Interest Problem Solving: Formulas, Notation, and Key Concepts

Effective rate of interest (iₜ)

The interest rate for period t, calculated as interest divided by the beginning-of-period value.

Effective rate of discount (dₜ)

The rate of what for period t, calculated as interest divided by the end-of-period value.

Discount factor (vₜ)

The factor that moves value from time t back to time t−1.

Force of interest (δₜ)

The instantaneous rate of interest at time t.

Effective interest formula

iₜ = [a(t) − a(t−1)] / a(t−1) for calculating effective interest.

Effective discount formula

dₜ = [a(t) − a(t−1)] / a(t) for calculating effective discount.

Interest ↔ discount relationship

iₜ = dₜ/(1−dₜ) and dₜ = iₜ/(1+iₜ) for converting between interest and discount rates.

Compound interest formula

a(t) = (1+i)ᵗ, representing the future value of an investment under

Simple interest formula

a(t) = 1 + it, representing the future value of an investment

Nominal-rate equivalence formula

[1 + i⁽ᵐ⁾/m]ᵐ = [1 − d⁽ⁿ⁾/n]⁻ⁿ, relating nominal rates to effective rates.

Forced interest relationship

δₜ = a′(t)/a(t), relating force of interest to the rate of increase in accumulated value.

Accumulation with force expression

a(t₁,t₂) = exp(∫ₜ₁ᵗ₂ δₜ dt), representing accumulation of value using force of interest.

Discounting with force expression

a⁻¹(t₁,t₂) = exp(−∫ₜ₁ᵗ₂ δₜ dt), expressing the

Compound-rate to force link

δ = ln(1+i) and i = eᵟ − 1, where

Core equation of value

Sets the time value of inflows equal to the time value of outflows.

Common trap: Mixing d and i

Recognizing that d is not equivalent to i, with v = 1−d and v = 1/(1+i).

Common trap: Understanding nominal rates

The monthly rate is i⁽ᵐ⁾/m, not the per-period effective rate.

Common trap: Using wrong interest type

Avoid using what formula when compounding is applied.

Core concept of time value

Understanding that accumulation moves cash flows forward and discounting moves them backwards.

Principal

Amount originally invested.

Length

Time the money remains invested.

Rate

Interest/yield rate that determines time value.

Accumulated value

Value of the principal at the ending time.

Yield rate

Rate that makes values at different times equivalent.

Basic problem of interest

Given any three of principal, length, rate, and accumulated value, solve for the fourth.

Equation of value

Accumulate or discount every payment to one comparison date for analysis.

Time value

Depends on the elapsed time since a past payment or the remaining time before a future payment.

Comparison date

Under compound interest, any date can be chosen as a reference for comparison of cash flows.

Steps to solve interest problems

1. Draw a timeline for cash flows.

2. Choose a comparison date.

3. Move cash flows to that date using accumulation or discount factors.

4. Set sides equal and solve for the unknown quantity.

Common trap: Equating amounts

Equating different time amounts without proper adjustment to a common time frame.

Common trap: Changing comparison dates

Changing comparison dates during the equation solving process.

Common trap: Misunderstanding yield rate

Mistaking the yield rate for a separate cash flow rather than a rate that ensures equivalence.