Asset Valuatio with kow cash flows Auities ad Perpetuities care loa, savig for retiremet, mortgage
Simple Perpetuity A perpetuity is a stream of cash flows each of the amout of dollars, that are received at the ed of each period forever Note: Cash flows are the same over time There is o cash flow today (i.e. you receive the first cash flow oe period from ow)
Simple Perpetuity $20 cash flow $10 $0 0 1 2 3 4 5 6 7 8 9 time
The PV of a perpetuity is, Valuig a perpetuity PV... 2 1 r (1 (1 3 (1 i 1 i r
Valuig a perpetuity (cot d) perpetuity P otice that t0 t1 t2 t3 P P t0 t1 P P P 1 r r
Example: You will receive $100 forever begiig the ext year. The aual iterest rate is 10%. Fid PV. PV $100/0.1 $1,000 Check: If we ivest $1,000 the we should be able to replicate the stream of cash flows geerated by the perpetuity. That is by ivestig $1,000 today we should receive a paymet of $100 each year forever. This is how we ca do this: 1) ivest $1,000 today for oe year 2) accumulate $1,000(1.1) $1,100 i our bak accout at time 1 3) Withdraw $100 from our bak accout (we re left with $1000) 4) Ivest the remaiig $1,000 at time 1 for oe additioal year. Do the same year after year
Simple Auity A auity is a stream of cash flows each of the amout of f dollars, that tare received at tthe ed of each period for the duratio of periods Note: Cash flows are the same over time There is o cash flow today (i.e. you receive the first cash flow oe period from ow)
Simple five year Auity $25 $20 cash flows $15 $10 $5 $0 0 1 2 3 4 5 6 7 8 9 10 time
Simple auity formula The PV of a auity for years is, PV 1 r (1 (1... 1 1 i i 1 (1 r (1 2 3 (1
Valuig a simple auity (cot d) auity a() t0 t1 t2 t3. t otice that P P t0 t2 t3.. t P P 1 1 a() a() P 1 1 (1 (1 r (1
Example: Fid the preset value of a auity that pays $500 for the duratio of 7 years (begiig at the ed of the first yea. The aual iterest rate is 5%. PV r 1 1 ( 1 ) r $500 0.05 1 1 1.05 7 $2,893.17
year auity versus perpetuity whe r10% $12 $10 $8 $6 $4 $2 $0 0 5 10 15 20 25 30 35 40 45 50
Growig perpetuity A growig perpetuity is a stream of cash flows that grows over time with growth rate g where cash flows are received at the ed of each period forever Note: Cash flows grow over time with rate g There is o cash flow today (i.e. you receive the first cash flow oe period from ow)
Growig perpetuity with growth rate of 8% $25 $20 ash flow c $15 $10 $5 $0 0 1 2 3 4 5 6 7 8 9 time
Growig perpetuity formula The first cash flow is received at the ed of the first period ad is growig at rate g afterwards I particular, cash flows look like: t0 t1 t2 t3.. t.. (1g) (1g) 2.. (1g) -1.. PV (1 g) 2 1 r (1 (1 g) (1 i 1 i-1 i (1 g) 3 (1 r - g 2...
Valuig a growig perpetuity (cot d) perpetuity PV otice that (1g) (1g) 2 t0 t1 t2 t3 (1g)PV PV t0 t1 (1 g)pv PV PV 1 r r - g
Growig perpetuity with growth rate g ad iterest rate r10% $250 $200 $150 $100 $50 $0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 g
Growig Auity A growig gauity is a stream of cash flows that grows over time with growth rate g where cash flows are received at the ed of each period for the duratio of years. Note: Cash flows grow over time with rate g There is o cash flow today (i.e. you receive the first cash flow oe period from ow)
Five year growig Auity with growth rate of 8% $20 $15 ca ash flow $10 $5 $0 0 1 2 3 4 5 6 7 8 9 10 time
Growig auity formula The PV of a growig auity for years is, PV 1 r (1 g) 2 (1 (1 g) 3 (1 2... i-1 (1 g) (1 g) 1 i i 1 (1 r - g (1 ) (1 g) (1-1
Valuig a growig auity (cot d) auity a() (1g) (1g) 2 (1g) -1 t0 t1 t2 t3. t otice that (1g) GP(1g) (1g) -1 GP t0 t1 t2.. t GP GP(1 g) (1 g) a() a() 1 (1 r - g (1
Growig auity with growth rate g ad iterest rate r10% $16 $14 $12 $10 $8 $6 $4 $2 $0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 g
Growig auity formula for rg g) (1 g) (1 g) (1 PV -1 2 (1... (1 (1 r 1 PV 3 2 (1... (1 (1 r 1 r 1
Example 1: if you save $1,000 each year for 35 years, how much will you have i your bak accout after 35 years if the iterest rate is 10%? PV 1,000 1 10% 1 9,644 FV PV(1.10) $271,024 35 (1.10) 10) $ 35 How much would you eed to save each year i order to accumulate $300,000 after 35 years? X 1 10% 1 (1.10) 10) $300,000000 X 35 (1.10) $ 35 $1,107107 $ Y 1 35 1 (1.10) 10) 300,000 000 271,024 28,976 Y $107 35 10% (1.10) $X $300,000 1.107107 X $1,107107 $1000 $271,024
What is the preset value of your istallmets if you save $1,000 each year for (a) 35 years ad (b) forever? $1,000 ( a) $9,644 ( b) 10% $10,000 What is the preset value of your istallmets if the iterest rate chages to 9%? PV $1,000 1 9% 1 (1.09) 35 10,566 What is the future value of your istallmets if the iterest rate chages to 9%? FV $ 1,000 1 1 (1.09) 9% (1.09) $ 35 35 $215,711
Example 2: You wat to ret a apartmet i Housto for oe year. The ladlord is ot willig to reduce the mothly ret of $1,000 but offers the first moth for o charge. You ca also stay i your old apartmet ad pay ret of $915 (at the begiig of each moth). What should you do? Assume a iterest rate of 1% per moth. $915 1 PV(curret ret paymets) $ 915 1 $10, 402 11 1% (1.01) PV(alterative ret paymets) $1,000 1 1% 1 (1.01) Would your choice be the same if you got the last moth free? 11 $10,368 $1,000 1 $ 1,000 1 $10,471 > 10 1% (1.01) $10,402
Example 3: You eed a parkig space for the period of two years. You ca either buy a parkig space for $10,000 ad the sell it i two years for $10,500, or ret a parkig space for the period of 2 years. The mothly ret is curretly $75 ad is expected to rise by 0.5% each moth (startig from the ext). What should you do? Assume a iterest rate of 1% per moth. PV(buy parkig space) $10,500 $10 10,000000 $1, 731 24 (1.01) 23 $75(1.005) 1.005 PV(ret parkig space) $75 1 $1, 701 1% 0.5% 1.01
Example 4: You have just eared a Federal tax retur ad are thikig to doate $2,000 to the Museum of Cotemporary Art i Housto. I retur Museum offers free aual membership ($100 per year paid at the begiig of the yea forever or a growig perpetuity of $70 with growth rate of 3% per year (the first paymet of $70 is i oe yea. What should you do? Assume a iterest rate of 7% per year. $100 PV(free membership offe $ 100 $1, 529 7% PV(growig perpetuity) $70 $1, 750 7% 3%
Example 5: 30 years ago, Adré Fraçois Raffray agreed to pay the 90 year old Jeae Calmet 2,500 fracs ($500) per moth (ed) util she dies. I retur he will receive her apartmet whe she dies. The apartmet is worth $184,000. Suppose the mothly iterest rate is 1%. Assumig M. Raffray thought this was a good deal, how log did he thik Jeae Calmet would live? Mr. Raffray will break eve if Jeae Clamet lives less tha additioal moths 500 1 184,000 01 0.01 (1.01) 184,000 0.01 1 1. 01 500 1 1. 01 184,000 0.0101 500 This implies that l 1 l(1.01). So... 155.1, which implies a age of 103 years old.
Example 6: A isurace aget offers you the followig cotract: you pay $5,000 per year (ed) for the ext 15 years ad i retur you will receive $7,000 a year (ed) for the followig 15 years. Suppose iterest rates are 9%. Should you buy this cotract? 7,000 7,000 7,000-5,000-5,000-5,000 t0 t1 t2 t15 t16 t17 t30 5000 1 1 7000 1 1 1 15 15 1.09 1.09 1.09 1.09 1.09 15 24,813
Example cot d: suppose that the isurace aget sweetes the deal ad says that the paymets that you receive will grow at 3% per year. Would you take the cotract ow? 15 5000 1 1 7000 1.03 1 1 15 0 09 03.09 1.09 1.09 0.09 0.03 1. 09 15 21,973