FNCE90060 Chap.2 Time Value and Interest Rates
Time Value and Interest Rates
Week 2 extends single-payment discounting to repeated cash flows and quoted interest rates. A perpetuity is an infinite fixed stream, a growing perpetuity increases at a constant rate forever, and an annuity is a finite fixed stream. Their formulas assume the first payment occurs one period after valuation, so timing language controls the setup. The rate must use the same interval as the cash flow.
An APR states an annual quote plus a compounding frequency; an effective annual rate states the actual one-year growth after compounding. Translating the quote into a periodic rate connects it to monthly, half-yearly or annual payments. The chapter also emphasises that a discount rate reflects timing and risk, so matching units without matching economic exposure is incomplete.
What this chapter covers
- 01
Compounding and discounting over several periods
- 02
Fixed perpetuities
- 03
Growing perpetuities and the r greater than g boundary
- 04
Finite annuities
- 05
APR and periodic rates
- 06
Effective annual rates and opportunity cost
Worked example · free
Value monthly loan repayments
- 1Convert the quote to a monthly rate: 0.06 divided by 12 equals 0.005.
- 1Classify thirty equal end-of-month payments as an ordinary annuity.
- 1Apply the annuity factor with C equal to 650, r equal to 0.005 and n equal to 30.
- 1The present value is approximately $18,066.14, below the $19,500 undiscounted total.
Key terms
- Perpetuity
- An infinite stream of fixed payments made at regular intervals.
- Growing Perpetuity
- An infinite payment stream that grows at a constant rate each period.
- Annuity
- A finite stream of equal payments at regular intervals.
- Periodic Rate
- The interest rate applying to one cash-flow or compounding interval.
- Annual Percentage Rate
- An annual quote that must be read with its compounding frequency.
- Effective Annual Rate
- The actual percentage growth over one year after within-year compounding.
Time Value and Interest Rates FAQ
How do I decide which stream formula applies?
Read the payment rule before the numbers. A finite equal stream is an annuity, an infinite equal stream is a perpetuity, and an infinite stream with constant growth is a growing perpetuity. Then confirm when the first payment occurs.
Why can APR not be compared directly with EAR?
APR is a quoted rate tied to a compounding frequency, while EAR includes the effect of compounding across the year. Convert both to the same effective horizon before deciding which return or borrowing cost is higher.
What happens when growth is at least the discount rate?
The constant-growth perpetuity expression has no finite economic value under that assumption. It signals that the steady-state model is inappropriate, not that the asset literally has an infinite usable price.
Why is the first-payment date so important?
The standard formulas assume the first payment arrives one period after valuation. A payment today is not discounted, so moving the stream forward or backward by one period changes the entire value.
How is an annuity due related to an ordinary annuity?
An annuity due pays at the beginning of each period, one interval earlier than the matching ordinary annuity. Value the ordinary stream first, then move the entire result one period toward the valuation date. With a positive periodic rate, the annuity-due value is the ordinary-annuity value multiplied by one plus that rate.
What units should appear beside a rate conversion?
Write the quotation, compounding frequency, periodic rate and number of cash-flow periods on separate lines. For a monthly APR problem, show APR divided by twelve, label the result per month and count months in the exponent. Unit labels expose a setup that mixes an annual rate with monthly payments.
How can I check a finite annuity value?
At a positive discount rate, the present value of end-of-period positive payments must be below their undiscounted sum. It should rise when the payment increases, rise when more payments are added and fall when the required return increases. These directions catch many calculator-entry errors without reproducing the full arithmetic.
Why is EAR usually above the matching APR?
A positive APR divided into several compounding periods earns interest on earlier interest within the year. EAR records that complete one-year growth, whereas APR is the periodic rate multiplied by the number of periods. With one annual compounding period they coincide; with positive more-frequent compounding, EAR exceeds the quoted APR.
Exam move
Create a classification card with three questions: finite or infinite, fixed or growing, first payment when. Add a rate card that translates APR to periodic rate and EAR. Practise writing units beside every rate and period count. Check annuity results against the undiscounted total, and check that positive compounding makes EAR exceed the corresponding APR quote.
For growing perpetuities, label the next payment and verify that the required return exceeds growth before calculating. Redo one example as an annuity due and explain the one-period shift. When comparing quoted rates, convert both to the same horizon, retain the quotation labels and state which alternative produces the larger actual accumulation.
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