1.2 Money Through Time
Chapter 1. Money Through Time
Financial decisions almost always involve time. We save money now to use later, borrow money now and repay it later, compare payments made at different dates, and plan for expenses that may occur many years in the future.
Because money can earn returns and because prices change over time, an amount of money does not have the same financial value at every date. Understanding this simple idea is the foundation of financial mathematics.
A financial value should always be associated with a point in time.
\[
\boxed{\text{Value is attached to a point in time.}}
\]
In this chapter, we develop the basic mathematics for moving money through time. We will study compound growth, present and future values, interest-rate conventions, inflation, and the distinction between nominal and real dollars.
These ideas will be used throughout the rest of the book when we study repeated saving, borrowing, investing, retirement withdrawals, Social Security, pensions, and other financial decisions.
- 1.1 Why Does Money Change with Time?
- 1.2 How Does Money Grow?
- 1.3 How Do We Move Money Backward and Forward in Time?
- 1.4 What Does “6% Interest” Actually Mean?
- 1.5 How Do We Compare Rates with Different Periods?
- 1.6 What Does Inflation Do to Money?
- 1.7 Nominal Dollars or Today’s Dollars?
- 1.8 Putting the Ideas Together
1.1 Why Does Money Change with Time?
Why This Matters
Suppose you are offered either $10,000 today or $10,000 five years from now. If there are no other differences between the two choices, most people would prefer the $10,000 today.
One reason is that money received today can be saved or invested and may grow before the five years have passed.
Suppose $10,000 can earn 5% per year. Without calculating exactly, would you expect it to be worth more than $12,000 after five years?
Make a prediction before continuing.
The Time Value of Money
The time value of money is the principle that the financial value of money depends on when the money is received or paid.
Several factors can make money at different dates financially different:
- money can earn interest or investment returns;
- inflation can reduce purchasing power;
- future payments involve uncertainty;
- having money earlier provides additional financial choices.
Financial mathematics gives us a systematic way to compare values occurring at different points in time.
A Financial Timeline
A timeline is one of the simplest and most useful tools in financial mathematics.
For example:
\[
\begin{array}{c|c|c|c|c|c}
0 & 1 & 2 & 3 & 4 & 5\\
\hline
V_0 & & & & & V_5
\end{array}
\]
Time \(0\) represents the starting date. The value at that date is \(V_0\). The value five periods later is \(V_5\).
Throughout this book, we use
\[
\boxed{V_t=\text{value at time }t.}
\]
Thus \(V_0\) is the value at the starting date, \(V_1\) is the value one period later, and \(V_n\) is the value after \(n\) periods.
In a problem beginning at time \(0\) and ending at time \(n\),
\[
PV=V_0,
\qquad
FV=V_n.
\]
The symbols \(PV\) and \(FV\) mean present value and future value. We use \(V_t\) because it allows us to describe values at any point on a financial timeline.
1.2 How Does Money Grow?
One Period of Growth
Suppose an initial amount \(V_0\) earns a return \(r\) during one period.
The interest or investment growth during that period is
\[
rV_0.
\]
Therefore, the value after one period is
\[
V_1=V_0+rV_0
=V_0(1+r).
\]
The quantity \(1+r\) is the growth factor.
\[
\boxed{V_1=V_0(1+r)}
\]
where \(r\) is the rate of return for one period.
Several Periods of Growth
If the same rate applies in the next period,
\[
V_2=V_1(1+r).
\]
Since \(V_1=V_0(1+r)\),
\[
V_2=V_0(1+r)^2.
\]
Continuing this process gives
\[
V_3=V_0(1+r)^3,
\]
and, after \(n\) periods,
\[
\boxed{V_n=V_0(1+r)^n}
\]
where
- \(V_0\) = initial value,
- \(V_n\) = value after \(n\) periods,
- \(r\) = return or interest rate per period,
- \(n\) = number of periods.
With compound growth, previously earned returns become part of the balance and can themselves earn future returns.
\[
\boxed{\text{Growth earns additional growth.}}
\]
Worked Example: Investing $10,000
Suppose $10,000 is invested for 10 years and earns 6% annually.
Here,
\[
V_0=10000,\qquad r=0.06,\qquad n=10.
\]
Therefore,
\[
V_{10}
=
10000(1.06)^{10}
\approx 17908.48.
\]
The investment will grow to approximately $17,908.
The original $10,000 earns approximately $7,908 over the ten-year period. Part of that growth comes from returns earned on earlier returns, not merely from returns on the original $10,000.
Why Time Matters So Much
Because \(n\) appears as an exponent in
\[
V_n=V_0(1+r)^n,
\]
financial growth is exponential rather than linear.
For example, $10,000 growing at 6% annually becomes approximately:
| Years | Value |
|---|---|
| 0 | $10,000 |
| 10 | $17,908 |
| 20 | $32,071 |
| 30 | $57,435 |
| 40 | $102,857 |
Notice that the increase during later decades is much larger than the increase during earlier decades.
Start with \(V_0=\$10,000\) and \(r=6\%\).
- Predict the values after 10, 20, 30, and 40 years.
- Calculate the actual values.
- Graph \(V_n\) against \(n\).
- Explain why the graph becomes steeper over time.
The Rule of 72
A useful mental approximation estimates how long an investment takes to double at a given annual compound return.
\[
\boxed{
\text{Approximate doubling time}
\approx
\frac{72}{\text{annual rate in percent}}
}
\]
For example, at 6% per year, the approximate doubling time is
\[
\frac{72}{6}=12\text{ years}.
\]
At 8%, the approximation gives
\[
\frac{72}{8}=9\text{ years}.
\]
The Rule of 72 is an approximation. It works well for many commonly encountered interest rates, but the compound-growth formula should be used when an accurate answer is needed.
Go Deeper: Where Does the Rule of 72 Come From?
To find the exact time required for an investment to double, start with the compound-growth formula
\[
V_n=V_0(1+r)^n.
\]
Doubling means \(V_n=2V_0\). Therefore,
\[
2V_0=V_0(1+r)^n,
\]
so
\[
2=(1+r)^n.
\]
Taking natural logarithms gives
\[
\ln 2=n\ln(1+r),
\]
and therefore the exact doubling time is
\[
\boxed{
n=\frac{\ln 2}{\ln(1+r)}
}.
\]
For relatively small \(r\),
\[
\ln(1+r)\approx r.
\]
Since \(\ln 2\approx0.693\),
\[
n\approx\frac{0.693}{r}.
\]
If the annual rate is written as a percentage \(R=100r\), then
\[
n\approx\frac{69.3}{R}.
\]
This suggests a “Rule of 69.3.” In practice, 72 is commonly used instead because it is easy to divide mentally by many common rates, including 3, 4, 6, 8, 9, and 12. It also gives a useful approximation over a range of typical interest rates.
The Rule of 72 is therefore not an unrelated financial trick. It is a convenient mental approximation derived from the same compound-growth model:
\[
\boxed{V_n=V_0(1+r)^n.}
\]
1.3 How Do We Move Money Backward and Forward in Time?
Moving Forward
Compound growth moves money from an earlier date to a later date:
\[
V_0
\longrightarrow
V_n.
\]
The relationship is
\[
V_n=V_0(1+r)^n.
\]
Moving Backward
Sometimes we know the future value and want to determine what that amount is worth at an earlier date.
Solving the compound-growth equation for \(V_0\) gives
\[
V_0
=
\frac{V_n}{(1+r)^n}.
\]
\[
\boxed{
V_0=\frac{V_n}{(1+r)^n}
}
\]
This process is called discounting.
The present value of a future amount is its equivalent value at the present valuation date, based on a specified rate of return.
Worked Example: $20,000 Ten Years from Now
How much would you need today to have $20,000 in 10 years if the money earns 5% annually?
\[
V_0
=
\frac{20000}{(1.05)^{10}}
\approx 12278.27.
\]
You would need approximately $12,278 today.
$12,278 today and $20,000 ten years from now are financially equivalent under the assumption of a 5% annual return.
The Common Valuation Date
Suppose one choice gives $8,000 today and another gives $10,000 five years from now. It is not enough simply to compare 8,000 with 10,000. The amounts occur at different dates.
Before adding or comparing financial amounts occurring at different dates, move them to the same point in time.
You may choose today as the valuation date and discount future amounts backward, or choose a future date and compound earlier amounts forward.
If the assumptions are consistent, either approach leads to the same financial comparison.
\[
\boxed{
\text{Present value and future value are two directions of the same process.}
}
\]
Values at Intermediate Times
The notation \(V_t\) becomes especially useful when a financial plan has several stages.
Instead of labeling every value as either a present value or future value, we simply specify its date:
\[
V_0
\longrightarrow
V_5
\longrightarrow
V_{10}
\longrightarrow
V_{20}.
\]
A value that is a future value from one perspective may become the starting value for the next stage.
This idea will become especially important when we connect retirement saving and retirement withdrawals.
1.4 What Does “6% Interest” Actually Mean?
A rate such as “6%” is not enough information by itself. We must also know what time period the rate describes and, when appropriate, how often interest is compounded.
The rate \(r\) and the number of periods \(n\) must use the same time unit.
\[
\boxed{
\text{Rate period}=\text{time unit counted by }n
}
\]
Nominal Annual Rate
Suppose an account quotes a nominal annual interest rate \(R\), compounded \(m\) times per year.
The rate used during each compounding period is
\[
r=\frac{R}{m}.
\]
If the investment lasts \(t\) years, the number of periods is
\[
n=mt.
\]
\[
\boxed{
r=\frac{R}{m},
\qquad
n=mt
}
\]
where \(R\) is a nominal annual rate and \(m\) is the number of compounding periods per year.
Common Compounding Frequencies
| Frequency | \(m\) |
|---|---|
| Annual | 1 |
| Semiannual | 2 |
| Quarterly | 4 |
| Monthly | 12 |
| Semimonthly | 24 |
| Biweekly | 26 |
| Weekly | 52 |
Semimonthly and biweekly are not the same.
- Semimonthly means 24 periods per year.
- Biweekly means 26 periods per year.
Worked Example: 6% Compounded Monthly
Suppose $10,000 earns a nominal annual rate of 6%, compounded monthly, for 10 years.
The monthly rate is
\[
r=\frac{0.06}{12}=0.005.
\]
The number of monthly periods is
\[
n=12(10)=120.
\]
Therefore,
\[
V_{120}
=
10000(1.005)^{120}
\approx 18193.97.
\]
The ending value is approximately $18,194.
Do not substitute \(r=0.06\) and \(n=120\). That would incorrectly treat 6% as the monthly rate.
If periods are months, the rate must also be a monthly rate.
1.5 How Do We Compare Rates with Different Periods?
Effective Annual Rate
A nominal rate does not by itself show the actual percentage growth over a full year when interest is compounded more than once per year.
For comparison, we often use the effective annual rate.
The effective annual rate, often abbreviated EAR, is the actual percentage growth over one year after accounting for compounding.
If the nominal annual rate is \(R\), compounded \(m\) times per year, then
\[
\boxed{
EAR=
\left(1+\frac{R}{m}\right)^m-1
}
\]
Worked Example: Nominal 6% Compounded Monthly
For a nominal annual rate of 6% compounded monthly,
\[
EAR
=
\left(1+\frac{0.06}{12}\right)^{12}-1.
\]
Thus,
\[
EAR\approx0.061678
=
6.1678\%.
\]
The account actually grows by about 6.17% over one year.
Converting an Effective Annual Rate to a Monthly Rate
Suppose instead that we know an effective annual rate \(R_{\text{eff}}\) and want an equivalent monthly rate \(r\).
The two rates must produce the same growth over one year:
\[
(1+r)^{12}=1+R_{\text{eff}}.
\]
Therefore,
\[
\boxed{
r=(1+R_{\text{eff}})^{1/12}-1
}
\]
General Effective Rate Conversion
The same principle works for any two periods.
If \(r_1\) is an effective rate for a period of length \(T_1\) and \(r_2\) is the equivalent effective rate for a period of length \(T_2\), then
\[
\boxed{
1+r_2=(1+r_1)^{T_2/T_1}
}
\]
Equivalent rates are rates that produce the same growth over the same amount of actual time.
Compare several accounts with different quoted rates and compounding frequencies.
- Convert each rate to an effective annual rate.
- Rank the accounts by actual annual growth.
- Check whether the account with the largest quoted nominal rate always has the largest EAR.
- Explain why comparing quoted rates directly can sometimes be misleading.
1.6 What Does Inflation Do to Money?
So far we have studied how an account value can grow. But financial planning involves another type of growth: the prices of goods and services may rise over time.
Inflation is a general increase in prices over time. When prices rise, a fixed number of dollars buys fewer goods and services.
Future Cost
If an item costs \(C_0\) today and its cost increases at rate \(i\) per period, its future cost after \(n\) periods is
\[
\boxed{
C_n=C_0(1+i)^n
}
\]
Worked Example: $60,000 of Annual Spending
Suppose a household needs $60,000 per year today. If inflation averages 3% annually, what nominal amount would provide approximately the same purchasing power 10 years from now?
\[
C_{10}
=
60000(1.03)^{10}
\approx 80634.98.
\]
Approximately $80,635 ten years from now would be needed to purchase what $60,000 buys today, under the 3% inflation assumption.
Receiving $80,635 in ten years does not necessarily mean having substantially greater purchasing power than $60,000 today. If prices rise by 3% per year, the two amounts represent approximately the same purchasing power.
Purchasing Power of a Future Amount
The process can also be reversed. A future nominal amount \(V_n\) can be expressed in today’s purchasing power:
\[
\boxed{
V_{\text{real}}
=
\frac{V_{\text{nominal}}}{(1+i)^n}
}
\]
A larger number of dollars in the future does not automatically mean greater purchasing power.
1.7 Nominal Dollars or Today’s Dollars?
When planning for the future, we can describe money in two different ways.
Nominal dollars are the actual dollar amounts expected to be received or spent at a future date.
Real dollars express amounts in terms of purchasing power at a selected base date, often today’s purchasing power.
Nominal Return and Real Return
Suppose an investment earns a nominal return \(r_{\text{nominal}}\), while inflation is \(i\).
The exact real return satisfies
\[
1+r_{\text{nominal}}
=
(1+r_{\text{real}})(1+i).
\]
Solving for the real return gives
\[
\boxed{
1+r_{\text{real}}
=
\frac{1+r_{\text{nominal}}}{1+i}
}
\]
or equivalently,
\[
\boxed{
r_{\text{real}}
=
\frac{1+r_{\text{nominal}}}{1+i}-1
}
\]
Worked Example: 7% Return and 3% Inflation
Suppose an investment earns a nominal return of 7% while inflation is 3%.
\[
r_{\text{real}}
=
\frac{1.07}{1.03}-1
\approx0.038835.
\]
Therefore,
\[
r_{\text{real}}\approx3.88\%.
\]
The investment’s purchasing power grows by approximately 3.88%, not 7%.
A frequently used approximation is
\[
r_{\text{real}}
\approx
r_{\text{nominal}}-i.
\]
This can be useful for quick estimates, but it is not exact. For precise calculations, use
\[
r_{\text{real}}
=
\frac{1+r_{\text{nominal}}}{1+i}-1.
\]
\[
\boxed{
\text{Nominal dollars with nominal rates;}
\qquad
\text{real dollars with real rates.}
}
\]
Mixing today’s dollars with nominal investment returns can produce misleading results.
Two Ways to Describe the Same Retirement Need
Suppose a household needs $60,000 per year in today’s purchasing power and expects to retire in 10 years. Assume inflation is 3%.
One approach is to work in nominal dollars. The first-year retirement need is approximately
\[
60000(1.03)^{10}
\approx80635.
\]
Another approach is to keep the retirement need at $60,000 in today’s dollars and use a real investment return.
These are not two different economic plans. They are two different ways to describe the same plan.
A correctly constructed nominal-dollar model and a correctly constructed real-dollar model should describe the same underlying purchasing power.
\[
\boxed{
\text{Nominal Model}
\longleftrightarrow
\text{Real Model}
}
\]
1.8 Putting the Ideas Together
The individual formulas in this chapter are useful, but the more important goal is to develop a consistent way of thinking about financial problems.
Four Questions to Ask Before Calculating
- At what point in time is the money being valued?
Identify the valuation date before comparing amounts. - What rate applies?
Determine whether the problem uses an investment return, interest rate, discount rate, inflation rate, or another growth rate. - What time period does the rate describe?
Make the rate period consistent with the number of periods. - Are the amounts nominal dollars or real dollars?
Keep dollar units and rates consistent.
Integrated Example: Planning for a Future Purchase
Suppose an expense costs $40,000 today and is expected to occur 12 years from now.
Assume its cost increases with inflation at 3% annually.
First determine the expected future cost:
\[
C_{12}
=
40000(1.03)^{12}
\approx57030.44.
\]
Now suppose money invested today can earn 6% annually. How much must be invested today to cover that expected future cost?
\[
V_0
=
\frac{57030.44}{(1.06)^{12}}
\approx28345.70.
\]
Under these assumptions, approximately $28,346 today could grow to the projected $57,030 cost in 12 years.
This example combines two different growth processes:
- the cost grows because of inflation;
- the invested money grows because of investment returns.
Financial planning often involves comparing several growth processes occurring at the same time.
Suppose you can invest either $25,000 today or wait three years and invest a larger amount.
How much would you need to invest three years from now to reach the same goal?
Before calculating, identify the common valuation date and decide whether the amounts should be compared at time 0, time 3, or the final goal date.
What the Simple Model Assumes
The compound-growth formula
\[
V_n=V_0(1+r)^n
\]
is powerful, but its simplicity depends on assumptions.
In its basic form, the model assumes:
- a known rate \(r\);
- a consistent definition of each period;
- no additional deposits or withdrawals;
- no taxes or fees unless they are already reflected in \(r\);
- compounding according to the stated model.
Actual investments may have changing returns, taxes, fees, and irregular cash flows. Later chapters will extend the basic model.
Digital Math Lab: Chapter Exploration
Use the Chapter 1 Math Lab to investigate the relationships among
\[
V_0,\qquad V_n,\qquad r,\qquad n,\qquad i.
\]
Suggested investigations:
- Keep \(V_0\) and \(r\) fixed and vary \(n\). Describe the shape of the growth curve.
- Keep \(V_0\) and \(n\) fixed and vary \(r\). Which changes have the greatest effect over long periods?
- Choose a future financial goal and calculate its present value under several assumed rates.
- Compare nominal annual rates with their effective annual rates.
- Choose an inflation rate and estimate the future cost of a major expense.
- Compare nominal investment growth with growth in purchasing power.
For each experiment, use the cycle
\[
\boxed{
\text{Predict}
\rightarrow
\text{Calculate}
\rightarrow
\text{Observe}
\rightarrow
\text{Explain}
}
\]
Common Mistakes in Chapter 1
- Comparing amounts from different dates directly.
Move them to a common valuation date first. - Using an annual rate with monthly periods.
Rates and periods must use consistent time units. - Confusing nominal and effective rates.
A quoted nominal annual rate may not equal the actual annual growth rate. - Ignoring inflation.
Future nominal dollars may have substantially less purchasing power than the same number of dollars today. - Subtracting inflation from investment return and treating the result as exact.
Use the exact real-return formula when precision matters. - Treating an assumed return as guaranteed.
A mathematical model may use a fixed rate even though actual investment returns vary.
Go Deeper: Why Compound Growth Is Exponential
Compound growth repeatedly multiplies the account by the same growth factor \(1+r\):
\[
\begin{aligned}
V_1 &= V_0(1+r),\\
V_2 &= V_0(1+r)^2,\\
V_3 &= V_0(1+r)^3,\\
&\ \vdots\\
V_n &= V_0(1+r)^n.
\end{aligned}
\]
Because the number of periods appears in the exponent, the relationship between value and time is exponential.
This explains why small differences in rates or time may eventually produce large differences in ending values.
Chapter 1 Quick Reference
Value notation:
\[
V_t=\text{value at time }t.
\]
Compound growth:
\[
\boxed{V_n=V_0(1+r)^n}
\]
Present value:
\[
\boxed{
V_0=\frac{V_n}{(1+r)^n}
}
\]
Nominal annual rate compounded \(m\) times per year:
\[
\boxed{
r=\frac{R}{m},
\qquad
n=mt
}
\]
Effective annual rate:
\[
\boxed{
EAR=
\left(1+\frac{R}{m}\right)^m-1
}
\]
General effective-rate conversion:
\[
\boxed{
1+r_2=(1+r_1)^{T_2/T_1}
}
\]
Future cost with inflation:
\[
\boxed{
C_n=C_0(1+i)^n
}
\]
Real return:
\[
\boxed{
r_{\text{real}}
=
\frac{1+r_{\text{nominal}}}{1+i}-1
}
\]
Two fundamental principles:
\[
\boxed{\text{Value is attached to a point in time.}}
\]
\[
\boxed{\text{Nominal dollars with nominal rates; real dollars with real rates.}}
\]
Questions to Explore
Check Your Understanding
- Explain in your own words why $10,000 today is generally not financially equivalent to $10,000 ten years from now.
- What does \(V_7\) represent?
- What is the relationship between \(V_0\), \(PV\), \(V_n\), and \(FV\)?
- Why should two cash flows occurring at different dates be moved to a common valuation date before they are compared?
- Explain the difference between a nominal annual rate and an effective annual rate.
- Explain the difference between nominal dollars and real dollars.
Calculate
- Find the value of $15,000 after 12 years at an annual return of 5%.
- How much must be invested today to have $50,000 in 15 years if the annual return is 6%?
- Find the effective annual rate corresponding to a nominal annual rate of 8% compounded monthly.
- If an expense costs $25,000 today, estimate its cost 20 years from now under 3% annual inflation.
- Find the real return when the nominal investment return is 8% and inflation is 2.5%.
Compare
- Compare $20,000 today with $30,000 ten years from now using annual valuation rates of 3%, 5%, and 7%. Does the preferred choice change?
- Compare two accounts:
one offers a nominal annual rate of 6% compounded monthly, and the other offers an effective annual rate of 6.1%.
Which provides greater annual growth? - Compare the future value of $10,000 invested for 30 years at 5%, 6%, and 7%. What does the comparison show about long-term sensitivity to return assumptions?
Explain and Decide
- A retirement projection says that a household may need $100,000 per year 20 years from now. Why is that number difficult to interpret without knowing whether it is expressed in nominal or real dollars?
- Someone says, “My investment earns 7% and inflation is 3%, so my real return is exactly 4%.” Evaluate the statement.
- Suppose two financial plans produce the same expected future value, but one requires substantially more money today. What additional questions would you ask before deciding between them?
Chapter Summary
This chapter introduced the central idea that financial value depends on time.
Compound growth moves money forward, while discounting moves money backward.
Interest rates must be matched to the periods they describe, and effective rates allow meaningful comparisons among differently quoted rates.
Inflation adds another dimension: the number of dollars may increase while purchasing power changes.
For that reason, financial calculations must distinguish carefully between nominal and real dollars.
Before performing a financial calculation, ask:
\[
\boxed{
\begin{array}{c}
\text{When is the money valued?}\\
\downarrow\\
\text{What rate applies?}\\
\downarrow\\
\text{What period does the rate describe?}\\
\downarrow\\
\text{Are the dollars nominal or real?}
\end{array}
}
\]
Looking Ahead
Chapter 1 focused mainly on moving a single amount of money through time.
Most real financial plans, however, involve money entering or leaving an account repeatedly.
We may contribute to retirement accounts every month, make mortgage payments, withdraw retirement income, or increase our contributions as income rises.
What happens when an existing balance is combined with a sequence of deposits, withdrawals, or payments?
Chapter 2 develops a unified cash-flow model for answering that question. It will show how many formulas that are traditionally taught separately can be understood as variations of one common mathematical structure.