Comprehensive Guide to Financial Mathematics: Formulas, Decision Trees, and Annuity Models

Decision Framework for Financial Mathematics

• Primary Branching Criterion:

  • Evaluate whether the financial scenario involves multiple payments/deposits or a single, one-time investment.

• Branch 1: One-Time Investments (\$ \rightarrow \):

  • Applies when there are no recurring deposits or disbursements; an initial principal is invested or borrowed once.

  • Simple Interest: Selected when the problem explicitly specifies the phrase "simple interest".

    • Characterized by linear growth.

    • Capital flow: Single lump sum to single final amount (\$ \rightarrow \).

  • Compound Interest: Selected when the problem contains the phrase "Compounding $x$".

    • Periodic Compounding: Applied when compounding occurs over distinct recurring intervals. Indicated by terms such as "annually", "semi-annually", "quarterly", "monthly", or "daily".

    • Continuous Compounding: Applied when compounding occurs continuously without discrete intervals. Indicated by the term "continuously".

• Branch 2: Multiple Payments or Deposits (Annuities):

  • Applies when equal payments ($PMT$) are made repeatedly at regular time intervals.

  • The formula selection depends on the timing of the principal balance (the question: "When is the Money?"):

    • Future Value Annuity (0 \rightarrow \):

      • The money accumulates into the future.

      • The balance begins at zero and grows toward a accumulated target (0 \rightarrow \).

      • Key identifying terms in word problems include: "future goal", "in $x$ years", and "savings".

    • Present Value Annuity ($\$ \rightarrow 0$):

      • The lump-sum pool of money is held in the present ("We have the money now").

      • The account begins with a principal balance and is systematically reduced to zero ($\$ \rightarrow 0$).

      • Key identifying terms in word problems include: "loan", "borrow", "pay off", "amortize", "withdrawals", and "retirement income".

Variable Definitions and Compounding Frequencies

• Core Financial Variables:

  • $I$: Total dollar amount of interest earned or owed.

  • $P$: Initial principal balance invested or borrowed.

  • $r$: Annual nominal interest rate (expressed as a decimal in calculations).

  • $t$: Time duration measured in years.

  • $SI$: Accumulated total value in a Simple Interest account (interchangeable with accumulated amount $A$).

  • $CI$: Accumulated total value in a Compound Interest account (interchangeable with accumulated amount $A$).

  • $m$: Number of compounding periods occurring within one calendar year.

  • $FV$: Future value of an annuity account at the conclusion of the term.

  • $PMT$: Fixed dollar value of each recurring regular payment.

  • $PV$: Present value of an annuity account at the inception of the term.

  • $UB$: Unpaid balance remaining on a loan or amortized account at an intermediate stage.

• Amortization-Specific Variable:

  • $p$: Time elapsed in years corresponding to the exact number of payments completed.

• Standard Compounding Frequencies ($m$ Values):

  • Annually: Compounded $1$ time per year ($m = 1$).

  • Semiannually: Compounded $2$ times per year ($m = 2$).

  • Quarterly: Compounded $4$ times per year ($m = 4$).

  • Monthly: Compounded $12$ times per year ($m = 12$).

  • Daily: Compounded $365$ times per year ($m = 365$).

Simple Interest Formulations

• Core Concepts:

  • Simple interest is calculated exclusively on the original principal once, with terms usually established at the beginning.

  • Models transitions where a single lump sum changes into another single lump sum (\$ \rightarrow \).

  • Relationship behavior is strictly linear; algebraic evaluation proceeds by directly isolating the missing variable.

• Governing Equations:

  • Interest Earned/Owed:         $I = P r t$

  • Total Accumulated Balance (Additive Form):         $SI = A = P + P r t$

  • Total Accumulated Balance (Factored Form):         $SI = A = P(1 + r t)$

Compound Interest Formulations

• Core Concepts:

  • Applies to a single investment or loan balance that grows exponentially over time by repeatedly accruing interest on both the principal and prior accumulated interest.

  • Represents a one-time investment transformation (\$ \rightarrow \).

• Periodic Compounding:

  • Accumulated amount over discrete compounding periods:         $CI = A = P\left(1 + \frac{r}{m}\right)^{m t}$

  • Solving for the total time duration ($t$) using natural logarithms:         $t = \frac{\ln\left(\frac{CI}{P}\right)}{\ln\left(1 + \frac{r}{m}\right)}$         (Note: When solving for time in years with a compounding frequency $m$, the total number of periods is given by $m t = \frac{\ln\left(\frac{CI}{P}\right)}{\ln\left(1 + \frac{r}{m}\right)}$)

• Continuous Compounding:

  • Accumulated amount when interest compounds instantaneously at every moment:         $CI = A = P e^{r t}$

Annuity Formulations: Future Value and Present Value

• Future Value ($FV$) of an Annuity:

  • Represents accounts built over time through repeated equal installments ($PMT$) starting from zero balance (0 \rightarrow \).

  • Accumulated Future Value Formula:         $FV = PMT \left[ \frac{\left(1 + \frac{r}{m}\right)^{m t} - 1}{\frac{r}{m}} \right]$

  • Solving for Total Number of Payment Periods ($m t$):         $m t = \frac{\ln\left(\frac{FV\left(\frac{r}{m}\right)}{PMT} + 1\right)}{\ln\left(1 + \frac{r}{m}\right)}$

• Present Value ($PV$) of an Annuity:

  • Represents an existing lump sum (such as a loan or retirement fund) that is paid down or systematically depleted to zero through regular recurring payments ($\$ \rightarrow 0$).

  • Present Value Formula:         $PV = PMT \left[ \frac{1 - \left(1 + \frac{r}{m}\right)^{-m t}}{\frac{r}{m}} \right]$

  • Solving for Total Number of Payment Periods ($m t$):         $m t = -\frac{\ln\left(1 - \frac{PV\left(\frac{r}{m}\right)}{PMT}\right)}{\ln\left(1 + \frac{r}{m}\right)}$

Loan Amortization and Unpaid Balance

• Definition and Application:

  • The unpaid balance ($UB$) calculates the remaining principal debt (the effective $PV$) midway through a loan's repayment term after a designated number of payments have already been executed.

• Explicit Unpaid Balance Formula:

  • Uses total scheduled payment periods ($m t$) and already completed payment periods ($m p$):         $UB = PMT \left[ \frac{1 - \left(1 + \frac{r}{m}\right)^{-(m t - m p)}}{\frac{r}{m}} \right]$

• Alternative Calculation Method:

  • Instead of evaluating $-(m t - m p)$, apply the standard Present Value ($PV$) formula directly, setting the negative exponent to the exact number of remaining payments left to be made on the loan.

Course Policy and Institutional Context

• Exam Regulation:

  • These reference formulas and summary sheets are strictly designated for study and homework (HW) use only.

  • This reference material is NOT LEGAL for use on any MATH 1330 examination.

• Institutional Attribution:

  • Authored by Dr. Jianna Davenport at Texas Tech University.