Question

An investment company invests the money it receives from investors on a collective basis, and each investor shares in the profits and losses. One company has an annual cash flow that has fluctuated in cycles of approximately 40 years since \(1920,\) when it was at a high point. The highs were approximately \(+20 \%\) of the total assets, while the lows were approximately \(-10 \%\) of the total assets. a) Model this cash flow as a cosine function of the time, in years, with \(t=0\) representing 1920 b) Graph the function from part a). c) Determine the cash flow for the company in 2008 d) Based on your model, do you feel that this is a company you would invest with? Explain.

Short answer

The cash flow in 2008 is approximately 0.365%. Investment risk due to high fluctuation in cash flows suggests cautious consideration.

Step-by-step solution

Identify the parameters

The function needs to represent cyclical cash flow changes over time since 1920. With highs of +20% and lows of -10%, the cosine function will be centered around 5% (midpoint of +20% and -10%) with an amplitude of 15% (half the distance from +20% to -10%). The period is 40 years.

Write the general cosine function

A general cosine function can be written as: \[ f(t) = A \, \cos(B(t - C)) + D \] where A is the amplitude, B is the frequency, C is the horizontal shift, and D is the vertical shift.

Determine the values of A, B, C, and D

For this problem: \[ A = 15\text{%} \] (from the amplitude calculation), \[ B = \frac{2\pi}{40} \] (since the period is 40 years), \[ C = 0 \] (no horizontal shift since t=0 represents 1920), and \[ D = 5\text{%} \] (centerline of the oscillation).

Write the specific cosine function

Substitute the values into the general cosine function: \[ f(t) = 15\text{%} \, \cos\left(\frac{2\pi}{40}t\right) + 5\text{%} \]

Graph the function from part a)

Plot the function \( f(t) = 15\text{%} \, \cos\left(\frac{2\pi}{40}t\right) + 5\text{%} \) over the desired time period. Mark the highs at +20% and the lows at -10% within the 40-year cycle.

Determine the cash flow for 2008

Calculate the number of years from 1920 to 2008: \( 2008 - 1920 = 88 \). Substitute \( t = 88 \) into the function: \[ f(88) = 15\text{%} \, \cos\left(\frac{2\pi}{40} \cdot 88\right) + 5\text{%} \]\

Simplify the calculation

Calculate the value inside the cosine: \( \frac{2\pi}{40} \cdot 88 = \frac{176\pi}{40} = 4.4\pi \). Since \( \cos(4.4\pi) = \cos(1.4\pi) = \cos(\pi + 0.4\pi) = -\cos(0.4\pi) \): \[ f(88) = 15\text{%} \, (-\cos(0.4\pi)) + 5\text{%} \]

Find the final cash flow

\( \cos(0.4\pi) \approx 0.309 \), thus: \[ f(88) = 15\text{%} \, (-0.309) + 5\text{%} = -4.635\text{%} + 5\text{%} = 0.365\text{%} \]

Evaluate investment potential

Analysis: Based on the mathematical model, the company has cash flows that vary significantly, suggesting potential risk. In 2008, the cash flow is very close to zero, indicating a poor performance that year.

Key concepts

Amplitude

The amplitude of a function describes the height of its peaks and the depth of its troughs. For the cosine function modeling the cash flow, the amplitude is crucial as it indicates the extent of the fluctuations in the company's cash flow. The amplitude here is 15%, which means the cash flow varies by 15% above and below the centerline (or average) value of 5%. This is calculated as half the distance between the highest point (+20%) and the lowest point (-10%) of the function.
This amplitude signifies the magnitude of the cyclical changes in cash flow experienced by the company over a period of time. It provides investors with a measure of the potential volatility they might expect.

Frequency

Frequency is key when modeling periodic functions like the cash flow example. It tells you how often the cycles repeat over a given period. For our cosine function, the period is 40 years. Frequency, in mathematical terms, is given by the formula \(B = \frac{2\pi}{T}\), where T is the period. Here, \(B = \frac{2\pi}{40}\). This signifies that the cash flow cycle repeats every 40 years.
Understanding frequency helps predict when certain patterns or behavior may recur. Investors would use this to anticipate future peaks and troughs, helping them to make informed decisions about their investments.

Cash Flow Analysis

Cash flow analysis involves examining the cash inflows and outflows of a business. In this exercise, it's particularly about understanding these flows as they fluctuate in a cyclical pattern over a significant period of time. With highs at +20% and lows at -10%, the company experiences major variations.
By modeling these cash flows with a cosine function, it becomes easier to visualize and predict the company's financial health over the years. This annual cash flow model aids in evaluating investment potential, helping investors understand how cash might fluctuate and assessing risk. In 2008, for example, the model calculates the cash flow to be nearly zero, indicating a performance dip.

Graphing Trigonometric Functions

Graphing the cosine function is essential to visually representing the cyclical nature of the company’s cash flows. For our function \[ f(t) = 15\% \, \cos\left(\frac{2\pi}{40}t\right) + 5\% \], the graph will oscillate between +20% and -10%, centered around 5%.
Graphing helps in identifying the trend and assists in understanding the period and amplitude intuitively. Investors can interpret how often cash flow reaches highs and lows and make decisions based on these visual insights. Clear markings of periods and amplitudes on the graph make these insights more accessible and comprehensible.