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Time value of money

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Time value of money

Some important concepts

Financial management: Lecture 3

- 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

- 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

- 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

- 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

- 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?

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

Financial management: Lecture 3

- 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

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

Financial management: Lecture 3

- 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

- 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

- Calculating the PV of the perpetuity could be hard

Financial management: Lecture 3

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

Financial management: Lecture 3

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

Financial management: Lecture 3

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- Using the formula for the annuity

Financial management: Lecture 3

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

Financial management: Lecture 3

- 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

- 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

- 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

- 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

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

- 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

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

Financial management: Lecture 3

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

- What is inflation?
- What is the real interest rate?
- What is the nominal interest rate?

Financial management: Lecture 3

- 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

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

Example - nominal figures

Financial management: Lecture 3

Example - real figures

Financial management: Lecture 3