1. Chapter 6
Discounted Cash Flow Valuation

2. 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 interest rates are quoted
Understand how loans are amortized or paid off

3. Chapter Outline Future and Present Values of Multiple Cash Flows
Valuing Level Cash Flows: Annuities and Perpetuities
Comparing Rates: The Effect of Compounding
Loan Types and Loan Amortization

4. Present and future value of multiple cash flows
Calculate PV(FV) of each cash flow and add them up: e.g. i=10%
PV = 100/(1+10%)+ 300/(1+10%) /(1+10%) formula way
PV = 100 PVIF1,10% PVIF2,10% PVIF3,10% table way
10 i CFi CFi CFi CFi NPV financial calculator

5. Valuing Level Cash Flows: Annuities and Perpetuities
We often deal with situations where cash flows are same
throughout the problem. For example, a car loan, rent
payment etc.
An annuity is a level stream of cash flows for a fixed period of
time. Cash flow must be the same in each period.
Ordinary annuity: Payments are at the end of period
Annuity due: Payments are at the beginning of period
Unless stated otherwise, assume you deal with ordinary annuity

6. Future value of an annuity
FVA3 = A (1+i)2 + A (1+i) + A formula way
= A {(1+i)2 + (1+i) + 1}
= A FVIFA3,i% table way
A PMT r i n FV financial calculator way
again given any 3, we can solve for the 4th

7. Present value of an annuity
PVA3 = A/(1+i)3 + A/(1+i)2 + A/(1+i) formula way
= A {1/(1+i)3 + 1/(1+i)2 + 1/(1+i)}
= A PVIFA3,i% table way
A PMT r i n PV financial calculator way

8. deriving the PVIFA3,i% formula
use sum of infinite geometric series formula:
asa one can show that

9. deriving the PVIFA3,i% formula

10. Perpetuities
A special case of an annuity is when the cash flows continue forever.
The most common application of perpetuities in finance is preferred stock
Preferred stock offers a fixed cash dividend every period (usually every quarter) forever.
The dividend never increases in value, so it’s similar to a bond with a fixed interest payment.
Present value of a perpetuity

11. Comparing Interest Rates
How do you compare interest rates?
Rates can be quoted monthly, annually or something in between,
and it quickly becomes confusing to try and determine the “real”
interest rate.
Stated Rate ( also called APR, Quoted Rate, Nominal Rate): rate
before considering any compounding effects
e.g. 10% APR quarterly compounding
Periodic Rate: APR/(# of times compounding occurs in a year)
It is the effective or “real” rate. It considers the compounding
effects.

12. Effective Annual Rate
Effective Annual Rate (EAR)
Rate on an annual basis that reflects all compounding
effects
EAR= (1+APR/n)n – 1
You can compare different interest rate quotations
by using EAR

13. Timing of cash flows tells you what the period is
Note: in TVM problems
Timing of cash flows tells you what the period is
Find and use the periodic rate that is consistent with the period definition

14. Loan Amortization: There are many different kinds of loans available
Pure discount loan
With such a loan, the borrower receives money today and repays a single lump sum at some time in the future.
Interest-only loans
This kind of loan repayment plan calls for the borrower to pay interest each period and repay
the entire principal at some point in the future.

15. different types of loans
Amortized loans
With a pure discount or interest-only loan, the principal is paid all in once. An alternative is an amortized loan where the lender may require the borrower to repay parts of the loan amount over time. The process of paying off a loan by making
regular principal reductions is called amortizing the loan.
Partially amortizing loan
Similar to amortized loan except the borrower makes a single, much larger final payment called a “balloon” to pay off the loan.

16. Example
You get a $10,000 car loan. It is a five year amortized loan with
annual installments. 12% is the interest rate charged by the bank.
Develop the amortization schedule.