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%. 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

Discounted Cash Flow Analysis(Time Value of Money)

  • Future value

  • Present value

  • Rates of return


Discounted cash flow analysis time value of money

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

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

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

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

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.


Discounted cash flow analysis time value of money

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.


Discounted cash flow analysis time value of money

After 3 years

FV3 = PV(1 + i)3

= 100(1.10)3

= $133.10.

In general,

FVn = PV(1 + i)n.


Discounted cash flow analysis time value of money

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

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


Discounted cash flow analysis time value of money

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

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.


Discounted cash flow analysis time value of money

Graphical Illustration:

FV

2

3.8

1

Years

0

1

2

3

4


Ordinary annuity and annuity due

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

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

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

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


Pvifa i n and fvifa i n

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 due1

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 the annuity were an annuity due

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 cash flow stream

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

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.


Discounted cash flow analysis time value of money

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.


Discounted cash flow analysis time value of money

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.


Discounted cash flow analysis time value of money

We will deal with 3

different rates:

iNom = nominal, or stated, or

quoted, rate per year.

iPer= periodic rate.

EAR= EFF% = effective

annual rate.


Discounted cash flow analysis time value of money

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

  • Examples:

    8%, Quarterly interest

    8%, Daily interest


Discounted cash flow analysis time value of money

  • 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%.


Discounted cash flow analysis time value of money

  • 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.


Discounted cash flow analysis time value of money

  • 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

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?


Discounted cash flow analysis time value of money

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%


Discounted cash flow analysis time value of money

  • 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.


Discounted cash flow analysis time value of money

How do we find EFF% for a

nominal rate of 10%, compounded

semiannually?


Ear eff of 10

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 be equal to the nominal rate

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

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.


Discounted cash flow analysis time value of money

Used in calculations,

shown on time lines.

iPer:

If iNom has annual compounding,

then iPer = iNom/1 = iNom.


Discounted cash flow analysis time value of money

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

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.


Discounted cash flow analysis time value of money

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


Discounted cash flow analysis time value of money

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

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


Compound each cf

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.


Discounted cash flow analysis time value of money

b. The cash flow stream is an annual

annuity whose EFF% = 10.25%.


What s the pv of this stream

What’s the PV of this stream?

0

1

2

3

5%

100

100

100

90.70

82.27

74.62

247.59


Amortization

Amortization

Construct an amortization schedule

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

with 3 equal payments.


Step 1 find the required 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 payments1

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

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

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.


Discounted cash flow analysis time value of money

BEGPRINEND

YRBALPMTINTPMTBAL

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

269840270332366

3366402373660

TOT1,206.34206.341,000

Interest declines. Tax implications.


Discounted cash flow analysis time value of money

$

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.


Discounted cash flow analysis time value of money

  • 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.


Discounted cash flow analysis time value of money

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.)


Discounted cash flow analysis time value of money

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.


Discounted cash flow analysis time value of money

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?


Discounted cash flow analysis time value of money

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


Discounted cash flow analysis time value of money

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.


Discounted cash flow analysis time value of money

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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