Question

The chief financial officer of a company reports that profits for the past fiscal year were \(\$ 893,000\). The officer predicts that profits for the next 5 years will grow at a continuous annual rate somewhere between \(3 \frac{1}{2} \%\) and \(5 \%\). Estimate the cumulative difference in total profit over the 5 years based on the predicted range of growth rates.

Short answer

Cumulative difference in total profit over the 5 years based on the predicted range of growth rates is \(P'_{\text{final}} - P_{\text{final}}\). Note that the net difference based on these estimates could be quite significant due to the power of exponential growth.

Step-by-step solution

Compute Accumulated Profit for Lower Estimate

First, we need to work out the cumulative profits after 5 years for the lower predicted continuous annual growth rate of \(3.5\%\). To do this, to use the formula for continuous compound interest which is \(P_{\text{final}} = P_{\text{initial}} \times e^{rt}\), where \(P_{\text{final}}\) is the final amount, \(P_{\text{initial}}\) is the initial amount, \(r\) is the rate, and \(t\) is the time in years. Thus, using the values provided: \(P_{\text{final}} = 893000 \times e^{0.035 \times 5}\)

Compute Accumulated Profit for Higher Estimate

Similarly, we need to work out the cumulative profits after 5 years for the higher predicted continuous annual growth rate of \(5\%\). Using similar formula, we get: \(P'_{\text{final}} = 893000 \times e^{0.05 \times 5}\)

Compute the Difference in Profits

Lastly, we need to work out the difference in total profits over the 5 years based on the two growth rates calculated. This can be done by finding the difference between \(P'_{\text{final}}\) and \(P_{\text{final}}\)

Key concepts

Exponential Growth

When we talk about exponential growth, we refer to the increase of a quantity at a rate that becomes ever more rapid in proportion to the growing total number or size. In financial terms, this is most commonly seen in contexts like interest accruing on an investment or the increasing value of a company's profits over time. The key feature of exponential growth is that it accelerates over time, unlike linear growth, which increases at a constant rate. Think of it as a snowball rolling downhill, gathering more snow and momentum as it goes – the descent represents passing time, and the growing snowball symbolizes the increasing value.

In the context of the CFO's report, if profits grow at an exponential rate, the amount of increase is not constant year over year; instead, it is proportionally larger with each passing year. For instance, even a consistent growth rate will result in a larger increase in profits as the base amount of profits that is growing becomes larger.

Profit Estimation

Profit estimation involves predicting the future financial gains of a company based on a set of assumptions or known growth rates. The CFO's job in our example involves forecasting profits by using historical data, industry trends, or in this case, a specific expected growth rate. Estimations can be fairly straightforward with linear growth models, but become more complex with exponential growth due to the variable change in growth amounts over time.

To estimate the range of possible future profits for the company, the CFO can employ continuous compounding, which allows them to calculate not just how much the profits are expected to increase each year, but also how each year's growth impacts the next, leading to a 'compounded' effect. The CFO uses a range – in this scenario between 3.5% to 5% – to reflect the uncertainty inherent in future predictions.

Compound Interest Formula

The compound interest formula is a key equation in finance for determining the amount of interest earned on an investment over time. Compound interest differs from simple interest in that it is calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. When compounding occurs continuously, as in our CFO's projections, we use the formula \(P_{\text{final}} = P_{\text{initial}} \times e^{rt}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828.

Within this formula:

• \(P_{\text{final}}\) represents the final amount after interest,
• \(P_{\text{initial}}\) is the initial amount before interest,
• \(r\) is the annual interest rate (expressed as a decimal), and
• \(t\) is the time the money is invested or borrowed for, in years.

The formula reflects the exponential nature of growth with continuous compounding, providing a means to calculate end values that include interest-on-interest effects.

Rate of Return

The rate of return is a measure of the profitability of an investment over a certain period of time. It is usually expressed as a percentage and can be calculated for assets ranging from stocks to savings accounts. In the context of our financial projection, the rate of return is the rate at which the company's profits are expected to grow each year, compounded continuously.

The rate is essential for estimating future profits; a higher rate of return would represent a more optimistic outlook on the market conditions or company's performance, while a lower rate might be used for more conservative estimates. When we discuss the rate of return in relation to compound interest, the continuous compounding formulates the return as part of the exponential growth equation – each year's growth expands upon the compounded profits from all previous years.