Lecture 2 Notes PDF

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Lecture 2: Principle of Finance     

Ordinary Perpetuities Deferred Perpetuities Ordinary Annuities Growing Perpetuities Growing Ordinary Annuities

Perpetuities: An equal, periodic cash flow that goes on forever  The first cash flow occurs at the end of period 1

 

Present Value of a Perpetuity C PV ( C ∈perpetuity )= r C = payment amount each period R = interest rate

Using value additivity, the present value of a perpetuity is: C C C C PV 0= + +… + 2 3 1+ r ( 1+r ) ( 1+r ) ( 1+r )n 1 1 1 1 PV 0=C∗( + +… ) + 2 3 n 1+r (1+r ) (1+r ) ( 1+r ) 1 1 1 1 1 + +… + As n approaches ∞, approaches n r 1+ r ( 1+r )2 ( 1+r )3 ( 1+r )

[(

)]

Deferred Perpetuity: An equal, periodic cash flow that starts at some future date and then goes on forever  1st cash flow start at some date  n=

This perpetuity starts 3 periods after and NO CASH until the end of 3rd period and start of 4th period 1. Pretend it is simply a perpetuity that starts at a later date 2. Discount it back to present value C ( ) r PV 0= n ( 1+r )

( )



NOTE: n is the year of perpetuity starting, NOT the date of the first cash pay-out

Ordinary Annuity: A series of equal, periodic cash flows occurring at the end of each period and lasting for n periods  The first cash flow occurs at the end of period 1  The last cash flow occurs at the end of period n

The present value of an ordinary annuity can be obtained as the difference between an ordinary perpetuities and deferred perpetuity PV 0 ( OA )=PV 10 −PV 02 1 C C C 1 PV 0 ( OA )= (1− − = ) r r (1+r ) n r ( 1+r )n

( ) ( )(

)( )

Future Value of Ordinary Annuity: obtained by compounding the present value above to the end n of time period n using ( 1+r )

( Cr )( 1− (1+r1 ) ) ( 1+r) C FV ( OA )= ( )[ ( 1+r ) −1] r

FV n( OA )=

n

n

n

n

#### r=interest rate= the rate at each compounding period, does not have to be annual or monthly Example: Suppose a prize guarantees you $10,000 per year for the next 3 years with the first payment to be made at the end of year 1. How much would you sell the prize for today if the interest rate is 10% p.a.?  3 Year Annuity

Present Value of Deferred Perpetuity

Present Value of Perpetuity

Ordinary Annuity =100,000-75,131.48 = $24,868.52

Example: John charges $2000 per month on his credit card without repayments for the next 24 months. Credit card interest is 18% p.a.. How much money will he be owing on the credit card 25 months from now when he receives her first credit card statement 25 months from now?

### NOT 24 months only, John doesn’t start paying until the 25th month Step 1: Monthly Interest Rates =

0.18 =0.015 % per month 12

Future Value of the 24 months annuity: C 2000 FV 24 = ∗[( 1+0.015 ) 24 −1 ]=$ 57,267 owingat the end of 24 months [ ( 1+r ) n−1] = 0.015 r

( )

(

)

FV at end of 24 months compounded to 25th: FV 25 =57267∗(1+0.015 ) =$ 58,126 owing at the end of 25 months Example 1 cont.: Your friend now owes $58,126 on her credit card has decided to start paying off in a monthly basis, starting next month. If she can only afford to repay $400 every month. How much would she still owe on her credit card by the end of 1 year? (12 months)

IDEA: work out the FV of the current balance after 1 year MINUS the FV of the 400 monthly repayment after 1 year Step 1: The future value of the current balance in 12 months compounding monthly is FV 12 =58126∗(1 +0.015 )12=$ 69497 Step 2: The future value of the $400 repaid every month after 12 months is: C n 400 12 FV 12= ∗[ ( 1+r) −1 ] = ∗[( 1+0.015 ) −1] =−$ 5216 r 0.015

( )

(

)

Step 3: The net amount after 12 months is: 69497− 5216=$ 64281 NOTE:  The beginning of period n is the same as the end of period n-1

E.g.: 31/6 and 1/7  The overall effect is to move the annuity back one period to our standard timeline

Example 2: Prize money of a $30 million dollar lottery. Two ways of taking the winnings: 1. $15 million paid today, or 2. $1 million per year starting today for 30 payments Method 1: Value the annuity as if it were an ordinary annuity and adjust it accordingly

1 ∗ 1− ( 1000000 ) 0.08 ( ( 1+ 0.08) )∗( 1+0.08 )=$ 12,158,406

PV 0 ( AD) =

30

Method 2: Calculate the PV of the 29 year annuity and then adds the cash flow in year 0

1 ∗ 1− +1,000,000=$ 12,158,406 ( 1000000 0.08 ) ( ( 1+ 0.08) )

PV 0( AD ) =

30

Growing Perpetuity A series of periodic cash flows occurring at the end of each period and growing at a constant rate forever

Growing Perpetuity: n−1 2 C2 =C1( 1+g ) ,C 3=C 1( 1+ g ) … … Cn =C1 (1+g )

#n-1 not n

The PV of a Growing Perpetuity n−1 C 1 C1 ( 1+g) C 1 (1+g ) 2 C 1 ( 1+g) PV 0= + + …+ + 1+ r ( 1+r )2 ( 1+r )3 ( 1+r )n 2

PV 0=C 1∗[

( 1+ g )n−1 1 + (1+ g ) (1+ g) + + …+ ] 1+ r (1+r ) 2 (1+r )3 ( 1+r ) n

as n approaches infinity, PV0 approaches

1 r−g

PV(growing perpetuity) =

C r−g

Note:  r>g, as it is impossible for an investment to grow faster than interest rate forever  When g=0, we get an ordinary perpetuity  It is possible for g...