Problem 16 - Calculus Concepts

Question

AT\&T (Historic) In \(1956,\) AT\&T laid its first underwater phone line. By \(1996,\) AT\&T Submarine Systems, the division of AT \& T that installs and maintains undersea communication lines, had seven cable ships and 1000 workers. On October \(5,1996,\) AT\&T announced that it was seeking a buyer for its Submarine Systems division. The Submarine Systems division of AT\&T was posting a profit of \(\$ 850\) million per year. (Source: "AT\&T Seeking a Buyer for Cable-Ship Business," Wall Street Journal, October 5,1996\()\) a. If AT\&T assumed that the Submarine Systems division's annual profit would remain constant and could be reinvested at an annual return of \(15 \%,\) what would AT\&T have considered to be the 20 -year present value of its Submarine Systems division? (Assume a continuous stream.) b. If prospective bidder A considered that the annual profits of this division would remain constant and could be reinvested at an annual return of \(13 \%,\) what would bidder \(A\) consider to be the 20 -year present value of AT\&T's Submarine Systems? (Assume a continuous stream.) c. If prospective bidder B considered that over a 20-year period, profits of the division would grow by \(10 \%\) per year (after which it would be obsolete) and that profits could be reinvested at an annual return of \(14 \%,\) what would bidder B consider to be the 20-year present value of AT\&T's Submarine Systems? (Assume a continuous stream.)

Step-by-Step Solution

Answer

AT&T values it at 5666.67 million, bidder A at 6538.46 million, and bidder B at 21250 million.

Step 1: Understanding the Problem

We are tasked with finding the present value of the Submarine Systems division's expected profits over 20 years under different conditions set by AT&T, bidder A, and bidder B. The calculations involve continuous compounding and different assumptions regarding profit growth and reinvestment rates.

Step 1: Present Value for AT&T's Assumptions

We need to calculate the present value of a continuous annuity with a rate of return of \(15\%\). The formula for the present value \(PV\) of a continuous income stream \(I\) is \(PV = \frac{I}{r}\), where \(r\) is the interest rate. Here, \(I = 850\) million and \(r = 0.15\).Calculate:\[ PV = \frac{850}{0.15} = 5666.67 \text{ million.} \]

Step 2: Present Value for Bidder A's Assumptions

Bidder A assumes a continuous income stream with no growth at \(13\%\) return. Using the same continuous annuity formula:\[ PV = \frac{850}{0.13} = 6538.46 \text{ million.} \]

Step 3: Present Value for Bidder B's Assumptions

Bidder B considers an annual profit growth of \(10\%\) and a \(14\%\) discount rate. The present value of a growing continuous annuity is \(PV = \frac{I}{r - g}\), where \(g\) is the growth rate. Here, \(I = 850\) million, \(r = 0.14\), and \(g = 0.10\).Calculate:\[ PV = \frac{850}{0.14 - 0.10} = \frac{850}{0.04} = 21250 \text{ million.} \]

Key Concepts

Continuous Compounding

Continuous compounding is a method used to calculate interest where the frequency of compounding is theoretically infinite. Unlike traditional compounding, where interest is compounded at regular intervals such as annually or quarterly, continuous compounding can be thought of as interest being added at every possible instant. This results in a slightly higher amount of accumulated interest compared to regular compounding.

In mathematical terms, the value of an investment using continuous compounding can be expressed as \( P = Pe^{rt} \), where:

• \( P \) is the future value of the investment,
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
• \( r \) is the annual interest rate,
• \( t \) is the time in years.

For the situation involving AT&T, continuous compounding is relevant in determining how the annual profits from the Submarine Systems division would appreciate over a period of time given a certain interest rate. This concept is particularly useful when considering long-term investments or funds that accrue interest continuously, as in this exercise.

Annuity Formula

An annuity formula is used to calculate the present or future value of a series of cash flows that occur at regular intervals. When we talk about a continuous annuity, it refers to payments received or made continuously over a period of time, rather than at discrete intervals.

To find the present value of a continuous annuity, the formula is: \( PV = \frac{I}{r} \), where:

• \( PV \) represents the present value,
• \( I \) is the continuous cash inflow or income,
• \( r \) is the constant interest or discount rate.

In this context, AT&T considers the continuous annuity as the continuous stream of profits from its Submarine Systems division. The present value calculation provides a snapshot of the worth of future profits, considering they are continuously compounded over a set period, which helps in decision-making when evaluating investment or sales proposals.

Profit Growth

Profit growth refers to the increase in a company's profit over time. In the scenario of AT&T's Submarine Systems division, this concept is crucial for Bidder B's assessment. Profit growth is expected at an annual rate of 10% before the anticipated obsolescence in 20 years.

When calculating present value with profit growth, the formula for a growing annuity is used: \( PV = \frac{I}{r - g} \), where:

• \( I \) is the initial profit,
• \( r \) is the discount rate,
• \( g \) is the growth rate of the profit.

This formula adjusts for both the expected increase in profits over time and the time value of money, making it possible for Bidder B to estimate an accurate present value considering future growth. This growth rate acknowledges the potential for profits to increase yearly, significantly affecting long-term financial projections for the division's performance.

Reinvestment Rates

Reinvestment rates refer to the rates at which profits from an investment are assumed to be reinvested to generate further returns. This concept plays a crucial role in determining an asset's present value, as seen with the Submarine Systems division.

For AT&T and the bidders, different reinvestment rates reflect varying expectations about how the profits can be leveraged to create additional value. In assessing the division's worth:

• AT&T assumes profits can be reinvested at a 15% return,
• Bidder A forecasts a 13% reinvestment rate,
• Bidder B estimates the rate at 14%.

These rates help in calculating the present value of future cash flows, emphasizing the potential earnings if profits are effectively reinvested. Different reinvestment rates provide a unique perspective on the opportunity cost and value creation potential associated with the investment in this division. Considering these rates allows each party involved to make informed decisions based on their financial strategies and expectations for future growth.