Introduction to Valuation

Chapter 5

Created by David Moore, PhD

Key Concepts

  1. Time Value of Money
  2. Compounding and Discounting
  3. You will be able to answer
    • How to determine the future value of an investment made today
    • How to determine the present value of cash to be received in the future
    • How to find the return on an investment
    • How long it takes for an investment to reach a desired value

Time Value of Money

This is the most important skill in this course.
  • First and most important tool in your toolbox
  • Foundation for more complex valuation (projects, bonds, stocks)
Time Value of Money

How important is this?

  • I’m not a finance major, will I really use this?
    • Use to make decision of purchasing a car. Loan vs buy? Cash vs finance?
    • Buying a house? Is bigger down payment worth it?
    • Saving for retirement. How much do I need to put away each month? What happens if I take money out early?
    • Credit cards. Should I charge this? What if it takes longer to pay off?
    • Saving for a big purchase.

How important is this?

  • I am a finance major, how often is it really used?
    • EVERYWHERE!!!!!
    • How do you value a…
    • Company….TVM
    • Stock…TVM
    • Project….TVM
    • Bond…TVM

Interest Rates and Time Value of Money

A dollar in hand is worth MORE than a dollar in the future.
  • Suppose you are given the following opportunity: Invest $\$100,000$ today and you will receive $\$105,000$ in one year.
  • Think this as depositing money in a bank account paying 5% interest in one year.
  • We call the difference in value between money today and money in the future the time value of money.

Lottery Example

You just won the powerball!! The prize is $\$$132 million!! However, that is paid out in 30 annual payments of $\$$4.4 million. Alternatively, you could take $\$$78.6 million today. What should you do???

Take the money!!!

Terminology

  • Present Value (PV): current value of money
  • Future Value (FV): Value of an investment after one or more periods (hours, day, month, year, etc.)
  • Interest rate (r or I): The rate at which we can exchange money today for money in the future.(the price of money).
  • AKA discount rate, opportunity cost of capital, cost of capital, cost of debt, cost of equity, required rate of return, rate of return, user cost.

Ways to solve TVM problems

  1. Formula
  2. Financial calculator
  3. Spreadsheet (Excel)

Future Value

Suppose you invest $\$$1,000 for one year at 5% per year?


What is the future value in one year?

Suppose you invest for another year. How much will you have in two years from now?

Future Value: formula

$FV=PV(1+r)^t$

FV = Future Value

PV = Present Value

r = Interest rate

t= number of periods

$(1+r)^t$= the future value factor

Effects of compounding


Simple interest: interest on the original principal only.
Compound interest: Interest on both the principal and reinvested interest.
Year Beginning Amount Simple Interest Compound Interest Total Interest Ending Amount
1 1000.00 50.00 0.00 50.00 1050.00
2 1050.00 50.00 2.50 52.50 1102.5
3 1102.50 50.00 5.13 55.13 1157.63
4 1157.63 50.00 7.88 57.88 1215.51
5 1215.51 50.00 10.78 60.78 1276.28
Total 250 26.28 276.28

Average Savings

Time Value of Money

Present Value

Present Value

If we can go forward in time to find a future value (FV), can we go backward in time to find a present value (PV)?

YES!

Present Value

  • How much do I have to invest today to reach certain amount of money in the future?
    • $FV = PV(1+r)^t$
    • Rearrange to solve for $PV = \frac{FV}{(1+r)^t}$
  • When we talk about discounting, we mean finding the present value of some future amount.
  • When we talk about the "value" of something, we mean the present value unless we specifically indicate that we are calculating the future value.

PV and FV

Finance uses "compounding" as the verb for going into the future and "discounting" as the verb to bring funds into the present.

Time Value of Money
Time Value of Money

FV, Time, and Rates

Time Value of Money

PV, Time, and Rates

Time Value of Money

PV and FV Examples

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?

Example 2

Your uncle wants to begin saving for his daughter's (your cousin) college education. He wants to know the amount of money he needs to invest today. He estimated that the total tuition cost will be $\$$150,000 in 17 years. He thinks it is safe to assume the annual return of 8%. How much does your uncle need to invest today?

Example 3

Suppose you had a relative deposit $\$$10 200 years ago. The money grew at an annual interest of 5.5% ever since. How much money do you have today?

Interest Rates and Time Periods

Finding the Discount Rate

  • Often we will want to know what the implied interest rate is on an investment
  • Rearrange the basic PV equation and solve for r
  • $FV = PV(1 + r)^t$
  • $r = {\frac{FV}{PV}}^{1/t} – 1$

Finding the Number of Periods

  • It is also possible to want to know how long it will take for an investment to grow to a certain value given a PV and rate.
  • Rearrange the basic PV equation and solve for t
  • $FV = PV(1 + r)^t$
  • $t=\frac{ln(\frac{FV}{PV})}{ln(1+r)}$

Example 1

You are looking at an investment that will pay $\$$1,200 in 5 years if you invest $1,000 today. What is the (annual ) rate of interest?

Example 2

Suppose you are offered an investment that will allow you to double your money in 6 years. What is the (annual) rate of interest?

Example 3

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?

Example 4

Suppose you want to buy a new house. You currently have $\$$15,000.You need to pay a 10% down payment plus an additional 5% of the loan amount for closing costs. Assume the type of house you want will cost about $\$$150,000 and you can earn 7.5% per year on your investment. How long will it be before you have enough money for the down payment and closing costs? Hint: The closing costs = 5% of loan. Loan = cost of house – down payment.

Financial Calculator

How to use Financial Calculator

  1. Go to APPS
  2. Select 1: Finance
  3. Select 1: TVM Solver
  4. Enter three of PV, FV, I%, and N
  5. Select the field you want to solve for and hit solve. (above enter key hit green alpha and then enter)
Revisit Examples

Excel

There are two ways to use Excel to calculate TVM.
  1. Code in formulas
  2. Use Excel built in functions:
    • = FV(rate, nper, pmt, PV, type)
    • = PV(rate, nper, pmt, fv)
    • =RATE(nper,pmt,pv,fv)
    • =NPER(rate,pmt,pv,fv)

Three Rules in Valuation

  • Rule #1: Compare and combine values at the same point in time.
  • Rule #2: Compound to calculate a cash flow's future value.
  • Rule #3: Discount to calculate the (present) value of a future cash flow at an earlier point in time.

Key Learning Outcomes

  • TIME VALUE OF MONEY
  • Calculate PV, FV, R, and N
  • Simple vs Compound interest
  • Be able to use formula, financial calculator or Excel.

Extra Practice

You just won $\$$50,000 on a scratch-off ticket!!! Given what you learned about the power of compound interest you invest the entire amount in an account earning 7% interest.

  1. In 15 years you withdraw $\$$60,000 for a down payment. How much do you have left in the account?
  2. In 45 years you want to retire. What rate do you have to earn to have $\$$1,000,000?
  3. If you continue to earn 7%, how long will it take to have $\$$1,000,000?

Next time

Chapter 6: Discounted Cash Flow Valuation