Discounted Cash Flow Analysis(Time Value of Money)

Future valuePresent valueRates of return

Time lines show timing of cash flows.

CF0 CF1 CF3CF2

0 1 2 3i%

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.

100

0 1 2 Years

i%

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

100 100100

0 1 2 3i%

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

1 through 3.

100 50 75

0 1 2 3i%

-50

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

FV = ?

0 1 2 310%

100

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.

FV3 = PV(1 + i)3

= 100(1.10)3

= $133.10.

In general,

FVn = PV(1 + i)n.

After 3 years

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

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

100

0 1 2 310%

PV = ?

PV = = FVn( ) .

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

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

= $100(0.7513) = $75.13.

FVn

(1 + i)n

11 + i

n

11.10

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 2n(0.1823) = 0.6931 n = 0.6931/0.1823 = 3.8 years.

Graphical Illustration:

01 2 3 4

1

2

FV

3.8

Years

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.

ordinary annuity and annuity due

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

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

PMT PMTPMT

0 1 2 3i%

PMT PMT

0 1 2 3

i%

PMT

Annuity Due

Ordinary Annuity

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

100 100100

0 1 2 310%

110

121

FV = 331

What’s the PV of this ordinary annuity?

100 100100

0 1 2 310%

90.91

82.64

75.13248.69 = PV

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

100 100

0 1 2 310%

100

300

3

What is the PV of this uneven cashflow stream?

0

100

1

300

210%

-50

4

90.91247.93225.39

-34.15530.08 = PV

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

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

Will the FV of a lump sum be larger or smaller if we compound more often, ho

lding the stated i% constant? Why?

LARGER!If compounding is more frequent than once a year--forexample, semiannually, quarterly,or daily--interest is earned on interest more often.

0 1 2 310%

0 1 2 3

5%

4 5 6

134.01

100 133.10

1 2 30

100

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

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

We will deal with 3different 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 dailyBank B: 15.5% compounded quarterlyBank C: 16% compounded annuallyWhich of these is the best if you are thinking of opening a sa

vings 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?

EFF% = 1 + i

m - 1Nom

m

= 1+ 0.10

2 - 1.0

= 1.05 - 1.0= 0.1025 = 10.25%.

2

2

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.

If iNom has annual compounding,then iPer = iNom/1 = iNom.

iPer:

EAR = EFF%:Used to compare returns on investments with different compounding patterns.

Also used for calculations if dealing with annuities where paymentsdon’t match interest compounding periods.

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

= $100(1.05)6 = $134.01.FV3Q = $100(1.025)12 = $134.49.

FV = PV 1 .+ imnNom

mn

FV = $100 1 + 0.10

23S

2x3

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?

0 1

100

2 35%

4 5 6 6-mos. periods

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

100

2 35%

4 5 6

100 100.00110.25121.55331.80

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

100

15%

2 3

100 100

90.7082.2774.62

247.59

Amortization

Construct an amortization schedulefor a $1,000, 10% annual rate loanwith 3 equal payments.

Step 1: Find the required payments.

PMT PMTPMT

0 1 2 310%

-1000

3 10 -1000 0

INPUTS

OUTPUT

N I/YR PV FVPMT

402.11

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 3to complete the amortization table.

Interest declines. Tax implications.

BEG PRIN ENDYR 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

$

0 1 2 3

402.11Interest

302.11

Level payments. Interest declines because outstanding balance declines. Lender earns10% on loan outstanding, which is falling.

Principal Payments

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.

FV=?

0 1 2 273

0.027397%

-100

Note: % in calculator, decimal in equation.

FV = $100 1.00027397 = $100 1.07765 = $107.77.

273273

273 -100 0

107.77

INPUTS

OUTPUT

N I/YR PV FVPMT

iPer = iNom/m= 10.0/365= 0.027397% per day.

Enter i in one step.Leave data in calculator.

Now suppose you leave your money in the bank for 21 months, which is 1.75 years or 273 + 365 = 638 days.

How much will be in your account at maturity?

Answer: Override N = 273 with N = 638. FV = $119.10.

iPer = 0.027397% per day.

FV = 119.10

0 365 638 days

-100

FV = $100(1 + 0.10/365)638

= $100(1.00027397)638

= $100(1.1910)= $119.10.

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?

3 Ways to Solve:

1. Greatest future wealth: FV2. Greatest wealth today: PV3. Highest rate of return: Highest EFF%

iPer = 0.019178% per day.

1,000

0 365 456 days

-850

1. Greatest Future Wealth

Find FV of $850 left in bank for15 months and compare withnote’s FV = $1000.

FVBank = $850(1.00019178)456

= $927.67 in bank.

Buy the note: $1000 > $927.67.

456 -850 0

927.67

INPUTS

OUTPUT

N I/YR PV FVPMT

Calculator Solution to FV:

iPer = iNom/m= 7.0/365= 0.019178% per day.

Enter iPer in one step.

2. Greatest Present Wealth

Find PV of note, and comparewith its $850 cost:

PV = $1000(1.00019178)456

= $916.27.

456 .019178 0 1000

-916.27

INPUTS

OUTPUT

N I/YR PV FVPMT

7/365 =

PV of note is greater than its $850 cost, so buy the note. Raises your wealth.

Find the EFF% on note and compare with 7.25% bank pays, which is your opportunity cost of capital:

FVn = PV(1 + i)n

1000 = $850(1 + i)456

Now we must solve for i.

3. Rate of Return

456 -850 0 1000

0.035646% per day

INPUTS

OUTPUT

N I/YR PV FVPMT

Convert % to decimal:

Decimal = 0.035646/100 = 0.00035646.

EAR = EFF% = (1.00035646)365 - 1 = 13.89%.

Using interest conversion:

P/YR = 365NOM% = 0.035646(365) = 13.01 EFF% = 13.89

Since 13.89% > 7.25% opportunity cost,buy the note.