Foundational Tools for Financial Mathematics: Timelines and Equations of Value

Improving your understanding of mathematics and its applications is a multifaceted process.

It is of utmost importance to become comfortable with abstract ideas, symbols, and the mental manipulation of these things. You should know the rules of algebra so well that they become automatic. Furthermore, visual representations of the concepts are often essential to seeing why facts are true and in making conjectures.

Abstraction in Financial Mathematics

This is certainly true in financial mathematics. The key visual representation for many situations in financial mathematics is the timeline (or “time diagram”). To make one, you should draw a number line and mark points out representing various moments in time, with the orientation of increasing time typically drawn to the right. You should then take financial amounts, whether they be deposits or withdrawals, and draw them above discrete moments in time on the number line. Deposits are typically shown as positive amounts and withdrawals as negative amounts.

In the most common situation, we assume that there is a constant effective periodic rate of interest $i$ . The period is most commonly taken to be a year (making $i$ an effective annual rate of interest). We also most commonly assume monetary amounts “accumulate” to “future values” using compound interest. If a monetary amount $A$ is deposited for amount of time $T$ (relative to the interest rate measurement period), the future value is $A(1+i)^{T}$ .

Equation of Value

In the video “Actuarial Exam 2/FM Prep: Use an Equation of Value to Solve for an Unknown Withdrawal“, I use these concepts to solve a problem where an unknown withdrawal must be found. The key idea is to relate financial transactions and future values, with the help of a timeline, to an equation of value.

In a general setting, suppose cashflow amounts $C_{0},C_{1},C_{2},\ldots,C_{n}$ , which can be positive or negative, are disbursed at times $t_{0}=0< t_{1}< t_{2}<\dots<t_{n}$ . Suppose further that we wish to find the future value of the entire cashflow at time $T$ , where $t_{n}<T$ . It turns out that this can be done by adding the future values of all the cashflow amounts, relative to how much time they accumulate in value.

In other words, the future value, assuming compound interest, is the expression

$C_{0}(1+i)^{T}+C_{1}(1+i)^{T-t_{1}}+C_{2}(1+i)^{T-t_{2}}+\cdots+C_{n}(1+i)^{T-t_{n}}$ .

A general term $C_{i}(1+i)^{T-t_{i}}$ in this sum can be visualized by taking the amount $C_{i}$ that is disbursed at time $t=t_{i}$ and “pushing it forward” in time to $t=T$ . This means that the amount of time it is “pushed forward” is $T-t_{i}.$ In so doing, it must be multiplied by the “growth factor” $(1+i)^{T-t_{i}}$ to obtain its future value $C_{i}(1+i)^{T-t_{i}}$ . Doing this for all the terms leads to the sum above. And the entire process can be visualized on a timeline like the one shown below.

Timeline showing the accumulation in value, to time T, of disbursed monetary amounts (positive = deposit, negative = withdrawal)

The timeline is helpful both in providing conceptual clarity to the abstract symbols and for solving problems. The equation of value then arises when this future value is set equal to some specific amount (a known future value, for example).

In the example from the video above, $C_{0}=10000$ , $C_{1}=-1.05K$ at $t_{1}=4$ , $C_{2}=-1.05K$ at $t_{2}=5$ , $C_{3}=-K$ at $t_{3}=6$ , $C_{4}=-K$ at $t_{4}=7$ , and $T=10.$ With $i=4\%=0.04$ and a “final balance” (future value) of 10000, the equation of value is

$10000\cdot 1.04^{10}-1.05K\cdot 1.04^{6}-1.05K\cdot 1.04^{5}-K\cdot 1.04^{4}-K\cdot 1.04^{3}=10000$

This final equation is easy to solve for the unknown $K$ because it is a linear equation in $K.$ This means we only need to isolate $K$ using basic properties from algebra to obtain the answer.

Algebraic Justification of the Method

Technically-speaking, as discussed in the last few minutes of the video above, we should algebraically justify that the sum of all the accumulated values is truly the future value of the entire cashflow.

What does this mean? It means that we should get the same future value if we imagine making our deposits and withdrawals as time goes by, rather than doing them all at once and adding the results. For example, assuming $C_{0}$ is a deposit at time $t_{0}=0$ (and therefore $C_{0}>0$ ), this amount will accumulate to $C_{0}(1+i)^{t_{1}}$ at time $t_{1}>t_{0}=0$ . If we then imagine that $C_{1}$ represents a withdrawal at time $t_{1}$ (and therefore $C_{1}<0$ ), say with $-C_{1}<C_{0}(1+i)^{t_{1}}$ so that our account has a positive balance, then the new balance $C_{0}(1+i)^{t_{1}}+C_{1}$ will accumulate in value to $(C_{0}(1+i)^{t_{1}}+C_{1})\cdot (1+i)^{t_{2}-t_{1}}$ at time $t=t_{2}$ .

By the distributive property and laws of exponents, this is the same as $C_{0}(1+i)^{t_{2}}+C_{1}(1+i)^{t_{2}-t_{1}}$ .

Continuing, image that $C_{2}$ represents another withdrawal at time $t=t_{2}$ . Assuming that this withdrawal is small enough, $C_{0}(1+i)^{t_{2}}+C_{1}(1+i)^{t_{2}-t_{1}}+C_{2}$ then represents a positive balance at time $t=t_{2}$ . This will then accumulate to $(C_{0}(1+i)^{t_{2}}+C_{1}(1+i)^{t_{2}-t_{1}}+C_{2})\cdot (1+i)^{t_{3}-t_{2}}$ at time $t=t_{3}$ .

Once again, by the distributive property and laws of exponents, this is the same as $C_{0}(1+i)^{t_{3}}+C_{1}(1+i)^{t_{3}-t_{1}}+C_{2}(1+i)^{t_{3}-t_{2}}$ .

Continuing with this line of algebraic reasoning, we see that the balance of $C_{0}(1+i)^{t_{n}}+C_{1}(1+i)^{t_{n}-t_{1}}+C_{2}(1+i)^{t_{n}-t_{2}}+\cdots+C_{n}$ at time $t=t_{n}$ accumulates to

$(C_{0}(1+i)^{t_{n}}+C_{1}(1+i)^{t_{n}-t_{1}}+C_{2}(1+i)^{t_{n}-t_{2}}+\cdots+C_{n})\cdot (1+i)^{T-t_{n}}$

at time $t=T$ . But this is the same as

$C_{0}(1+i)^{T}+C_{1}(1+i)^{T-t_{1}}+C_{2}(1+i)^{T-t_{2}}+\cdots+C_{n}(1+i)^{T-t_{n}}$

This concludes our algebraic justification of the method.