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### Chapter 5

Discounted Cash Flow Valuation

Key Concepts and Skills

- Be able to compute the future value of multiple cash flows
- Be able to compute the present value of multiple cash flows
- Be able to compute loan payments
- Be able to find the interest rate on a loan
- Understand how loans are amortized or paid off
- Understand how interest rates are quoted

Future Value (FV)

- Amount to which an investment will grow after earning interest.

FV = C0×(1 + r)T

Where C0 is cash flow today (time zero) and

r is the appropriate interest rate.

Ex: Suppose you invest $500 in a bank today and $600 in one year. If the fund pays 9% annually, how much will you have in two years?

FV=500(1.09)2 + 600(1.09)=1,248.05

Present Value (PV)

- Value today of a future cash flow

PV=Future Value of t Period/(1 + r)T

Ex: You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?

PV=3,000/(1 + 0.08)2 = $2,572

Annuity

- Annuity is the finite series of equal payments that occur at regular intervals

-If the first payment occurs at the end of the period, it is called an ordinary annuity

-If the first payment occurs at the beginning of the period, it is called an annuity due.

Annuity

Ex1: You are purchasing a car. You are scheduled to make 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)?

Ex2: You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account?

C

C

0

1

2

3

PerpetuityA constant stream of cash flows that lasts forever.

…

The formula for the present value of a perpetuity is:

Perpetuity

Ex1: An investment offers a perpetual cash flow of $500 every year. The return you require on such an investment is 8 percent. What is the value of this investment?

Ex2: Suppose the Fellini Co wants to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend will Fellini have to offer if the preferred stock is going to sell?

The issue that is already out has a present value of $40 and a cash flow of $1 every quarter forever.

Effective Annual Rates and Compounding

- EAR is the actual rate paid (or received) after accounting for compounding that occurs during the year.
- If you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison.
- Compounding is the process of calculating interest on an investment overtime to earn more interest.
- Stated or Quoted Interest Rate: is the interest rate expressed in terms of the interest payment made each period. Also, quoted interest rate.

Effective Annual Rates and Compounding

Where m is the number of periods per year

- You are looking at two saving accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use?

EAR= (1+ 0.0525/365)365=5.39%

Second account:

EAR= (1+ 0.053/2)2-1=5.37%

Which account should you choose and why?

Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year?

First account:

Daily rate= 0.0525/365=0.00014383562

FV=100(1.00014383562)365=105.39

Second account:

Semiannual rate=0.0530/2=0.0265

FV=100(1.0265)2 =105.37

You have more money in the first account.

Annual Percentage Rate (APRs)

Annual percentage rate (APR) is the interest rate charged per period multiplied by the number of periods per year. The quoted rate is the same as an APR.

For example. An APR of 12 percent on a loan calling for monthly payments is really 1 percent per month. The EAR on such a loan is thus:

Annual Percentage Rate (APRs)

Ex: Suppose if you put it in another account, you earn 3% per quarter.

What is the APR? 3(4)=12%

How much are you effectively earning?

FV=(1.03)4=1.1255

Rate=(1.1255-1)/1=0.1255=12.55%

If you have effective rate, how can you compute the APR?

Rearrange the EAR equation you get:

APR=m[(1+EAR)1/m -1]

LOAN TYPES AND LOAN AMORTIZATION

- Pure Discount Loans

Borrower receives money today and repays a single lump sum at some time in the future. Treasury bills are a common example of pure discount loans.

- Interest-Only Loans

Borrower pays interest only each period and the entire principal at maturity. Corporate bonds are a common example of interest-only loans.

LOAN TYPES AND LOAN AMORTIZATION

Consider a 5-year, interest only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually.

What would the stream of cash flows be?

Years 1-4: Interest payments of 0.07(10,000)=700

Year 5: Interest + Principal=10,700

Amortized Loans

Borrowers repay part or all of principal over the life of the loan. Two methods are (1) fixed amount of principal to be repaid each period, which results in uneven payments, and (2) fixed payments, which results in uneven principal reduction. Traditional auto and mortgage loans are example of the second type of amortized loans.

For example. suppose a business takes out a $5,000, five-year loan at 9 percent. The loan agreement calls for the borrower to pay the interest on the loan balance each year and to reduce the loan balance each year by $1,000. Since the loan amount declines by $I,000 each year, it is fully paid in five years.

Probably the most common way of amortizing a loan is to have the borrower make a single fixed payment every period.

- For example, suppose our five-year, 9 percent, $5,000 loan was amortized this way. How would the amortization schedule look?

In the first year. the interest is $450. Since the total payment is $1,285.46. the principal paid in the first year must be:

Principal paid =$1,285.46 - 450 = $835.46

The ending loan balance is thus:

Ending balance =$5,000 - 835.46 = $4,164.54

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