1.2 Cash Flows: One Model, Many Problems
Chapter 2. Cash Flows: One Model, Many Problems
Most real financial decisions involve more than a single amount of money.
We deposit money regularly, make loan payments, withdraw retirement income, increase contributions over time, and combine existing balances with new cash flows.
At first, these situations may appear to require many separate formulas.
A major goal of this chapter is to show that they can often be understood through one common mathematical structure.
Many saving, borrowing, and withdrawal problems can be viewed as the combination of:
\[
\boxed{
\text{Growth of Existing Money}
+
\text{Growth of New Cash Flows}
}
\]
The chapter develops a general cash-flow model using the notation
\[
\boxed{
V_0,\quad V_n,\quad P,\quad r,\quad g,\quad n.
}
\]
This model will later become the mathematical foundation for retirement accumulation, retirement withdrawals, mortgages, annuities, and many other financial applications.
- 2.1 What Is a Cash Flow?
- 2.2 How Do Repeated Payments Accumulate?
- 2.3 One General Cash-Flow Model
- 2.4 What Happens When Payments Are Constant?
- 2.5 What Happens When Payments Grow?
- 2.6 How Do Withdrawals Fit the Same Model?
- 2.7 How Do Loans and Mortgages Fit the Same Model?
- 2.8 What Changes When Payments Occur at the Beginning of a Period?
- 2.9 Solving for Different Unknowns
- 2.10 Putting the Model Together
2.1 What Is a Cash Flow?
A cash flow is an amount of money that enters or leaves a financial account at a particular time.
A cash flow is money added to or removed from an account at a specified point in time.
Examples include:
- a monthly retirement contribution;
- a mortgage payment;
- a retirement withdrawal;
- a tuition payment;
- a recurring investment deposit;
- a pension payment.
Cash Flows on a Timeline
Suppose an account begins with value \(V_0\), and an amount \(P\) is added at the end of each period.
A simple timeline is:
\[
\begin{array}{c|c|c|c|c}
0 & 1 & 2 & \cdots & n\\
\hline
V_0 & P & P & \cdots & P
\end{array}
\]
The account value at the end of period \(n\) is denoted by \(V_n\).
We use \(P\) for a periodic payment or cash flow.
In the general growing-payment model, \(P\) denotes the cash flow occurring at the end of period 1.
Standard financial calculators and spreadsheets often use
\[
PV,\qquad FV,\qquad PMT.
\]
In this book,
\[
V_0=PV,
\qquad
V_n=FV,
\qquad
P\leftrightarrow PMT.
\]
When payments are level, \(P\) corresponds directly to the usual \(PMT\).
When payments grow, \(P\) represents the first payment in the sequence.
Our Sign Convention
To use one model for both deposits and withdrawals, we adopt a consistent sign convention.
\[
\boxed{
P>0:\text{ money added to the account}
}
\]
\[
\boxed{
P<0:\text{ money removed from the account}
}
\]
Thus:
- retirement contributions usually have \(P>0\);
- retirement withdrawals usually have \(P<0\);
- loan payments usually have \(P<0\) when viewed from the borrower’s account.
2.2 How Do Repeated Payments Accumulate?
Consider an account with no initial balance and equal payments \(P\) made at the end of each period.
Suppose the account earns a return \(r\) per period.
The first payment has the longest time to grow, while the final payment is made at time \(n\) and therefore has no additional time to grow before the ending value is measured.
The accumulated value is
\[
P(1+r)^{n-1}
+
P(1+r)^{n-2}
+
\cdots
+
P(1+r)
+
P.
\]
Factoring out \(P\),
\[
V_n
=
P\left[
(1+r)^{n-1}
+
(1+r)^{n-2}
+
\cdots
+
1
\right].
\]
The quantity in brackets is a finite geometric series.
\[
\boxed{
V_n
=
P\frac{(1+r)^n-1}{r}
}
\]
This formula assumes:
- equal payments \(P\);
- payments at the end of each period;
- a constant return \(r\) per period;
- no initial balance.
Worked Example: Saving $500 per Month
Suppose you deposit $500 at the end of each month for 10 years.
The account earns a nominal annual rate of 6% compounded monthly.
The monthly rate is
\[
r=\frac{0.06}{12}=0.005,
\]
and the number of monthly payments is
\[
n=12(10)=120.
\]
Therefore,
\[
V_{120}
=
500
\frac{(1.005)^{120}-1}{0.005}
\approx
81939.67.
\]
The account grows to approximately $81,940.
The total amount contributed is
\[
120(500)=60000.
\]
The remaining approximately $21,940 comes from investment growth.
Different payments grow for different lengths of time.
Earlier payments have more time to compound than later payments.
2.3 One General Cash-Flow Model
We now combine two ideas:
- an initial value \(V_0\) that grows through time;
- a sequence of payments that may also grow over time.
Suppose:
- the initial account value is \(V_0\);
- the first payment is \(P\), made at the end of period 1;
- payments grow at rate \(g\) per period;
- the account earns rate \(r\) per period;
- there are \(n\) periods.
The payment sequence is
\[
P,\quad
P(1+g),\quad
P(1+g)^2,\quad
\ldots,\quad
P(1+g)^{n-1}.
\]
Building the Ending Value
The initial balance grows to
\[
V_0(1+r)^n.
\]
The first payment grows for \(n-1\) periods:
\[
P(1+r)^{n-1}.
\]
The second payment is \(P(1+g)\) and grows for \(n-2\) periods:
\[
P(1+g)(1+r)^{n-2}.
\]
Continuing in this way, the final payment is
\[
P(1+g)^{n-1}.
\]
Thus,
\[
\begin{aligned}
V_n
={}&
V_0(1+r)^n\\
&+
P(1+r)^{n-1}\\
&+
P(1+g)(1+r)^{n-2}\\
&+\cdots\\
&+
P(1+g)^{n-1}.
\end{aligned}
\]
The payment terms form a geometric pattern.
After simplifying, we obtain the central cash-flow model.
For \(r\ne g\),
\[
\boxed{
V_n
=
V_0(1+r)^n
+
P
\frac{(1+r)^n-(1+g)^n}{r-g}
}
\]
where:
- \(V_0\) = initial value;
- \(V_n\) = ending value;
- \(P\) = payment at the end of period 1;
- \(r\) = return or interest rate per period;
- \(g\) = growth rate of the payment per period;
- \(n\) = number of periods.
\[
\boxed{
\text{Ending Value}
=
\text{Growth of Initial Value}
+
\text{Accumulated Value of Cash Flows}
}
\]
The Special Case \(r=g\)
The formula above has \(r-g\) in the denominator, so it cannot be used directly when \(r=g\).
In that case, the cash-flow terms simplify differently.
\[
\boxed{
V_n
=
V_0(1+r)^n
+
nP(1+r)^{n-1}
}
\]
When \(r=g\), do not substitute directly into the general fraction because both numerator and denominator become zero.
Use the special-case formula instead.
Go Deeper: Why Does the General Formula Work?
The cash-flow portion is
\[
P(1+r)^{n-1}
+
P(1+g)(1+r)^{n-2}
+
\cdots
+
P(1+g)^{n-1}.
\]
Factor out \(P(1+r)^{n-1}\):
\[
P(1+r)^{n-1}
\left[
1+
\frac{1+g}{1+r}
+
\left(\frac{1+g}{1+r}\right)^2
+
\cdots
+
\left(\frac{1+g}{1+r}\right)^{n-1}
\right].
\]
The bracketed expression is a geometric series with ratio
\[
q=\frac{1+g}{1+r}.
\]
Using
\[
1+q+q^2+\cdots+q^{n-1}
=
\frac{1-q^n}{1-q},
\]
and simplifying gives
\[
P
\frac{(1+r)^n-(1+g)^n}{r-g}.
\]
Adding the accumulated initial value \(V_0(1+r)^n\) produces the general model.
The general cash-flow model is fundamentally an application of the geometric-series formula.
2.4 What Happens When Payments Are Constant?
If every payment is the same, then the payment growth rate is
\[
g=0.
\]
Substituting \(g=0\) into the general model gives
\[
\boxed{
V_n
=
V_0(1+r)^n
+
P\frac{(1+r)^n-1}{r}
}
\]
If there is no initial balance, \(V_0=0\), then
\[
\boxed{
V_n
=
P\frac{(1+r)^n-1}{r}
}
\]
which is the standard future-value formula for an ordinary annuity.
The ordinary annuity formula is not a separate model.
It is the special case
\[
V_0=0,\qquad g=0
\]
of the general cash-flow model.
Worked Example: Existing Savings Plus Monthly Contributions
Suppose you already have $25,000 invested and add $400 at the end of each month for 15 years.
Assume a nominal annual return of 6% compounded monthly.
Here,
\[
V_0=25000,
\qquad
P=400,
\qquad
r=\frac{0.06}{12}=0.005,
\qquad
g=0,
\qquad
n=180.
\]
The model gives
\[
V_{180}
=
25000(1.005)^{180}
+
400\frac{(1.005)^{180}-1}{0.005}.
\]
This calculation separates naturally into:
\[
\boxed{
\text{Growth of Existing Savings}
+
\text{Growth of New Contributions}
}.
\]
This way of writing the model helps distinguish two sources of the final balance:
- money already accumulated before the new saving plan begins;
- new contributions made during the plan.
2.5 What Happens When Payments Grow?
Many real financial plans do not use constant payments.
For example, a person may increase retirement contributions as income rises.
Suppose the first payment is \(P\), and each later payment grows by rate \(g\).
Then the payment sequence is
\[
P,\quad
P(1+g),\quad
P(1+g)^2,\quad
\ldots.
\]
The variable \(g\) is the rate at which the periodic payment changes from one period to the next.
Worked Example: Increasing Retirement Contributions
Suppose you contribute $6,000 at the end of the first year and increase the annual contribution by 3% each year.
The account earns 7% annually for 20 years.
Assume no initial balance.
Here,
\[
V_0=0,
\qquad
P=6000,
\qquad
r=0.07,
\qquad
g=0.03,
\qquad
n=20.
\]
Therefore,
\[
V_{20}
=
6000
\frac{(1.07)^{20}-(1.03)^{20}}{0.07-0.03}.
\]
Increasing contributions can substantially increase the ending balance.
The benefit comes from both larger later contributions and the investment growth earned on earlier contributions.
Compare the same saving plan using
\[
g=0,\quad 1\%,\quad 2\%,\quad 3\%,\quad 5\%.
\]
- Predict how increasing \(g\) will affect \(V_n\).
- Calculate the ending balances.
- Compare the additional contributions with the additional investment growth.
- Explain why small annual increases may have a large long-term effect.
2.6 How Do Withdrawals Fit the Same Model?
A withdrawal is simply a negative cash flow.
Therefore, we do not need a new formula.
\[
\boxed{
\text{Saving and withdrawing use the same model.}
}
\]
The difference is the sign of \(P\).
Fixed Withdrawals
For constant withdrawals,
\[
P<0,
\qquad
g=0.
\]
The account value is
\[
V_n
=
V_0(1+r)^n
+
P\frac{(1+r)^n-1}{r}.
\]
Worked Example: Retirement Withdrawals
Suppose a retirement account begins with $500,000.
The retiree withdraws $30,000 at the end of each year, and the account earns 5% annually.
What is the balance after 10 years?
Use
\[
V_0=500000,
\qquad
P=-30000,
\qquad
r=0.05,
\qquad
g=0,
\qquad
n=10.
\]
Then
\[
V_{10}
=
500000(1.05)^{10}
–
30000\frac{(1.05)^{10}-1}{0.05}.
\]
The investment return partly offsets the withdrawals.
Whether the account grows, declines, or remains roughly stable depends on the relationship among \(V_0\), \(P\), \(r\), and \(n\).
Growing Withdrawals
Retirement withdrawals may increase over time to help maintain purchasing power.
If withdrawals grow at rate \(g\), then
\[
P<0, \qquad g>0.
\]
If the increase matches inflation, then
\[
g=i,
\]
where \(i\) is the inflation rate per withdrawal period.
If withdrawals increase at the same rate as inflation,
\[
g=i,
\]
then the withdrawals are approximately constant in purchasing-power terms.
2.7 How Do Loans and Mortgages Fit the Same Model?
A loan begins with a positive balance owed and is reduced by periodic payments.
From the borrower’s account perspective, the payments are negative cash flows.
For a fixed-payment loan,
\[
g=0,
\qquad
P<0.
\]
A fully paid loan has
\[
V_n=0.
\]
Substituting these conditions into the general level-payment model gives
\[
0
=
V_0(1+r)^n
+
P\frac{(1+r)^n-1}{r}.
\]
Solving for \(P\),
\[
\boxed{
P
=
-V_0
\frac{r(1+r)^n}{(1+r)^n-1}
}
\]
The negative sign reflects money leaving the borrower’s account.
The standard mortgage-payment formula is another special case of the same cash-flow model:
\[
\boxed{
g=0,\qquad V_n=0.
}
\]
Worked Example: A Mortgage Payment
Suppose a borrower finances $240,000 with a 30-year mortgage at a nominal annual rate of 6% compounded monthly.
Then
\[
V_0=240000,
\qquad
r=\frac{0.06}{12}=0.005,
\qquad
n=360,
\qquad
V_n=0.
\]
The monthly payment is
\[
P
=
-240000
\frac{0.005(1.005)^{360}}{(1.005)^{360}-1}.
\]
The magnitude \(|P|\) is the monthly amount paid by the borrower.
Mathematically, these situations differ mainly in their assumptions:
| Application | Typical Conditions |
|---|---|
| Saving | \(P>0\) |
| Growing saving | \(P>0,\ g>0\) |
| Withdrawal | \(P<0\) |
| Level withdrawal | \(P<0,\ g=0\) |
| Loan or mortgage | \(P<0,\ g=0,\ V_n=0\) |
2.8 What Changes When Payments Occur at the Beginning of a Period?
So far, the standard assumption has been that each payment occurs at the end of a period.
This is called an ordinary annuity timing convention.
Sometimes payments occur at the beginning of each period.
Examples may include rent, certain savings plans, or other scheduled payments.
Shifting Every Payment One Period Earlier
If every payment occurs one period earlier, every payment receives one additional period of growth.
Therefore, the accumulated cash-flow portion is multiplied by
\[
1+r.
\]
If all payments are shifted one period earlier, then
\[
\boxed{
V_n
=
V_0(1+r)^n
+
(1+r)
P
\frac{(1+r)^n-(1+g)^n}{r-g}
}
\]
for \(r\ne g\).
\[
\boxed{
\text{Earlier cash flows have more time to grow.}
}
\]
Do not automatically multiply the entire model by \(1+r\).
Only the cash-flow sequence shifts when payment timing changes.
The initial value \(V_0\) remains at time 0.
2.9 Solving for Different Unknowns
A major advantage of a unified model is that the same equation can answer many different questions.
The six main quantities are
\[
\boxed{
V_0,\quad V_n,\quad P,\quad r,\quad g,\quad n.
}
\]
If five are known, the sixth may be the quantity we want to determine.
Solving for the Ending Value
The model is already written directly for \(V_n\):
\[
V_n
=
V_0(1+r)^n
+
P
\frac{(1+r)^n-(1+g)^n}{r-g}.
\]
Solving for the Initial Value
For \(r\ne g\),
\[
\boxed{
V_0
=
\frac{
V_n
–
P\frac{(1+r)^n-(1+g)^n}{r-g}
}{(1+r)^n}
}
\]
Solving for the Payment
\[
\boxed{
P
=
\left[
V_n-V_0(1+r)^n
\right]
\frac{r-g}{(1+r)^n-(1+g)^n}
}
\]
for \(r\ne g\).
This formula can answer questions such as:
- How much should I save each period?
- How much can I withdraw?
- What loan payment will reduce the balance to zero?
Solving for \(r\), \(g\), or \(n\)
The variables \(r\), \(g\), and \(n\) appear in exponents and multiple parts of the formula.
In many cases, they cannot be isolated using simple algebra.
When a variable cannot be isolated algebraically, the same model can still be solved numerically using a calculator, spreadsheet, or Digital Math Lab.
The Digital Math Lab can treat the general model as a six-variable solver:
\[
\boxed{
V_0,\quad V_n,\quad P,\quad r,\quad g,\quad n.
}
\]
Enter five values and leave one blank.
The Lab can solve for the missing quantity.
Some variables have direct algebraic formulas.
Others may require numerical methods.
2.10 Putting the Model Together
Recognize the Structure Before Choosing a Formula
Instead of asking,
“Which formula should I memorize for this problem?”
ask:
- What is the initial value \(V_0\)?
- What ending value \(V_n\) is being considered?
- What is the first periodic cash flow \(P\)?
- Is money entering or leaving the account?
Determine the sign of \(P\). - Does the payment remain constant or change over time?
Identify \(g\). - What return or interest rate applies?
Identify \(r\). - Are the rate and period units consistent?
- When do payments occur?
End of period or beginning of period?
One Model, Several Familiar Problems
| Problem | Conditions |
|---|---|
| Lump-sum growth | \(P=0\) |
| Level saving | \(P>0,\ g=0\) |
| Growing saving | \(P>0,\ g>0\) |
| Level withdrawals | \(P<0,\ g=0\) |
| Growing withdrawals | \(P<0,\ g>0\) |
| Loan or mortgage payoff | \(P<0,\ g=0,\ V_n=0\) |
| Beginning-of-period payments | Shift cash flows one period earlier |
These are not fundamentally different mathematical problems.
They are different financial interpretations of the same cash-flow structure.
\[
\boxed{
V_n
=
V_0(1+r)^n
+
P
\frac{(1+r)^n-(1+g)^n}{r-g}
}
\]
Integrated Example: Saving and Then Withdrawing
Suppose a person saves for retirement and later begins withdrawing from the same account.
During the saving stage,
\[
P>0.
\]
During the retirement stage,
\[
P<0.
\]
The account value at the transition point is the same financial quantity:
\[
\boxed{
V_{T_1}
=
\text{ending value of the saving stage}
=
\text{starting value of the withdrawal stage}.
}
\]
The notation \(V_t\) makes multi-stage planning natural:
\[
\boxed{
V_0
\longrightarrow
V_{T_1}
\longrightarrow
V_{T_2}
}
\]
We do not need to rename \(V_{T_1}\) from a “future value” in one stage to a “present value” in the next.
It is simply the value at time \(T_1\).
Common Mistakes in Chapter 2
- Using the wrong sign for \(P\).
Deposits are positive; withdrawals or payments out are negative. - Forgetting what \(P\) represents when \(g\ne0\).
\(P\) is the first payment, not every payment. - Using inconsistent rate and payment periods.
Monthly cash flows require rates and periods expressed on a compatible monthly basis. - Ignoring payment timing.
End-of-period and beginning-of-period payments do not have the same accumulated value. - Using the \(r\ne g\) formula when \(r=g\).
Use the special-case formula instead. - Treating every familiar finance formula as unrelated.
First check whether the problem is a special case of the general cash-flow model.
Go Deeper: A General Cash-Flow View
The model in this chapter assumes a particular pattern of cash flows.
A still more general financial principle is:
\[
\boxed{
V_T
=
\sum_k
C_k(1+r)^{T-k}
}
\]
where \(C_k\) is the cash flow occurring at time \(k\), and every cash flow is moved to the common valuation date \(T\).
The formulas in this chapter arise because the cash flows follow a regular geometric pattern, allowing the sum to be simplified into a closed-form expression.
Chapter 1 introduced the principle that values must be moved to a common valuation date.
Chapter 2 applies that principle to many cash flows at once.
Digital Math Lab: Chapter Exploration
Use the general cash-flow model to explore
\[
V_0,\quad V_n,\quad P,\quad r,\quad g,\quad n.
\]
Suggested investigations:
- Compare a lump-sum investment with a regular saving plan.
- Compare level contributions with contributions that grow at 2% or 3% per year.
- Change \(P\) from positive to negative and observe how the same model changes from accumulation to withdrawal.
- Set \(V_n=0\) and explore loan-payment problems.
- Compare end-of-period and beginning-of-period payments.
- Leave one of the six variables blank and solve for it.
Use the exploration cycle
\[
\boxed{
\text{Predict}
\rightarrow
\text{Calculate}
\rightarrow
\text{Change}
\rightarrow
\text{Observe}
\rightarrow
\text{Explain}
}
\]
Questions to Explore
Check Your Understanding
- Explain what the symbols \(V_0\), \(V_n\), \(P\), \(r\), \(g\), and \(n\) represent.
- Why is \(P\) described as the first payment when \(g\ne0\)?
- What sign should \(P\) have for a retirement contribution? For a retirement withdrawal?
- Explain why earlier payments have more influence on \(V_n\) than later equal payments.
- What conditions turn the general cash-flow model into the ordinary annuity formula?
- What conditions turn the model into a fixed-payment loan problem?
Calculate
- Find the value after 20 years of an account that begins with $10,000 and receives $3,000 at the end of each year, assuming a 6% annual return.
- Repeat the previous problem if the annual payment increases by 2% per year.
- A retirement account begins with $600,000 and pays out $35,000 at the end of each year.
If the account earns 5%, find the balance after 15 years. - Find the payment needed to amortize a $200,000 loan over 20 years at 5% annually with annual payments.
Compare
- Compare saving $5,000 per year for 25 years with saving $5,000 in the first year and increasing the contribution by 3% annually.
- Compare the ending values of equal payments made at the beginning and end of each year.
- Compare a retirement withdrawal plan with constant withdrawals to one in which withdrawals increase with inflation.
Explain and Decide
- A person says, “A mortgage and a retirement savings plan have completely different mathematics.”
Explain why this statement is misleading. - Why might increasing contributions gradually be more realistic than assuming the same contribution for 30 years?
- Suppose a retirement plan appears sustainable when \(g=0\) but not when \(g=3\%\).
What does this tell you about the effect of inflation-adjusted withdrawals?
Chapter 2 Quick Reference
Core variables:
\[
\boxed{
V_0,\quad V_n,\quad P,\quad r,\quad g,\quad n
}
\]
Cash-flow sign convention:
\[
P>0:\text{ money added},
\qquad
P<0:\text{ money removed}.
\]
General cash-flow model, \(r\ne g\):
\[
\boxed{
V_n
=
V_0(1+r)^n
+
P
\frac{(1+r)^n-(1+g)^n}{r-g}
}
\]
Special case \(r=g\):
\[
\boxed{
V_n
=
V_0(1+r)^n
+
nP(1+r)^{n-1}
}
\]
Level payments, \(g=0\):
\[
\boxed{
V_n
=
V_0(1+r)^n
+
P\frac{(1+r)^n-1}{r}
}
\]
Fixed loan payment, \(V_n=0,\ g=0\):
\[
\boxed{
P
=
-V_0
\frac{r(1+r)^n}{(1+r)^n-1}
}
\]
Beginning-of-period payments:
Multiply the accumulated cash-flow portion by \(1+r\).
Main interpretation:
\[
\boxed{
\text{Ending Value}
=
\text{Growth of Existing Money}
+
\text{Growth of New Cash Flows}
}
\]
Chapter Summary
This chapter developed a unified model for financial problems involving repeated cash flows.
The same structure can describe saving, growing contributions, retirement withdrawals, loans, mortgages, and other applications.
The key step is not to memorize a different formula for each financial story.
Instead, identify the values of
\[
V_0,\quad V_n,\quad P,\quad r,\quad g,\quad n
\]
and interpret the signs, timing, and growth assumptions correctly.
\[
\boxed{
\text{One Cash-Flow Model}
\longrightarrow
\text{Many Financial Problems}
}
\]
The financial story changes, but the underlying mathematics often remains the same.
Looking Ahead
Chapter 2 developed the general mathematical engine.
The next step is to apply that engine across an entire retirement life cycle.
How do the saving years and the retirement-withdrawal years fit together as two stages of one financial plan?
Chapter 3 will connect accumulation and decumulation through the value of the same account at the retirement date.