Nominal and Effective Interest Rates

Transcription

1 Nominal and Effective Interest Rates Effective interest rates tell you how much interest accrues over some integer number of interest periods with the effects of compounding included. Nominal interest rates ignore the effects of compounding, so they can t be used in interest rate formulas. r = i m i r m i 1 i 1 eff r ieff 1 1 m n m converts an interest rate per compounding period into a nominal rate covering m periods converts a nominal rate covering m periods into an interest rate per compounding period converts an interest rate per compounding period into an effective rate covering n periods converts a nominal rate covering m periods into an effective rate covering the same m periods 1

2 Decoding Interest Rate Statements 1. Assume that all interest rates are nominal rates unless you re told otherwise. 2. Assume the compounding frequency is m = 1 unless you re told otherwise. 3. Assume that the time period of interest is a year unless you re told otherwise. 2

3 Reconciling Compounding Periods and Payment Periods Individual Cash Flows F = P ( F P, effective i per interest period, number of periods ) P = F ( P F, effective i per interest period, number of periods ) Cash Flow Series P = A ( P A, effective i per payment period, number of payments ) F = A ( F A, effective i per payment period, number of payments ) A = P ( A P, effective i per payment period, number of payments ) A = F ( A F, effective i per payment period, number of payments ) P = G ( P G, effective i per payment period, number of payments ) F = G ( F G, effective i per payment period, number of payments ) A = G ( A G, effective i per payment period, number of payments ) 3

4 MARR & WACC Sources of Capital Sell part of the company to investors in the form of stock (equity capital) Use retained earnings from previous profitable investments (equity capital) Borrow money from investors by selling bonds (debt capital) Borrow money from a bank in the form of a loan (debt capital) Debt:Equity Ratio D:E = debt capital / equity capital WACC WACC f c f c f c f c 1 t s s re re b b fx = fraction of capital pool from source x cx = cost of capital from source x (%) t = tax rate (if needed) 4

5 MARR & WACC MARR MARR based on WACC + profit MARR based on opportunity cost of first unfunded project on list 5

6 Evaluating Projects Using Present Worth If PW 0, the project earns more than the MARR If PW = 0, the project earns exactly the MARR If PW 0, the project earns less than the MARR INDEPENDENT VS. MUTUALLY EXCLUSIVE Independent means investing in one has no bearing on whether you invest in the other Independent = select all alternatives with PW/AW/FW/CC 0 Mutually exclusive means something prevents you from investing in both at the same time Mutually Exclusive = select the one best alternative COST (SERVICE) VS. REVENUE Revenue = Choose Do Nothing if all PW/AW/FW/CC < 0 Cost = Do Nothing is not an option; choose the least costly alternative Cost projects can only be fairly compared over equal service lives Revenue projects don t require equal service lives to compare 6

7 EVALUATION OF A BOND PURCHASE Coupon Rate (r) is an APR Coupon Frequency (m) is usually 2 per year in the U.S. r Coupon Amount Face Value m Project life (n) = number of coupon periods remaining to maturity Consider the MARR to be an APR unless stated otherwise If PW 0, you earn at least your MARR If PW < 0, PW = change in price needed to earn exactly your MARR 7

8 Evaluating Projects Using Future Worth If FW 0, the project earns more than the MARR If FW = 0, the project earns exactly the MARR If FW 0, the project earns less than the MARR Worth values must be compared at the same point in time so if the alternatives have unequal lives you must use a common global future worth not the individual local future worth values of the alternatives 8

9 Evaluating Cost Projects with Unequal Lives In order to fairly compare cost alternatives, they must all provide equal service. Equal service means they must fulfill the same goal for the same time period. Common study period approach Compare alternatives over a common time period Account for unequal lives using salvage values, rentals, etc. Can use PW, AW, FW since all lives are the same Replacement chain approach Assume a common study period = LCM of service lives Use the repeatability assumption to replicate costs out to the LCM AW of replacement chain = AW of first instance so AW is the most efficient method 9

10 Evaluating Projects Using Capitalized Cost CC = PW of project with infinite life (P A,i%, ) = 1/i P = A / i If PP > CP pair each A with an effective i per PP or use (A F) to convert each A into an equivalent A per CP P A i PP PP or PP A A F,i,PP CP i CP CP To compare cost projects with a mix of finite and infinite lives, use the replacement chain approach with LCM = 10

11 Evaluating Independent (Revenue) Projects 2 k bundles possible (including do nothing ) 1. Calculate present worth of each project 2. Eliminate all projects with PW < 0 3. Enumerate all 2 k possible bundles of remaining projects 4. Discard those whose investment cost exceeds the capital pool 5. Select the one bundle with the highest present worth 11

12 Evaluating Individual Projects Using Annual Worth If AW 0, the project earns more than the MARR If AW = 0, the project earns exactly the MARR If AW 0, the project earns less than the MARR AW = CR + AOC + Annual Revenue CR = P(A P,i%,n) + S(A F,i%,n) 12

13 Evaluating Alternatives Using Annual Worth For revenue projects with unequal lives use present worth analysis. For cost projects with unequal lives use annual worth analysis. 13

14 AW OF A LONG LIFE OR INFINITE LIFE INVESTMENT (A P,i%, ) = i A = P i AW = PW MARR To compare cost projects with a mix of finite and infinite lives, use the replacement chain approach with LCM = 14