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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
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 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 $50 at the end of Years 1 through 3.$100 after 3 years if i = 10%?

0

1

2

3

10%

100

FV = ?

Finding FVs is compounding.


After 1 year $50 at the end of Years 1 through 3.

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 $50 at the end of Years 1 through 3.

FV3 = PV(1 + i)3

= 100(1.10)3

= $133.10.

In general,

FVn = PV(1 + i)n.


PVIF $50 at the end of Years 1 through 3.i,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 $50 at the end of Years 1 through 3.$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 FV $50 at the end of Years 1 through 3.n = 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? $50 at the end of Years 1 through 3.

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: $50 at the end of Years 1 through 3.

FV

2

3.8

1

Years

0

1

2

3

4


Ordinary annuity and annuity due
ordinary annuity $50 at the end of Years 1 through 3.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 $50 at the end of Years 1 through 3.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%? $50 at the end of Years 1 through 3.

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? $50 at the end of Years 1 through 3.

0

1

2

3

10%

100

100

100

90.91

82.64

75.13

248.69 = PV


Pvifa i n and fvifa i n
PVIFA $50 at the end of Years 1 through 3.i,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 $50 at the end of Years 1 through 3.

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 the $50 at the end of Years 1 through 3.annuity 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 cash $50 at the end of Years 1 through 3.flow 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.


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 compound more often, holding the stated i% constant? Why?

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 compound more often, holding the stated i% constant? Why?

different rates:

iNom = nominal, or stated, or

quoted, rate per year.

iPer = periodic rate.

EAR = EFF% = effective

annual rate.


  • i compound more often, holding the stated i% constant? Why?Nom is stated in contracts. Periods per year (m) must also be given.

  • Examples:

    8%, Quarterly interest

    8%, Daily interest


  • Periodic rate compound more often, holding the stated i% constant? Why?= 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 compound more often, holding the stated i% constant? Why?(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 example effective annual rates
An Example: Effective Annual Rates with quarterly compounding. Must put on EFF% basis to compare rates of return.

  • 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 with quarterly compounding. Must put on EFF% basis to compare rates of return. 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 with quarterly compounding. Must put on EFF% basis to compare rates of return. 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 with quarterly compounding. Must put on EFF% basis to compare rates of return.

nominal rate of 10%, compounded

semiannually?


Ear eff of 10
EAR = EFF% of 10% with quarterly compounding. Must put on EFF% basis to compare rates of return.

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 be with quarterly compounding. Must put on EFF% basis to compare rates of return.equal 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? with quarterly compounding. Must put on EFF% basis to compare rates of return.

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, with quarterly compounding. Must put on EFF% basis to compare rates of return.

shown on time lines.

iPer:

If iNom has annual compounding,

then iPer = iNom/1 = iNom.


EAR = EFF%: with quarterly compounding. Must put on EFF% basis to compare rates of return.

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.


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



Compound each cf
Compound Each CF months.

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 months.

annuity whose EFF% = 10.25%.


What s the pv of this stream
What’s the PV of this stream? months.

0

1

2

3

5%

100

100

100

90.70

82.27

74.62

247.59


Amortization
Amortization months.

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. months.

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. months.

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. months.

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 months. 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.


BEG PRIN END months.

YR BAL PMT INT PMT BAL

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

2 698 402 70 332 366

3 366 402 37 366 0

TOT 1,206.34 206.34 1,000

Interest declines. Tax implications.


$ months.

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.



On January 1 you deposit $100 in an account that pays a auto loans, business loans, retirement plans, etc. They are very important!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.)


i auto loans, business loans, retirement plans, etc. They are very important!Per = 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?


i 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.Per = 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 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.

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 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.

Find PV of note, and compare

with its $850 cost:

PV = $1000(1.00019178)456

= $916.27.


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