Time value of money
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Time value of money. Some important concepts. Today’s agenda. Review of what we have learned in the last lecture Continue to discuss the concept of the time value of money present value (PV) discount rate (r) net present value (NPV) Learn how to draw cash flows of projects

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Time value of money

Time value of money

Some important concepts

Financial management: Lecture 3


Today s agenda

Today’s agenda

  • Review of what we have learned in the last lecture

  • Continue to discuss the concept of the time value of money

    • present value (PV)

    • discount rate (r)

    • net present value (NPV)

  • Learn how to draw cash flows of projects

  • Learn how to calculate the present value of annuities

  • Learn how to calculate the present value of perpetuities

  • Inflation, real interest rates and nominal interest rates, and their relationship

Financial management: Lecture 3


What have we learned in the last lecture

What have we learned in the last lecture

  • The motivation for the study of the financial market

  • The seven functions of a financial market

  • The cost of capital

  • The present value concept

  • The NPV rule

  • The difference between capital budgeting and the investment in the financial market (simply called investment)

Financial management: Lecture 3


Example 1

Example 1

  • John got his MBA from SFSU. When he was interviewed by a big firm, the interviewer asked him the following question:

  • A project costs 10 m and produces future cash flows, as shown in the next slide, where cash flows depend on the state of the economy.

  • In a “boom economy” payoffs will be high

    • over the next three years, there is a 20% chance of a boom

  • • In a “normal economy” payoffs will be medium

    • over the next three years, there is a 50% chance of normal

  • In a “recession” payoffs will be low

    • over the next 3 years, there is a 30% chance of a recession

  • In all three states, the discount rate is 8% over all time horizons.

  • Tell me whether to take the project or not

Financial management: Lecture 3


Cash flows diagram in each state

Cash flows diagram in each state

  • Boom economy

  • Normal economy

  • Recession

$3 m

$8 m

$3 m

-$10 m

$7 m

$2 m

$1.5 m

-$10 m

$6 m

$1 m

$0.9 m

-$10 m

Financial management: Lecture 3


Example 1 continues

Example 1 (continues)

  • The interviewer then asked John:

    • Before you tell me the final decision, how do you calculate the NPV?

      • Should you calculate the NPV at each economy or take the average first and then calculate NPV

      • Can your conclusion be generalized to any situations?

Financial management: Lecture 3


Calculate the npv at each economy

Calculate the NPV at each economy

  • In the boom economy, the NPV is

    • -10+ 8/1.08 + 3/1.082 + 3/1.083=$2.36

  • In the average economy, the NPV is

    • -10+ 7/1.08 + 2/1.082 + 1.5/1.083=-$0.613

  • In the bust economy, the NPV is

    • -10+ 6/1.08 + 1/1.082 + 0.9/1.083 =-$2.87

      The expected NPV is

      0.2*2.36+0.5*(-.613)+0.3*(-2.87)=-$0.696

Financial management: Lecture 3


Calculate the expected cash flows at each time

Calculate the expected cash flows at each time

  • At period 1, the expected cash flow is

    • C1=0.2*8+0.5*7+0.3*6=$6.9

  • At period 2, the expected cash flow is

    • C2=0.2*3+0.5*2+0.3*1=$1.9

  • At period 3, the expected cash flows is

    • C3=0.2*3+0.5*1.5+0.3*0.9=$1.62

  • The NPV is

    • NPV=-10+6.9/1.08+1.9/1.082+1.62/1.083

    • =-$0.696

Financial management: Lecture 3


Perpetuities

Perpetuities

  • We are going to look at the PV of a perpetuity starting one year from now.

  • Definition: if a project makes a level, periodic payment into perpetuity, it is called a perpetuity.

  • Let’s suppose your friend promises to pay you $1 every year, starting in one year. His future family will continue to pay you and your future family forever. The discount rate is assumed to be constant at 8.5%. How much is this promise worth?

C

C

C

C

C

C

PV

???

Yr2

Yr3

Yr4

Yr5

Time=infinity

Yr1

Financial management: Lecture 3


Perpetuities continue

Perpetuities (continue)

  • Calculating the PV of the perpetuity could be hard

Financial management: Lecture 3


Perpetuities continue1

Perpetuities (continue)

  • To calculate the PV of perpetuities, we can have some math exercise as follows:

Financial management: Lecture 3


Perpetuities continue2

Perpetuities (continue)

  • Calculating the PV of the perpetuity could also be easy if you ask George

Financial management: Lecture 3


Calculate the pv of the perpetuity

Calculate the PV of the perpetuity

  • Consider the perpetuity of one dollar every period your friend promises to pay you. The interest rate or discount rate is 8.5%.

  • Then PV =1/0.085=$11.765, not a big gift.

Financial management: Lecture 3


Perpetuity continue

Perpetuity (continue)

  • What is the PV of a perpetuity of paying $C every year, starting from year t +1, with a constant discount rate of r ?

C

C

C

C

C

C

Yr0

t+2

t+3

t+4

T+5

Time=t+inf

t+1

Financial management: Lecture 3


Perpetuity continue1

Perpetuity (continue)

  • What is the PV of a perpetuity of paying $C every year, starting from year t +1, with a constant discount rate of r ?

Financial management: Lecture 3


Perpetuity alternative method

Perpetuity (alternative method)

  • What is the PV of a perpetuity that pays $C every year, starting in year t+1, at constant discount rate “r”?

    • Alternative method: we can think of PV of a perpetuity starting year t+1. The normal formula gives us the value AS OF year “t”. We then need to discount this value to account for periods “1 to t”

  • That is

Financial management: Lecture 3


Annuities

Annuities

  • Well, a project might not pay you forever. Instead, consider a project that promises to pay you $C every year, for the next “T” years. This is called an annuity.

  • Can you think of examples of annuities in the real world?

C

C

C

C

C

C

PV

???

Yr2

Yr3

Yr4

Yr5

Time=T

Yr1

Financial management: Lecture 3


Value the annuity

Value the annuity

  • Think of it as the difference between two perpetuities

    • add the value of a perpetuity starting in yr 1

    • subtract the value of perpetuity starting in yr T+1

Financial management: Lecture 3


Example for annuities

Example for annuities

  • you win the million dollar lottery! but wait, you will actually get paid $50,000 per year for the next 20 years if the discount rate is a constant 7% and the first payment will be in one year, how much have you actually won (in PV-terms) ?

Financial management: Lecture 3


My solution

My solution

  • Using the formula for the annuity

Financial management: Lecture 3


Example

Example

You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease?

Financial management: Lecture 3


Solution

Solution

Financial management: Lecture 3


Lottery example

Lottery example

  • Paper reports: Today’s JACKPOT = $20mm !!

    • paid in 20 annual equal installments.

    • payment are tax-free.

    • odds of winning the lottery is 13mm:1

  • Should you invest $1 for a ticket?

    • assume the risk-adjusted discount rate is 8%

Financial management: Lecture 3


My solution1

My solution

  • Should you invest ?

  • Step1: calculate the PV

  • Step 2: get the expectation of the PV

  • Pass up this this wonderful opportunity

Financial management: Lecture 3


Mortgage style loans

Mortgage-style loans

  • Suppose you take a $20,000 3-yr car loan with “mortgage style payments”

    • annual payments

    • interest rate is 7.5%

  • “Mortgage style” loans have two main features:

    • They require the borrower to make the same payment every period (in this case, every year)

    • The are fully amortizing (the loan is completely paid off by the end of the last period)

Financial management: Lecture 3


Mortgage style loans1

Mortgage-style loans

  • The best way to deal with mortgage-style loans is to make a “loan amortization schedule”

  • The schedule tells both the borrower and lender exactly:

    • what the loan balance is each period (in this case - year)

    • how much interest is due each year ? ( 7.5% )

    • what the total payment is each period (year)

  • Can you use what you have learned to figure out this schedule?

Financial management: Lecture 3


My solution2

My solution

Ending

balance

Total

payment

Interest

payment

Principle

payment

year

Beginning

balance

0

$20,000

$1,500

$6,191

$7,691

$13,809

1

7,154

13,809

1,036

6,655

7,691

2

7,154

7,691

0

7,154

537

3

Financial management: Lecture 3


Future value

Future value

  • The formula for converting the present value to future value:

    = present value at time zero

    = future value in year i

    = discount rate during the i years

Financial management: Lecture 3


Manhattan island sale

Manhattan Island Sale

Peter Minuit bought Manhattan Island for $24 in 1629. Was this a good deal? Suppose the interest rate is 8%.

Financial management: Lecture 3


Manhattan island sale1

Manhattan Island Sale

Peter Minuit bought Manhattan Island for $24 in 1629. Was this a good deal?

To answer, determine $24 is worth in the year 2003, compounded at 8%.

FYI - The value of Manhattan Island land is well below this figure.

Financial management: Lecture 3


Inflation

Inflation

  • What is inflation?

  • What is the real interest rate?

  • What is the nominal interest rate?

Financial management: Lecture 3


Inflation rule

Inflation rule

  • Be consistent in how you handle inflation!!

  • Use nominal interest rates to discount nominal cash flows.

  • Use real interest rates to discount real cash flows.

  • You will get the same results, whether you use nominal or real figures

Financial management: Lecture 3


Example1

Example

You own a lease that will cost you $8,000 next year, increasing at 3% a year (the forecasted inflation rate) for 3 additional years (4 years total). If discount rates are 10% what is the present value cost of the lease?

Financial management: Lecture 3


Inflation1

Inflation

Example - nominal figures

Financial management: Lecture 3


Inflation2

Inflation

Example - real figures

Financial management: Lecture 3


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