Home » Algebra » Time Value of Money
Interest Calculation
Introduction to interest calculation formulas that determine how money balances deposited into an interest earning account grow over time.
Effective Annual Rate (EAR)
The effective annual rate (EAR) is a conversion of interest rates to an equivalent rate with annual compounding.
Derivation of Continuous Compounding
Derivation of continuous compounding shows how compound interest converges to an exponential function as compounding intervals tend to zero.
Annuity Calculation
Introduction to fixed interest annuity calculation with formulas for present and future value as well as usage examples.
Derivation of Annuity Formulas
Easy to understand derivation of annuity formulas for the calculation of fixed interest series of constant payments and perpetuities.

# Time Value of Money

The time value of money takes account of changes to the financial value of money over time. In order to understand this concept it may help to first consider the analogy of the more commonly known phenomenon of inflation. When the rate of inflation is high money loses real purchasing power quickly. Similarly, the higher the interest rates in financial markets, the less valuable is future money in terms of money of today.

Inflation is a change in the ratio of the amount of money chasing goods and services. Where money growth exceeds the one of productivity, the real value of money drops. In contrast, time value of money is exclusively concerned with the reproduction of money through interest within the financial system.

Figure 1 shows an example bank account earning 4% of interest. While its nominal value keeps growing along the green line, its present value shown in black is flat-lining. This is because the conversion of future or past money balances to present time value reverts the effects of compound interest.

## Present and Future Value

Reverting compound interest effects by discounting back to present value is the essence of the concept of time value of money. Present and future values of current account balances follow from compound interest calculation. Assuming constant interest rates with discrete compounding, the relevant formulas are:

\begin{gather*}
FV(t)=PV \cdot (1+\frac{1}{m}\cdot r_i)^{t \cdot m}\\
\\
PV=FV(t) \cdot (1+\frac{1}{m}\cdot r_d)^{-t \cdot m}\\
\end{gather*}
\begin{alignat*}{2}
FV(t):~&& &\mathrm{future ~ value}\\
&& &\mathrm{at ~ time} \space t\\
PV:~&& &\mathrm{present ~ value}\\
&& &(\mathrm{at ~ time} \space 0)\\
m: ~&& & \mathrm{compounding}\\
&& & \mathrm{frequency}\\
r_i: ~&& & \mathrm{interest \space rate}\\
r_d: ~&& & \mathrm{discount \space rate}\\
\end{alignat*}

While the future value FV is an exponential growth, the present value PV is the inverse, an exponential decline[1]. Therefore, when the interest rate ri is same as the discount rate rd, then the present value of an interest earning balance remains constant through time.

For instance, the pricing of annuities is an application of discounting to present value. Though most textbook examples assume the equality of investment interest ri and discount interest rd, they almost always differ in retail banking. Just compare offers on high interest savings accounts and mortgages to see this.

## Time Value versus Real Value

Since time value of money is a matter of interest rates within the financial system, its relation to the real economy is often neglected. However, it’s real purchasing power that should matter most to retail banking clients.

As of writing this article in November 2022, the US and the EU are experiencing a period of financial repression[2]. During normal economic times the real interest rate, which is defined as the difference between nominal interest and inflation, is positive. Without real interest, incentives for lending and saving are lacking. Therefore, financial markets should naturally adjust interest rates to top inflation. And politics are repressive if they don’t allow this to happen.

Nevertheless, Figure 2 depicts the situation that citizens throughout the world currently find themselves in. Even though the financial system considers a savings account earning 4% of interest as having a stable time value, real values keep dropping.

Moreover, the financial view of time value is not without contradictions in itself. Arbitrage free pricing rules enforce a depreciation of future money balances when interest rates are rising. But rising interest rates limit the speed of money creation and thus counter inflation. Consequently, adjustments between real and financial economy can temporarily diverge from long term trends.

## References

[1] Present Value, Wikipedia.org

[2] The Price of Time: The Real Story of Interest, Edward Chancellor, 2022, Allen Lane

Published: November 26, 2022
Updated: November 29, 2022

Financial Algebra
Financial Algebra