1. Discounted Cash Flow Valuation Multiple Cash Flows We have dealt with lump sums We have dealt with lump sums What if there is more than one cash flow? What if there is more than one cash flow? Different amounts at different times Similar amounts at different times ordinary annuity ordinary annuity annuity due annuity due First Future Values - Multiple Cash Flows First Future Values - Multiple Cash Flows Table A-4 (ordinary annuity) Calculator (ordinary annuity vs. annuity due)

2. Discounted Cash Flow Valuation How the cash flows can be viewed Series of lump sums Series of lump sums Series of payments Series of payments Some examples that are standard ordinary annuities for Future Value Calculations Savings Savings Retirement Retirement Some ordinary annuities with Present Value car loans car loans mortgages mortgages

3. Discounted Cash Flow Valuation Back to the basics of the equation with annuities: PV = PMT x (PVIFA, n, k) where n is number of payments and k is the interest rate for the period PV = PMT x (PVIFA, n, k) where n is number of payments and k is the interest rate for the period FV = PMT x (FVIFA, n, k) FV = PMT x (FVIFA, n, k) PMT = PV / (PVIFA, n, k) PMT = PV / (PVIFA, n, k) Remember must know all on the right side... Remember must know all on the right side... What about n and k? What about n and k?

4. Discounted Cash Flow Valuation Formula for PVIFA PVIFA = {1 - (1/ [1 + k]) n }/ k PVIFA = {1 - (1/ [1 + k]) n }/ k Text has PVIFA = ∑ 1/(1 + k) n Text has PVIFA = ∑ 1/(1 + k) n These are the same! These are the same! Formula for FVIFA FVIFA = {[1 + k]) n - 1}/ k FVIFA = {[1 + k]) n - 1}/ k Text has FVIFA = ∑ (1 + k) n-1 Text has FVIFA = ∑ (1 + k) n-1 These are the same! These are the same! Add these two formulas and now you can “solve” for k or n as well. Find FVIFA in Problem 17

5. Three Methods for Annuities Table in back of the book… PV = Payment x PVIFA (n, k) PV = Payment x PVIFA (n, k)Formula PV = Payment x {1 - (1/ [1 + k]) n }/ k PV = Payment x {1 - (1/ [1 + k]) n }/ kCalculator TVM keys TVM keys Example – Problem 18

6. Discounted Cash Flow Valuation Using the calculator for k (I) or n (N) some tricks to remember… some tricks to remember… direction of the cash flow is important for determining the k or n…usually PV is negative be sure and have payments per year correct and corresponding to compound per year…these are the P/Y and C/Y functions Example Example $1,000 PMT, -$6710 PV, N=10 (P/Y and C/Y =1) Compute I…answer is 8%

7. Discounted Cash Flow Valuation Perpetuities The never ending interest payment... The never ending interest payment... Special formula from limits in math Special formula from limits in math Problem 23 Problem 23

8. Discounted Cash Flow Valuation Amortization Paying off loans Paying off loans lump sum payoff (pure discount) interest only and principal at the end… equal payment series (interest and some principal at each payment) Amortization Schedule Amortization Schedule the principal reduction schedule typical ordinary annuity problem… calculator on the remaining balance

9. Amortization Interest is accumulated based on the outstanding principal Think of a loan where each period you pay it off and then “re-borrow” the reduced principal Table for Amortization Find payment first Find payment first Beginning balance x interest rate Beginning balance x interest rate Portion of Payment to interest, remainder to principal Portion of Payment to interest, remainder to principal Remaining principal is Ending Balance Remaining principal is Ending Balance Start again next period Start again next period Problem 42 annual payments and monthly Via Calculator for Problem 42