FINC2011 Chap.3 Rates, Loans and Uneven Cash Flows
Rates, Loans and Uneven Cash Flows
This chapter turns quoted rates into comparable effective returns and applies periodic rates to loans and irregular streams. It distinguishes APR from EAR, handles monthly and quarterly compounding, introduces continuous compounding, solves level-payment loans, and reconciles each payment between interest and principal.
The recurring exam move is to match the rate period with the cash-flow period before solving for a value or payment.
What this chapter covers
- 01
APR and periodic rate conventions
- 02
Effective annual rate comparison
- 03
Continuous and discrete compounding
- 04
Level-payment loan valuation
- 05
Interest and principal decomposition
- 06
Deferred repayment structures
- 07
Uneven cash flows split into valuation blocks
Worked example · free
Solve and reconcile a monthly loan payment
- 1Convert the quote to a monthly rate: 0.078 divided by 12 equals 0.0065.
- 2Use 30 periods in the annuity payment formula to obtain a monthly payment of $662.47.
- 1First-month interest is $18,000 times 0.0065, which equals $117.00.
- 1Principal reduction is $662.47 minus $117.00, leaving $17,454.53 outstanding.
Key terms
- Annual Percentage Rate
- A quoted annual rate formed from the periodic rate multiplied by the number of compounding periods per year.
- Periodic Rate
- The interest rate applying over one payment or compounding interval.
- Amortising Loan
- A loan repaid through scheduled payments that cover periodic interest and gradually reduce principal.
- Continuous Compounding
- A convention that models interest as compounding continuously and uses an exponential growth factor.
Rates, Loans and Uneven Cash Flows FAQ
Why is APR different from EAR?
APR is a quoting convention that annualises a periodic rate without adding the within-year interest-on-interest effect. EAR measures actual growth over a year after compounding. When the APR compounds more than annually, EAR is normally higher than the quoted APR.
How is a loan payment split?
Interest equals the opening balance times the periodic rate. Principal reduction equals the fixed payment minus that interest. As the balance falls, interest falls and more of the same payment reduces principal. The closing balance reconciles opening balance plus interest minus payment.
How should uneven cash flows be valued?
Discount each cash flow separately or partition the stream into genuine patterns such as an annuity followed by a perpetuity. Calculate each block at its natural valuation date, then move all block values to the same comparison date before adding them.
Exam move
Create a conversion ladder from APR to periodic rate to EAR and use it on monthly, quarterly, and semiannual quotes. Write m for compounding frequency and n for the actual payment count so they cannot be confused. For loans, build the first two rows of an amortisation schedule by hand and reconcile each closing balance.
For deferred loans, locate the annuity value one period before the first payment and determine whether the principal must grow or be discounted to that date. For uneven streams, circle the segments that share a pattern and value those segments separately instead of forcing an average cash flow. Add a unit label to every interest rate: quoted per year, effective per year, or effective per payment period.
When comparing products, convert them to a common effective horizon and keep fees outside the rate conversion until their cash-flow dates are known. Do not obtain a monthly rate by simply dividing an effective annual rate; solve for the periodic growth factor that compounds to the annual factor. For each loan question, calculate the instalment and then prove it with a balance roll-forward.
Opening balance plus periodic interest minus payment must equal closing balance, while payment minus interest must equal principal reduction. Reconcile the identities using unrounded internal amounts. Practise balances both immediately before and immediately after a payment, because the remaining number of instalments changes across that date.
For payment holidays or balloon amounts, build dated rows rather than stretching the level-annuity formula. Include any origination fee at its actual date and compare the present value of borrower cash flows with the amount advanced. Finish rate questions by explaining which quote is cheaper or more rewarding on the common basis. Finish loan questions by explaining whether principal falls, remains level, or grows and why.
These interpretations expose a periodic-rate mistake even when the calculator output appears plausible. Practise one loan twice: first solve prospectively from the present value of remaining payments, then solve retrospectively from the original advance and payments already made. Agreement provides an independent timing check.
Record any difference caused solely by displayed rounding instead of changing the rate or period count to force equality.
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