Chapter 6 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 interest rates are quoted Understand how loans are amortized or paid off

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

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%)2 + 400/(1+10%)3 formula way PV = 100 PVIF1,10% + 300 PVIF2,10% + 400 PVIF3,10% table way 10 i 0 CFi 100 CFi 300 CFi 400 CFi NPV financial calculator

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

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 3 n FV financial calculator way again given any 3, we can solve for the 4th

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 3 n PV financial calculator way

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

deriving the PVIFA3,i% formula

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

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.

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

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

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.

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.

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.