Discounted Cash Flow Analysis (Time Value of Money) Future value Present value Rates of return

Time lines show timing of cash flows. CF 0 CF 1 CF 3 CF i% 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 Years i%

Time line for an ordinary annuity of $100 for 3 years i%

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

What’s the FV of an initial $100 after 3 years if i = 10%? FV = ? % 100 Finding FVs is compounding.

After 1 year FV 1 = PV + INT 1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $ After 2 years FV 2 = FV 1 (1 + i) = PV(1 + i) 2 = $100(1.10) 2 = $

FV 3 = PV(1 + i) 3 = 100(1.10) 3 = $ In general, FV n = 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 % PV = ?

PV = = FV n ( ). Solve FV n = PV(1 + i ) n for PV: PV = $100 ( ) = $100(PVIF i, n ) = $100(0.7513) = $ FV n (1 + i) n i n

If sales grow at 20% per year, how long before sales double? Solve for n: FV n = 1(1 + i) n 2= 1(1.20) n (1.20) n = 2 n ln (1.20)= ln 2 n(0.1823)= n= / = 3.8 years.

Graphical Illustration: 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 0123 i% PMT 0123 i% PMT Annuity Due Ordinary Annuity

What’s the FV of a 3-year ordinary annuity of $100 at 10%? % FV = 331

What’s the PV of this ordinary annuity? % = PV

Find the FV and PV if the annuity were an annuity due % 100

300 3 What is the PV of this uneven cash flow stream? % = PV

What interest rate would cause $100 to grow to $ in 3 years? $100 (1 + i ) 3 = $

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.

% % Annually: FV 3 = 100(1.10) 3 = Semiannually: FV 6 = 100(1.05) 6 =

We will deal with 3 different rates: i Nom = nominal, or stated, or quoted, rate per year. i Per = periodic rate. EAR= EFF% = effective annual rate.

i Nom is stated in contracts. Periods per year (m) must also be given. Examples: 8%, Quarterly interest 8%, Daily interest

Periodic rate = i Per = i Nom /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: i Per = 8/4 = 2%. 8% daily (365): i Per = 8/365 = %.

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 = EFF%=10.25% because (1.1025) 1 = 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 = At this rate, an investment of $1 for 365 periods would grow to ($1x )= $ So, EAR = 16.18% Bank B is paying 0.155/4 = or 3.875% per quarter. At this rate, an investment of $1 of 4 quarters would grow to: ($1x )= 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% EAR Annual = 10%. EAR Q =( /4) 4 - 1= 10.38%. EAR M =( /12) = 10.47%. EAR D(360) =( /360) = 10.52%.

Can the effective rate ever be 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? i Nom : 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 i Nom has annual compounding, then i Per = i Nom /1 = i Nom. i Per :

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? = $100(1.05) 6 = $ FV 3Q = $100(1.025) 12 = $ FV = PV1.+ i m n Nom mn FV = $ S 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? % mos. periods 100

Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques.

Compound Each CF % FVA 3 = 100(1.05) (1.05) =

b. The cash flow stream is an annual annuity whose EFF% = 10.25%.

What’s the PV of this stream? %

Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

Step 1: Find the required payments. PMT % INPUTS OUTPUT NI/YRPVFV PMT

Step 2: Find interest charge for Year 1. INT t = Beg bal t (i) INT 1 = 1,000(0.10) = $100. Step 3: Find repayment of principal in Year 1. Repmt = PMT - INT = = $

Step 4: Find ending balance after Year 1. End bal= Beg bal - Repmt = 1, = $ Repeat these steps for Years 2 and 3 to complete the amortization table.

Interest declines. Tax implications. BEGPRINEND YRBALPMTINTPMTBAL 1$1,000$402$100$302$ TOT1, ,000

$ Interest Level payments. Interest declines because outstanding balance declines. Lender earns 10% 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.)

i Per = 10.0% / 365 = % per day. FV=? % -100 Note: % in calculator, decimal in equation.   FV = $ = $ = $

INPUTS OUTPUT N I/YRPVFV PMT i Per =i Nom /m =10.0/365 = % 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 = 638 days. How much will be in your account at maturity? Answer:Override N = 273 with N = 638. FV = $

i Per = % per day. FV = days -100 FV=$100( /365) 638 =$100( ) 638 =$100(1.1910) =$

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 % 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: FV 2. Greatest wealth today: PV 3. Highest rate of return: Highest EFF% i Per = % per day. 1, days -850

1. Greatest Future Wealth Find FV of $850 left in bank for 15 months and compare with note’s FV = $1000. FV Bank =$850( ) 456 =$ in bank. Buy the note: $1000 > $

INPUTS OUTPUT NI/YRPVFV PMT Calculator Solution to FV: i Per =i Nom /m =7.0/365 = % per day. Enter i Per in one step.

2. Greatest Present Wealth Find PV of note, and compare with its $850 cost: PV=$1000( ) 456 =$

INPUTS OUTPUT NI/YRPVFV PMT 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: FV n = PV(1 + i) n 1000 = $850(1 + i) 456 Now we must solve for i. 3. Rate of Return

% per day INPUTS OUTPUT NI/YRPV FV PMT Convert % to decimal: Decimal = /100 = EAR = EFF%= ( ) = 13.89%.

Using interest conversion: P/YR=365 NOM%= (365)= EFF%=13.89 Since 13.89% > 7.25% opportunity cost, buy the note.