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Discounted Cash Flow Analysis (Time Value of Money). Future value Present value Rates of return. Time lines show timing of cash flows. 0. 1. 2. 3. i%. CF 0. CF 1. CF 2. CF 3.

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Discounted Cash Flow Analysis (Time Value of Money)

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Discounted Cash Flow Analysis(Time Value of Money)

  • Future value

  • Present value

  • Rates of return


Time lines show timing of cash flows.

0

1

2

3

i%

CF0

CF1

CF2

CF3

Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1, or the beginning of Period 2; and so on.


Time line for a $100 lump sum due at the end of Year 2.

0

1

2 Years

i%

100


Time line for an ordinary annuity of $100 for 3 years.

0

1

2

3

i%

100

100

100


Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3.

0

1

2

3

i%

-50

100

75

50


What’s the FV of an initial$100 after 3 years if i = 10%?

0

1

2

3

10%

100

FV = ?

Finding FVs is compounding.


After 1 year

FV1= PV + INT1 = PV + PV(i)

= PV(1 + i)

= $100(1.10)

= $110.00.

After 2 years

FV2 = FV1(1 + i)

= PV(1 + i)2

= $100(1.10)2

= $121.00.


After 3 years

FV3 = PV(1 + i)3

= 100(1.10)3

= $133.10.

In general,

FVn = PV(1 + i)n.


PVIFi,n and FVIFi,n

  • FVIFi,n = (1+i)n

    EX. i=10%, n=7

    = (1+.1)7

    =1.94871

  • PVIFi,n = 1/(1+i)n


What’s the PV of $100 due in 3 years if i = 10%?

Finding PVs is discounting, and it’s the reverse of compounding.

0

1

2

3

10%

PV = ?

100


Solve FVn = PV(1 + i )n for PV:

PV = = FVn( ).

n

FVn

(1 + i)n

1

1 + i

PV = $100( ) = $100(PVIFi, n)

= $100(0.7513) = $75.13.

1

1.10

3


If sales grow at 20% per year, how long before sales double?

Solve for n:

FVn= 1(1 + i)n

2= 1(1.20)n

(1.20)n= 2

n ln(1.20)= ln 2

n(0.1823)= 0.6931

n= 0.6931/0.1823 = 3.8 years.


Graphical Illustration:

FV

2

3.8

1

Years

0

1

2

3

4


ordinary annuity and annuity due

  • An ordinary or deferred annuity consists of a series of equal payments made at the end of each period.

  • An annuity due is an annuity for which the cash flows occur at the beginning of each period.

  • Annuity due value=ordinary due value x (1+r)


What’s the difference between an ordinary annuity and an annuity due?

Ordinary Annuity

0

1

2

3

i%

PMT

PMT

PMT

Annuity Due

0

1

2

3

i%

PMT

PMT

PMT


What’s the FV of a 3-year ordinary annuity of $100 at 10%?

0

1

2

3

10%

100

100

100

110

121

FV = 331


What’s the PV of this ordinary annuity?

0

1

2

3

10%

100

100

100

90.91

82.64

75.13

248.69 = PV


PVIFAi,n and FVIFAi,n

  • PVIFAi,n = [1 - (1+i)-n ] / i

    • Ex. i=10%, n=3

    • = [1 - (1+0.1)-3 ] / 0.1

    • = 2.48685

  • FVIFAi,n = [(1+i)n – 1] / i


Ordinary Annuity and Annuity Due

Ordinary Annuity

PVAn (OA)= PMT(PVIFAi,n)

FVAn (OA)= PMT(FVIFAi,n)

Annuity Due

PVAn (AD)= PMT(PVIFAi,n)(1+i)

FVAn (AD)= PMT(FVIFAi,n)(1+i)


Find the FV and PV if theannuity were an annuity due.

0

1

2

3

10%

100

100

100


What is the PV of this uneven cashflow stream?

0

1

2

3

4

10%

100

300

300

-50

90.91

247.93

225.39

-34.15

530.08 = PV


What interest rate would cause $100 to grow to $125.97 in 3 years?

$100 (1 + i )3 = $125.97.

(1+i)3 = 1.2597

(1+i) = 1.07999

i = 8% approx.


Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated i% constant? Why?

LARGER!If compounding is more frequent than once a year--for

example, semiannually, quarterly,

or daily--interest is earned on interest more often.


0

1

2

3

10%

100

133.10

Annually: FV3 = 100(1.10)3 = 133.10.

0

1

2

3

4

5

6

0

1

2

3

5%

100

134.01

Semiannually: FV6 = 100(1.05)6 = 134.01.


We will deal with 3

different rates:

iNom = nominal, or stated, or

quoted, rate per year.

iPer= periodic rate.

EAR= EFF% = effective

annual rate.


  • iNom is stated in contracts. Periods per year (m) must also be given.

  • Examples:

    8%, Quarterly interest

    8%, Daily interest


  • Periodic rate = iPer = iNom/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.

  • Examples:

    8% quarterly: iPer = 8/4 = 2%.

    8% daily (365): iPer = 8/365 = 0.021918%.


  • Effective Annual Rate (EAR = EFF%):

    The annual rate which causes PV to grow to the same FV as under multiperiod compounding.

    Example: EFF% for 10%, semiannual:

    FV=(1.05)2 = 1.1025.

    EFF%=10.25% because

    (1.1025)1 = 1.1025.

    Any PV would grow to same FV at 10.25% annually or 10% semiannually.


  • An investment with monthly compounding is different from one with quarterly compounding. Must put on EFF% basis to compare rates of return.

  • Banks say “interest paid daily.” Same as compounded daily.


An Example: Effective Annual Rates

  • Suppose you’ve shopped around and come up with the following three rates:

  • Bank A: 15% compounded daily

  • Bank B: 15.5% compounded quarterly

  • Bank C: 16% compounded annually

  • Which of these is the best if you are thinking of opening a savings account?


Bank A is compounding every day. = 0.15/365

= 0.000411

At this rate, an investment of $1 for 365 periods would grow to ($1x1.000411365)= $1.1618

So, EAR = 16.18%

Bank B is paying 0.155/4 = 0.03875 or 3.875% per quarter.

At this rate, an investment of $1 of 4 quarters would grow to: ($1x1.038754)= 1.1642

So, EAR = 16.42%


  • Bank C is paying only 16% annually.

    In summary, Bank B gives the best offer to savers of 16.42%.

    The facts are (1) the highest quoted rate is not necessarily the best and (2) the compounding during the year can lead to a significant difference between the quoted rate and the effective rate.


How do we find EFF% for a

nominal rate of 10%, compounded

semiannually?


EAR = EFF% of 10%

EARAnnual= 10%.

EARQ=(1 + 0.10/4)4 - 1= 10.38%.

EARM=(1 + 0.10/12)12 - 1= 10.47%.

EARD(360)=(1 + 0.10/360)360 - 1= 10.52%.


Can the effective rate ever beequal to the nominal rate?

  • Yes, but only if annual compounding is used, i.e., if m = 1.

  • If m > 1, EFF% will always be greater than the nominal rate.


When is each rate used?

iNom:

Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines unless compounding is annual.


Used in calculations,

shown on time lines.

iPer:

If iNom has annual compounding,

then iPer = iNom/1 = iNom.


EAR = EFF%:

Used to compare returns on investments with different compounding patterns.

Also used for calculations if dealing with annuities where payments

don’t match interest compounding periods.


FV of $100 after 3 years under 10% semiannual compounding? Quarterly?

mn

i

Nom

FV

=

PV

1 .

+

n

m

2x3

0.10

FV

=

$100

1

+

3S

2

= $100(1.05)6 = $134.01.

FV3Q = $100(1.025)12 = $134.49.


What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semi-annually?

4

5

0

1

2

3

6 6-mos.

periods

5%

100

100

100


  • Payments occur annually, but compounding occurs each 6 months.

  • So we can’t use normal annuity valuation techniques.


Compound Each CF

0

1

2

3

4

5

6

5%

100

100.00

100

110.25

121.55

331.80

FVA3= 100(1.05)4 + 100(1.05)2 + 100

= 331.80.


b. The cash flow stream is an annual

annuity whose EFF% = 10.25%.


What’s the PV of this stream?

0

1

2

3

5%

100

100

100

90.70

82.27

74.62

247.59


Amortization

Construct an amortization schedule

for a $1,000, 10% annual rate loan

with 3 equal payments.


Step 1: Find the required payments.

0

1

2

3

10%

-1000

PMT

PMT

PMT

3 10 -1000 0

INPUTS

N

I/YR

PV

FV

PMT

OUTPUT

402.11


Step 1: Find the required payments.

PVAn (OA)= PMT(PVIFAi,n)

PMT=PVA3 /(PVIFA10%,3)

=1000/2.4869

= 402.107


Step 2: Find interest charge for Year 1.

INTt= Beg balt (i)

INT1= 1,000(0.10) = $100.

Step 3: Find repayment of principal in

Year 1.

Repmt = PMT - INT

= 402.11 - 100

= $302.11.


Step 4: Find ending balance after Year 1.

End bal= Beg bal - Repmt

= 1,000 - 302.11 = $697.89.

Repeat these steps for Years 2 and 3

to complete the amortization table.


BEGPRINEND

YRBALPMTINTPMTBAL

1$1,000$402$100$302$698

269840270332366

3366402373660

TOT1,206.34206.341,000

Interest declines. Tax implications.


$

402.11

Interest

302.11

Principal Payments

0

1

2

3

Level payments. Interest declines because outstanding balance declines. Lender earns

10% on loan outstanding, which is falling.


  • Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, etc. They are very important!

  • Financial calculators (and spreadsheets) are great for setting up amortization tables.


On January 1 you deposit $100 in an account that pays a nominal interest rate of 10%, with daily compounding (365 days).

How much will you have on October 1, or after 9 months (273 days)? (Days given.)


iPer= 10.0% / 365

= 0.027397% per day.

0

1

2

273

0.027397%

FV=?

-100

273

(

)

FV

=

$100

1.00027397

273

(

)

=

$100

1.07765

=

$107.77.

Note: % in calculator, decimal in equation.


You are offered a note which pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank which pays a 7.0% nominal rate, with 365 daily compounding, which is a daily rate of 0.019178% and an EAR of 7.25%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless.

Should you buy it?


iPer =0.019178% per day.

0

365

456 days

-850

1,000

2 Ways to Solve:

1. Greatest future wealth: FV

2. Greatest wealth today: PV


1. Greatest Future Wealth

Find FV of $850 left in bank for

15 months and compare with

note’s FV = $1000.

FVBank=$850(1.00019178)456

=$927.67 in bank.

Buy the note: $1000 > $927.67.


2. Greatest Present Wealth

Find PV of note, and compare

with its $850 cost:

PV=$1000(1.00019178)456

=$916.27.


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