = 8 I/Y = 6 PV = 0 FV = 10000 CPT PMT

Compound Annuities Example If you deposit $2,000 in an individual retirement account (IRA) at the end of each year and it grows at a rate of 10% per year, how much will you have when you retire in 30 years?

Calculating the Future Value of an annuity Chapter 2 Calculating the Future Value of an annuity Calculator N = 30 I/Y = 10 PV = 0 PMT = 2000 CPT FV

Amortized Loans Loans paid off in equal installments. (mortgage and car payments) You borrow $16,000 at 8% interest to buy a car and repay it in 4 equal payments at the end of each of the next 4 years. What are the annual payments?

Calculating the Future Value of a Single Sum Chapter 2 Calculating the Future Value of a Single Sum Calculator N = 4 I/Y = 8 PV = -16000 FV = 0 CPT PMT FIN 261 Personal Finance

Let’s work more problems

Problem 1 How much would you have if you invest $100 today, earning 6% annually, you hold the investment for 5 years? What if the interest is compounded monthly?

Problem 2 Let’s say your aunt gave you a EE savings bond that will be worth $100 in 5 years earning 4.5% annually. What is its value today?

Problem 3 What is the future value of an ordinary annuity if $100 payments occur for 10 years, and you can earn a 7% return?

Problem 4 You just won the lottery and will receive $100,000 at the end of each of the next 20 years. Alternatively, you can take the value in a lump sum today. If the appropriate rate is 8%, what should the lump sum be?

Problem 5 An investment advisor recommends you buy an annuity that will pay $1,000 at the end of each year for the next 10 years. What should you pay for this annuity if you can earn a 9% return?

Problem6 What is the monthly payment for principal and interest on a $150,000 mortgage priced at 7.2% for 30 years?

Problem 7 An investor expects to earn a 9.1% return. How long will it take to double the investment?