Unit 5 Discounted Cash Flow Valuation

Annuities, Perpetuities, APR & EAR: What is an annuity?

A series of equal cash flows paid or received at regular intervals for a fixed amount of time.
Example: $500 paid every month for 5 years.

Annuities, Perpetuities, APR & EAR: What are the TWO key characteristics of an annuity?

• The cash flows are equal.

• They occur at regular intervals for a finite/fixed number of periods.

Annuities, Perpetuities, APR & EAR: What is an ordinary annuity?

An annuity where the cash flows occur at the END of each period.

Annuities, Perpetuities, APR & EAR: What is an annuity due?

An annuity where the cash flows occur at the BEGINNING of each period.

Annuities, Perpetuities, APR & EAR: Ordinary annuity vs. annuity due

Ordinary = END of each period.
Annuity Due = BEGINNING of each period.

Annuities, Perpetuities, APR & EAR: How can you recognize an annuity due in a question?

Look for phrases such as "first payment today," "beginning of each period," or "payments in advance."

Annuities, Perpetuities, APR & EAR: How can you recognize an ordinary annuity?

Payments occur at the end of each period. If a problem doesn't specify otherwise, finance problems commonly treat an annuity as an ordinary annuity.

Annuities, Perpetuities, APR & EAR: Why is an annuity due worth more than an otherwise identical ordinary annuity?

Each payment occurs one period earlier, giving it one extra period to earn interest.

Annuities, Perpetuities, APR & EAR: ⭐ Annuity Due Value Formula

Annuity Due Value = Ordinary Annuity Value × (1 + r)

Annuities, Perpetuities, APR & EAR: What does r represent in annuity formulas?

The interest rate per period.

⭐ Future Value of an Ordinary Annuity Formula

FV = C[(1 + r)ᵗ − 1] / r

Where:
C = cash flow/payment each period
r = interest rate per period
t = number of periods

What does the future value of an annuity tell you?

How much a series of equal payments will be worth at a future point in time after earning interest.

Example: $300 deposited monthly for 30 years at a 13% APR. What are N and I/Y?

N = 30 × 12 = 360
I/Y = 13 ÷ 12 = 1.08333% per month

The time periods and interest rate must match.

Why do we use 360 periods for 30 years of monthly payments?

30 years × 12 months per year = 360 monthly periods.

Why is 13% divided by 12 in a monthly annuity problem?

Because the 13% APR must be converted into a monthly periodic rate to match monthly cash flows.

BA II Plus setup for FV of $300 monthly for 30 years at 13% APR

N = 360
I/Y = 13 ÷ 12
PV = 0
PMT = −300
CPT → FV

The slide's result is approximately $1,311,980.94.

Why might PMT be entered as negative on the BA II Plus?

Because deposits/payments represent cash flowing out of you. The resulting FV is cash flowing back to you, so it appears positive.

What is a perpetuity? ⭐ IMPORTANT DEFINITION

A series of equal cash flows that continues forever (infinitely).

Annuity vs. perpetuity ⭐

Annuity: equal payments for a finite/fixed number of periods.
Perpetuity: equal payments forever.

What word in a problem strongly suggests a perpetuity?

"Forever," "indefinitely," or wording indicating payments never end.

⭐ Perpetuity Formula

PV = C / r

What do the variables in the perpetuity formula mean?

PV = present value
C = constant cash flow per period
r = discount/interest rate per period

What value does the basic perpetuity formula calculate?

Present value, not future value.

Example: An investment pays $2,000 every year forever and the discount rate is 11.5%. What is its PV?

PV = $2,000 ÷ 0.115
= $17,391.30

Why is 11.5% entered as 0.115 in the perpetuity formula?

Percentages must be converted to decimals when using the formula directly.

What is APR (Annual Percentage Rate)? ⭐

The quoted annual interest rate. APR does not account for the effect of compounding within the year.

Another name for APR in this unit

The quoted rate.

⭐ APR Formula

APR = Periodic Rate × Number of Periods per Year

⭐ Periodic Rate Formula

Periodic Rate = APR ÷ Number of Periods per Year

If the monthly rate is 0.50%, what is the APR?

0.50% × 12 = 6.00%

If the semiannual rate is 0.50%, what is the APR?

0.50% × 2 = 1.00%

If APR = 12% with monthly compounding, what is the monthly rate?

12% ÷ 12 = 1% per month.

What does "semiannual compounding" mean?

Interest compounds 2 times per year.

What does "quarterly compounding" mean?

Interest compounds 4 times per year.

What does "monthly compounding" mean?

Interest compounds 12 times per year.

What does "daily compounding" mean in your professor's examples?

Interest compounds 365 times per year.

What is EAR (Effective Annual Rate)? ⭐

The actual annual rate earned or paid after accounting for compounding within the year.

APR vs. EAR ⭐ IMPORTANT

APR = quoted annual rate; ignores within-year compounding.
EAR = actual/effective annual rate; includes within-year compounding.

Why is EAR useful?

It lets you fairly compare investments or loans that have different compounding frequencies.

What is EAR sometimes called for a bank deposit?

APY (Annual Percentage Yield).

⭐ EAR Formula

EAR = [1 + (APR / m)]ᵐ − 1

Where m = number of compounding periods per year.

What does m mean in the EAR formula?

The number of times interest compounds per year.

What are common values for m?

Annual = 1
Semiannual = 2
Quarterly = 4
Monthly = 12
Daily = 365

When does EAR = APR? ⭐

When interest is compounded annually (m = 1).

If compounding occurs more than once per year, what is normally true about EAR and APR for a positive rate?

EAR > APR because EAR includes the effect of compounding.

Example: APR = 10%, compounded quarterly. What is EAR?

EAR = [1 + (.10/4)]⁴ − 1
= 10.381%

Example: APR = 10%, compounded monthly. What is EAR?

EAR = [1 + (.10/12)]¹² − 1
= 10.471%

With the same APR, what happens when compounding becomes more frequent?

The EAR increases.

Why does more frequent compounding increase EAR?

Interest is added more often, allowing you to earn interest on previously earned interest sooner.

If two savings accounts have different APRs AND different compounding frequencies, which rate should you compare? ⭐

Compare their EARs, not just their APRs.

Account A: 5.25% APR compounded daily. What is its EAR?

EAR = (1 + .0525/365)³⁶⁵ − 1
≈ 5.390%

Account B: 5.30% APR compounded semiannually. What is its EAR?

EAR = (1 + .0530/2)² − 1
≈ 5.370%

Why does the 5.25% daily account outperform the 5.30% semiannual account in the professor's example?

Although its APR is lower, its more frequent compounding produces a higher EAR: about 5.390% vs. 5.370%.

What is continuous compounding?

Compounding that occurs continuously rather than a fixed number of times per year.

⭐ Continuous Compounding EAR Formula

EAR = eᵠ − 1

Here q is the quoted annual rate (APR) expressed as a decimal.

What is e in the continuous-compounding formula?

A mathematical constant, approximately 2.71828, available as the eˣ function on a calculator.

If the quoted rate is 10% and compounding is continuous, what is EAR?

EAR = e^.10 − 1
≈ .10517
= 10.517%

⭐ Most important rule when working with interest rates and time periods
THE INTEREST RATE AND TIME PERIOD MUST MATCH.

Monthly periods → monthly rate.
Annual periods → annual rate.

If cash flows occur monthly but you're given an APR, what should you usually do?
Convert APR into a monthly periodic rate:

APR ÷ 12

and express N in months.

Example: 5 years with monthly payments. What should N be?

5 × 12 = 60 periods.

If payments are quarterly for 5 years, what should N be?

5 × 4 = 20 periods.

Can you divide EAR by 12 to obtain a monthly periodic rate? ⭐

NO. Your professor specifically warns against this. EAR already incorporates compounding, so EAR ÷ 12 is not the correct monthly rate.

What rate CAN you divide by the number of periods to obtain the periodic rate?

APR.

Periodic Rate = APR ÷ m.