Notation Decisions for Financial Mathematics for Life

The goal is to use notation that is mathematically clean, easy to understand, and still connected to standard financial notation used in textbooks, spreadsheets, and financial calculators.

1. Core Value Notation

Use

\[
\boxed{V_t=\text{value at time }t}
\]

This is preferred over using only \(PV\) and \(FV\) because it emphasizes that value is associated with a particular point in time.

For a problem running from time \(0\) to time \(n\),

\[
V_0=\text{initial value}, \qquad V_n=\text{ending value}.
\]

Traditional finance notation can then be introduced as

\[
\boxed{PV=V_0,\qquad FV=V_n}
\]

when the problem begins at time \(0\) and ends at time \(n\).

2. Why \(V\) Is a Good Choice

The letter \(V\) stands naturally for Value. It is more general than alternatives such as \(B\) for balance or \(A\) for amount.

It also works especially well for multi-stage problems:

\[
V_0 \longrightarrow V_{T_1} \longrightarrow V_{T_2}.
\]

For example, in retirement planning, \(V_{T_1}\) can be the value at retirement. It does not need to be renamed from a “future value” in the saving stage to a “present value” in the retirement stage.

This supports the general principle:

\[
\boxed{\text{Value is attached to a point in time.}}
\]

3. Core Variables

The preferred core notation is

\[
\boxed{V_0,\quad V_n,\quad PMT,\quad r,\quad g,\quad n.}
\]

Symbol	Meaning	Notes
\(V_0\)	Initial value	Equivalent to \(PV\) in standard finance notation
\(V_n\)	Ending value after \(n\) periods	Equivalent to \(FV\)
\(PMT\)	First periodic payment or cash flow	Very common financial-calculator notation
\(r\)	Return or interest rate per period	Must use the same time period as \(n\)
\(g\)	Growth rate of the periodic payment	Preferred over \(s\); \(g\) more naturally suggests “growth”
\(n\)	Number of periods	Standard and widely used

4. Change from \(s\) to \(g\)

The earlier notation used \(s\) for the growth rate of periodic payments. The preferred notation is now

\[
\boxed{g=\text{growth rate of the periodic cash flow}}
\]

because \(g\) is more intuitive and commonly associated with a growth rate.

The sequence of payments is therefore

\[
PMT,\quad PMT(1+g),\quad PMT(1+g)^2,\quad \ldots,\quad PMT(1+g)^{n-1}.
\]

5. General Cash-Flow Formula

With the preferred notation, the general formula becomes

\[
\boxed{
V_n
=
V_0(1+r)^n
+
PMT
\frac{(1+r)^n-(1+g)^n}{r-g}
}
\]

for \(r\ne g\), assuming the first periodic payment occurs at the end of period 1.

The special case \(r=g\) is

\[
\boxed{
V_n
=
V_0(1+r)^n
+
n\,PMT(1+r)^{n-1}
}
\]

6. Sign Convention for \(PMT\)

Use one consistent cash-flow convention:

\[
\boxed{
PMT>0:\text{ money added to the account}
}
\]

\[
\boxed{
PMT<0:\text{ money removed from the account}
}
\]

This makes the same formula useful for both accumulation and decumulation.

Examples:

• \(PMT>0\): retirement contributions or savings deposits
• \(PMT<0\): retirement withdrawals or loan payments

7. Important Special Cases

The general notation makes several familiar financial formulas appear as special cases.

Lump-sum growth:

\[
PMT=0
\]

so

\[
V_n=V_0(1+r)^n.
\]

Level periodic saving:

\[
g=0,\qquad PMT>0.
\]

Growing periodic saving:

\[
PMT>0,\qquad g>0.
\]

Fixed retirement withdrawals:

\[
PMT<0,\qquad g=0.
\]

Inflation-adjusted retirement withdrawals:

\[
PMT<0,\qquad g=i,
\]

where \(i\) is the inflation rate per period.

Mortgage or loan payoff:

\[
PMT<0,\qquad g=0,\qquad V_n=0.
\]

8. Multi-Stage Use of \(V_t\)

One major advantage of \(V_t\) notation is that the same value variable can be carried naturally from one stage to another.

For example:

\[
\boxed{
V_0
\longrightarrow
V_{T_1}
\longrightarrow
V_{T_2}
}
\]

where

• \([0,T_1]\) is the retirement saving stage, and
• \([T_1,T_2]\) is the retirement withdrawal stage.

The ending value of the first stage is simply the beginning value of the second stage:

\[
\boxed{
V_{T_1}^{\text{ending}}
=
V_{T_1}^{\text{beginning of retirement}}
}
\]

This is cleaner than writing

\[
FV_1=PV_2.
\]

9. Connection to Standard Financial Calculators

Students should still learn the standard financial-calculator variables:

\[
\boxed{N,\quad I/Y,\quad PV,\quad PMT,\quad FV}
\]

The book can explicitly connect them to the general notation:

\[
PV=V_0,\qquad FV=V_n.
\]

The additional variable

\[
g
\]

extends the traditional five-variable time-value-of-money framework by allowing periodic payments to grow or decline over time.

10. Final Recommended Notation

The working notation for the book is therefore:

\[
\boxed{
V_t=\text{value at time }t
}
\]

with the principal variables

\[
\boxed{
V_0,\quad V_n,\quad PMT,\quad r,\quad g,\quad n.
}
\]

This notation is intended to balance three goals:

• mathematical clarity,
• continuity across multiple financial stages, and
• easy connection to conventional \(PV\), \(FV\), and \(PMT\) notation.