Lecture_03-04_Time Value of Money 2

8/17/2019 Lecture_03-04_Time Value of Money 2

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Professor Sang Byung [email protected]

The Time Value of Money 2



Last time

• The time value of money (1 nd part)• Basic concept• FV vs PV•

Compound vs Discount• Discount factor• Timeline



Today

•

The time value of money (2 nd part)• PV short cuts• Compounding frequencies

• 1 st problem set• Will be posted at 7 PM today on Blackboard• Due by Feb 1 st (Monday) 11:59 PM CST



Present value

1 1 ⋯

1 =

=1

== ×

0 1 2 T-1 T ……

3

−



Important short cuts

•

Annuity• Perpetuity• Growing annuity/perpetuity• Delayed annuity/perpetuity

*** Do not just memorize the formulas!

*** Instead, use the following math fact!



The sum of a geometric series

•

Geometric sequence• First term ( )• Common ratio ( )

• How to calculate its sum• Identify its first term and common ratio .•

Identify the number of terms (N) you add .

, , , , , …,

a×11

( ≠ 1)



Examples

•

Examples• 2 4 8 16 32 ⋯ 1024•

⋯

• 1 .

.

. ⋯ .

First term

# of termsa×11



Infinite sum•

We can only calculate this when < <

• Therefore

•

Examples• ⋯• 1

.

. . ⋯

a×11 Approach

to zero

1



Annuity

• Definition• A stream of equal cash flows paid at regular intervals

• Here, assume that the first cash flow arrives at the end ofthe first period !



PV of Annuity

• A geometric sequence!• First term +• Common ratio

+• # of terms

1 ×1 111 11

× 1 1 1

Annuity factor (AF)



Example

•

Somebody tells you that she has a 10%, 30 yearmortgage and her annual payment is $6,000.

•

What is the value of her mortgage? (Assumefor simplicity, yearly mortgage payments .)



Answer



Perpetuity

•

Cash flows last forever!

11 1

1



History of perpetuity•

This is an odd-looking cash flow stream!• Yes, but nevertheless exists in the real world .

• The Napoleonic wars• The British government build up an enormous debt in termsof the values of the day, which consisted of several different

issues of securities• They decided to consolidate this debt by making a single

issue of perpetuities, i.e. bonds with no maturity date, which

promises to pay a fixed sum every year forever.• They were called “ consols .”

• Almost 200 years later these are still going strong!



Example

•

Suppose there is a consol paying £5 per yearand the interest rate is 10%. What is the valueof the consol?



Growing annuity

1 ×1 111 11

× 1 1 11



Example

•

Suppose you have just won the first prize in alottery.

• The lottery offers you two possibilities forreceiving your prize.

• (1) Receive a payment of $10,000 at the end of the year,and then, for the next 15 years this payment will berepeated, but it will grow at a rate of 5%.

•

(2) Receive $100,000 right now.

• If the discount rate is 12%, which of the two

possibilities would you take?



Answer • First option

• 10,000• 0.12 0.05 0.07• 16• $10000×

.

... $91,989.41

• Second option• $100,000

• You do not need to memorize the formula.•

You can still use the formula in slide #6!



Growing perpetuity

11 11



Example

•

What is the value of a growing perpetuity thatpays every year. Assume its growth rate is 1%and the discount rate is 11%. The first paymentoccurs today and is $100.



Answer

• $. − . $1,000(Incorrect)

• What is the right answer?

PV 100 100×1.011.11 100×1.011.11 ⋯ −.

. $1,110

• Another way• The sum of $100 today and a growing perpetuity with C = $101 , r =

11%, and g = 1%.

100 101

0.11 0.01 $1,110



Delayed annuity/perpetuity

•

PV of an asset that pays cash flows beginning3 years from now.

0 1 2 3 4 5

0 0 C C C

...

(1) Calculate the PV as of year 2 (using the formulas)

(2) Discount the result for two more years!



Example

•

What is the present value of the following cashflow stream?

• Year 01 to 10: $100• Year 11 to 20: $200• From year 21: $300

• Assume the 10% discount rate.



Answer 1

•

Method 1

• PV1 100×

.

. . 614.46

• PV2 . ×200× .

. . 473.80

• 3 . × . 445.93

• 1534.19



Answer 2

•

Method 2: As the sum of three perpetuities

• A perpetuity with $100 payment•

A 10-year delayed perpetuity with $100 payment• A 20-year delayed perpetuity with $100 payment

• . . × .

. × . $1534.19



Compounding frequency

•

So far, we have assumed that we are dealingwith compounding only once a year.• But what happens when the compounding is done more

than once in a year?



Different rates

•

Example• Suppose that a bank offers a stated annual interest rate

(or SAIR) of 8%, compounded semi-annually.• This means that 4% is paid twice a year.

• After 6 months, $100 $100(1.04) = 104• After another 6 months, $104 $104(1.04) = $108.16

• Here 4% is called a period rate .

• Note that $108.16>$108• 8.16% effective annual interest rate (or EAR)



SAIR• Notation

• SAIR =• Compounding frequency =• Period rate =

• Suppose that the SAIR ( ) is 12%.• Annual ( m 1) 12% period rate• Semi-annually ( m 2) 6% period rate• Quarterly (

m 4) 3% period rate

• Monthly ( m 12) 1% period rate

• This rate is also called an annual percentage rate(or APR ).



EAR•

Year-1 balance if we deposit $1 today$1× 1•

Definition of EAR• The rate that, when compounded annually, produces the

same return as , when is compounded m times ayear.

1 1• This implies

1 1



Example

•

18% SAIR

• Annually EAR = 18%• Semi-annually EAR = 18.81%• Quarterly EAR = 19.25%• Monthly EAR = 19.56%

• Most credit cards use monthly or daily!• Daily (m = 365) EAR = 19.7164%



Continuous compounding

•

What if (m → ∞)?• “continuously” compounding

• There is no actual such thing as compoundinginterest.

• Mathematical abstraction (what is the upper bound?)• Mathematical convenience

lim→ 1

Where

2 718282an irrational number.



Example

•

18% SAIR with continuous compounding

• $1 deposit . $1.197217•

EAR = 19.7217%

Lecture_03-04_Time Value of Money 2

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