Discounting. Finance 100

Transcription

1 Discoutig Fiace 100 Prof. Michael R. Roberts 1 Topic Overview The Timelie Compoudig & Future Value Discoutig & Preset Value Multiple Cash Flows Special Streams of Cash Flows» Perpetuities» Auities Iterest Rates 2 1 1

2 The Timelie Timelie: a liear represetatio of the timig of potetial cash flows. Two types of cash flows: 1. Iflows (i.e., moey we get) are represeted by positive umbers 2. Outflows (i.e., moey we give) are represeted by egative umbers Example:» Assume that you are ledig $10,000 today ad that the loa will be repaid i two aual $6,000 paymets. 3 Moey s Time Uits Thik of moey as havig a time uit deotig whe it is received (or paid)» Just like currecy We ca oly compare moey i the same time uits:» It does t make sese to add $50 US to 50; ad» It does t make sese to add $50 received today with $50 received ext year. Discoutig ad Compoudig are the tools to maipulate moey s time uits» Discoutig coverts moey s time uits back i time» Compoudig coverts moey s time uits forward i time 4 2 2

3 Compoudig The Future Value (FV ) of a cash flow T-years from today is: ( 1 ) FV = C + r» C = Cash Flow (or CF )» r = discout rate Example:» Would you rather receive $1,000 today or $1,210 i two years if you ca ear 10% per year o the $1,000? Timelie ad Future Value =? T 5 Discoutig The Preset Value (PV) of a cash flow T-years from today is: C PV = = C + r ( 1+ r) T ( 1 ) Example:» What is the price of a savigs bod that will pay $15,000 i te years if the aual iterest rate is 6%? Timelie =? T Preset Value =? 6 3 3

4 Multiple Cash Flows Preset Value (PV) ad Future Value (FV) are liear operators» PV(C 1 + C 2 ) = PV(C 1 )+PV(C 2 )» FV(C 1 + C 2 ) = FV(C 1 )+FV(C 2 ) Example: If we ca ear a 10% aual iterest rate ad save $1000 today, ad $1000 at the ed of each of the ext two years how much will we have i 3 years? Timelie =? FV =? 7 Geeral Stream of Cash Flows Preset Value PV = PV ( C ) = = 0 = 0 C (1 + r) The PV of a stream of cash flows is just the sum of the PVs. Future Value (same idea): = 0 = 0 ( ) ( ) FV = FV ( C ) = C 1+ r = PV 1+ r 8 4 4

5 Perpetuities A perpetuity is a stream of cash flows with o ed: Cash Flows 0 C 1 C 2 C 3 C 4 C 5 C 6 Periods Examples:» Cecus Agreemets issued i 12 th cetury i Italy, Frace, ad Spai to circumvet usury laws of Catholic Church (o pricipal = o loa)» Hoogheemraadschap Lekdijk Bovedams 17 th cetury Dutch Water Board to upkeep local dikes (they still pay iterest!)» British cosol bods» Paama Caal perpetuities How do we compute PV? 9 Valuig Perpetuities Step 1: Write out the PV of the perpetuity Step 2: Pull out the cash flow, C Step 3: Multiply both sides by 1/(1+r) Step 4: Subtract (3) from (2) Step 5: Do some algebra

6 Perpetuity Example What does the timelie look like? The stream of cash flows is a? with a PV =? 11 Growig Perpetuities A growig perpetuity is a stream of cash flows that grow at a costat periodic rate, g, with o ed. Cash Flows 0 C C(1+g) C(1+g) 2 C(1+g) 3 C(1+g) 4 C(1+g) 5 Periods Agai, ifeasible to calculate by brute force so is there a shortcut? C PV = r g» You should be able to derive this

7 Auities A auity is a level stream of regular paymets that last for a fixed umber of periods Cash Flows 0 C C C C C Periods Examples:» Mortgages» Lottery prizes (sometimes )» Retiremet savigs plas How do we compute PV? 13 Valuig Auities Part I A auity is just the differece i two perpetuities startig at differet times!» Perpetuity #1 starts today: Cash Flows 0 CF CF CF CF CF CF Periods It has preset value at time 0 equal to C/r.» Perpetuity #2 starts i period : Cash Flows CF Periods It has preset value at time equal to C/r ad at time 0 equal to (C/r)(1+r)

8 Valuig Auities Part II Subtractig the cash flow streams of the two perpetuities gives us the cash flow stream for our auity Cash Flows 0 CF CF CF CF CF 0 Periods Therefore, differece i preset values for the two perpetuities must equal the preset value of our auity ( Perpetuity #1) ( Perpetuity #2) PV = PV PV = What s the future value of a auity?? 15 Auity Example PV of Optio A:» What is the timelie?» What is the preset value of all the cash flows? PV of Optio B =?

9 Valuig Growig Auities A growig auity is a costat growig stream of regular paymets that last for a fixed umber of periods Cash Flows 0 CF (1+g)CF (1+g) 2 CF (1+g) -2 CF (1+g) -1 CF 0 Periods The preset value of this stream is T C 1+ g PV = 1 r 1+ r» You should be able to derive this 17 Iteral Rate of Retur (IRR) The Iteral Rate of Retur (IRR) is the oe iterest rate that sets the et preset value of the cash flows equal to zero C Iitial Cost = 0 = 0 (1 + IRR) Example 1:» The IRR of a security (e.g., bod, stock, CD, etc.) is just the oe iterest rate that sets the preset value of all the cash flows equal to the price (a.k.a. PV) of the security: C Price = 0 = 0 (1 + IRR) Example 2:» The IRR of a ivestmet project (e.g., acquisitio, merger, capital expediture, etc.) is just the oe iterest rate that sets the preset value of all the cash flows equal to the iitial outlay (a.k.a. PV) of the ivestmet: C Iitial Outlay = 0 (1 + IRR) =

10 Computig the Iteral Rate of Retur Example The Timelie =? Preset Value =? IRR =? 19 Effective Aual Rate (EAR) The Effective Aual Rate (EAR) idicates the total amout of iterest that will be eared at the ed of oe year» Cosiders the effect of compoudig» Also referred to as the effective aual yield (EAY) or aual percetage yield (APY)» We ca use this to discout cash flows, as log as we express time i aual uits (i.e., years) So far everythig was o a aual basis» Cash flows were every year» Iterest was o a aual bases (i.e., compouded oce a year)» Therefore, distictio was irrelevat: EAR = r

11 Adjustig the Discout Rate to Differet Time Periods Earig 5% aually is ot the same as earigs 2.5% every six moths because of compoudig So, if the EAR is 5% but we have semi-aual discoutig the Equivalet Periodic Rate (EPR) is 2 1/2 1 + EPR 1 = 5% EPR = = = 2.47% < 2.5% ( ) ( ) More geerally, ( ) ( ) $1 $1.025 $ EPR = + 1/ m (1 EAR) 1» where m = # of compoudig periods per year (e.g., semi-aual m = 2, quarterly m = 4, mothly m = 12, )» EPR is just a -period discout rate 21 EAR ad EPR Examples If the EAR is 10% ad we have quarterly compoudig, what is the EPR? If the EPR is 0.6% ad we have mothly compoudig, what is the EAR?

12 Valuig Mothly Cash Flows Example Timelie:? Mothly EPR = Periodic Cash Flow =?? 23 Aual Percetage Rate (APR) The Aual Percetage Rate (APR), idicates the amout of simple iterest eared i oe year.» Simple iterest is the amout of iterest eared without the effect of compoudig.» The APR is typically less tha the effective aual rate (EAR) which icorporates the effect of compoudig Couterexample? The APR itself caot be used as a discout rate.» The APR with m compoudig periods is a way of quotig the actual iterest eared each compoudig period: APR Iterest Rate per Compoudig Period = i = m periods / year

13 EAR vs APR How do I covert a APR (ot a discout rate) to a EAR (a discout rate)? m APR 1 + EAR = 1 + m» EAR icreases with the frequecy of?» If compoudig is oce per year (m=1) the EAR=?» Cotiuous Compoudig: I limit as m, (1+APR/m) m exp(apr) Some otatio» R = APR (ot a discout rate!)» i = APR/m = iterest rate per compoudig period 25 Valuig Mothly Cash Flows Revisited Example Recall the problem o slide 22:» Mothly iterest with a EAR of 6% What is the APR (R) o this accout? How much iterest is eared each period?» Same as before so How much do you have to save at the ed of each moth to accumulate $100,000 i 10 years?» Same as before!

14 Covertig the APR to a Discout Rate Example Strategy: Compute the PV of the lease ad compare it with the $150,000 Timelie:? This cash flow stream is a? with? periodicity 27 Covertig the APR to a Discout Rate Example (Cot.) Computig the mothly discout rate:» Method 1: We re give a APR of 5% with semiaual compoudig, which implies the EAR =? Covert aual discout rate ito mothly discout rate?» Method 2: Compute a effective periodic iterest rate from the APR,? Covert six-moth discout rate ito mothly periodic rate:?

15 Covertig the APR to a Discout Rate Example (Cot.) With the mothly discout rate i had, the PV of the auity is? The PV of the lease is greater tha the upfrot paymet of $150,000 so purchase the system outright 29 omial Versus Real Iterest Rates omial Iterest Rate: The rates quoted by fiacial istitutios ad used for discoutig or compoudig cash flows, r Real Iterest Rate: The rate of growth of your purchasig power, after adjustig for iflatio, rr 1 + r Growth of Moey Growth i Purchasig Power = 1 + rr = = 1 + π Growth of Prices rr r π = r π 1 + π

16 US Iterest Rates ad Iflatio 31 What Formulas Should I Kow? otatio:» r = discout rate» R = APR» T = # of years» m = umber of compoudig periods per year» = Total umber of periods = T * m (years * periods/year)» i = R/m = effective periodic iterest rate Give a discout rate r for oe period, covert to a -period discout rate: period discout rate = ( 1+ r) 1 Covertig from a APR to a EAR: APR 1+ EAR = 1+ m m

17 What Formulas Should I Kow? (Cot.) Real Iterest Rates 1 + R rr = π Preset ad Future Value of a Sigle Cash Flow Preset Value of a Growig Auity ( 1 ) ( 1 ) PV = FV + i PV = FV + r g PV = C 1 ( i g) (1 i) +» Implies: 1) Future Value of a Growig Auity, 2) Future Value of a Auity, ad 3) Preset Value of a Auity Preset Value of a Growig Perpetuity C PV = ( i g)» Implies: 1) Future Value of a Growig Perpetuity, 2) Future Value of a Perpetuity, ad 3) Preset Value of a Perpetuity T 33 Summary Moey has a time uit» Ca oly compare moey i same uits!» Compoud to get future values» Discout to get preset values Future ad Preset Values are liear» Use them o streams of cash flows Special streams of cash flows» Perpetuity» Auity Iterest Rates» APR vs. EAR» Real vs. omial