Foundations of Financial Management: An Analysis (part 31)

 “Happiness is not in the mere possession of money; it lies in the joy of achievement, in the thrill of creative effort.” 

– Franklin D. Roosevelt

Valuation and Rates of Return

by

Charles Lamson

The factors that determine valuation are varied, and you will be exposed to many of them in the next several posts.

In the last several posts we considered the basic principles of the time value of money. In the next few posts we will use many of those concepts to determine how financial assets (bonds, preferred stock, and common stock) are valued and how investors establish the rates of return they demand. We will use material from these next few posts to determine the overall cost of financing to the firm. We merely turn The coin over. Once we know how much bondholders and stockholders demand in the way of rates of return, we will then observe what the corporation is required to pay them to attract their funds. The cost of corporate financing capital is subsequently used in analyzing whether a project is acceptable for investment or not. These relationships are depicted in Figure 1.

Valuation Concepts

The valuation of a financial asset is based on determining the present value of future cash flows. Thus we need to know the value of future cash flows and the discount rate to be applied to the future cash flows to determine the current value.

The market-determined required rate of return, which is the discount rate, depends on the market's perceived level of risk associated with the individual security. Also important is the idea that required rates of return are competitively determined among the many companies seeking financial capital. Investors are willing to accept low return for low risk and vice versa. The market allocates capital to companies based on risk, efficiency, and expected returns---which are based to a large degree on past performance. The reward to the financial manager for efficient use of capital in the past is a lower required return for investors than that of the competing companies that did not manage their the resources as well.

In the next few posts, we apply concepts of evaluation to corporate bonds, preferred stock, and common stock. Although we describe the basic characteristics of each form of security as part of the valuation process, extended discussion of each security is deferred until later posts.

Valuation of Bonds

As previously stated, the value of a financial asset is based on the concept of the present value of future cash flows. Let's apply this approach to bond valuation. A bond produces an annuity stream of interest payments and a $1,000 principal payment at maturity. These cash flows are discounted at Y, the yield to maturity. The value of Y is determined in the bond market and represents the required rate of return for bonds of a given risk and maturity. More will be said about the concept of yield to maturity in the next section. 

 The price of a bond is thus equal to the present value of regular interest payments discounted by the yield to maturity added to the present value of the principal (also discounted by the yield to maturity).

This relationship can be expressed mathematically as follows.

Although the price of the bond could be determined with extensive calculations, it is much simpler to use present value tables. We take the present value of the interest payments and then add this value to the present value of the principal payment at maturity.

Present Value of Interest Payments In this case we determine the present value of a $100 annuity for 20 years. The discount rate is 10 percent. Using Table 1 above, the present value of an annuity, we find the following:

Present Value of Principal Payment (Par Value) at Maturity This single value of $1,000 will be received after 20 years. Note the term principal payment at maturity is used interchangeably with par value or face value of the bond. We discount $1,000 back to the present at 10 percent. Using Table 2, present value of a single amount, we find the following:

Table 2 Present value of a single amount

The current price of the bond, based on the present value of interest payments and the present value of the principal payment at maturity, is $1,000.40.

The price of this bond in this case is essentially the same as its par, or stated, value to be received at maturity of $1,000. This is because the annual interest rate is 10 percent (the annual interest rate of $100 divided by $1,000) the and the yield to maturity, or discount rate, is also 10 percent. When the interest rate on the bond and the yield to maturity are equal, the bond will trade at par value. Later we shall examine mathematical effects of varying the yield to maturity above or below the interest rate on the bonnd. But first we will examine more fully the concept of yield to maturity, which is where we turn our attention in the next post.

*MAIN SOURCE: BLOCK & HIRT, 2005, FOUNDATIONS OF FINANCIAL MANAGEMENT, 11TH ED., PP. 268-271*

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Foundations of Financial Management: An Analysis (part 30)

“Money is only a tool. It will take you wherever you wish, but it will not replace you as the driver.” 

– Ayn Rand

 The Time Value of Money (part E)

 by

 Charles Lamson

Special Considerations in Time Value Analysis

In the previous posts covering the time value of money, we assumed interest was compounded or discounted on an annual basis. This assumption will now be relaxed. Contractual arrangements, such as an installment purchase agreement or a corporate bond contract, may call for semiannual, quarterly, or monthly compounding periods. The adjustment to the normal formula is quite simple. To determine n (number of periods), simply multiply the number of years by the number of compounding periods during the year. The factor for i (interest rate) is then determined by dividing the quoted annual interest rate by the number of compounding periods.

Case 1---Find the future value of a $1,000 investment after 5 years at 8% annual interest, compounded semi-annually.

n = 5 X 2 = 10      i = 8%/2 = 4 %

*Again, the IF in the subscript indicates we are using the interest factor from the table.

Case 2---Find the present value of 20 quarterly payments of $2,000 each to be received over the next five years. The stated interest rate is 8% per annum. The problem calls for the present value of an annuity. We again follow the same procedure as in Case 1 in regard to n and i.

*Again, the A in the subscript indicates we are talking about annuities rather than a single amount.

Patterns of Payment

Time value of money problems may evolve around a number of different payment or receipt patterns. Not every situation will involve a single amount or an annuity. For example a contract may call for the payment of a different amount each year over a three-year period. To determine present value, each payment is discounted (Table 2 from Part 26 of this analysis) to the present and then summed.

(Assume 8% discount rate)

A more involved problem might include a combination of single amounts and an annuity. If the annuity will be paid at some time in the future, it is referred to as a deferred annuity and it requires a special treatment. Assume the same problem as above, but with an annuity of $1,000 that will be paid at the end of each year from the fourth through the eighth year. With a discount rate of 8%, what is the present value of the cash flows?

We know the present value of the first three payments is $5,022, from our calculation above, but what about the annuity? Let's diagram the 5 annuity payments.

The information source is Table 4, the present value of an annuity of $1 in Part 27 of this analysis. For n = 5, i = 8 percent, the discount factor is 3.993---leaving a "present value" of the annuity of $3,993. However, tabular values only discount to the beginning of the first stated period of an annuity---in this case the beginning of the fourth year as diagrammed below.

The $3,993 must finally be discounted back to the present. Since this single amount falls at the beginning of the fourth period---in effect, the equivalent of the end of the third period---we discount back for 3 periods at the stated 8 percent interest rate. Using Table 2, we have:

A second method for finding the present value of a deferred annuity is to:

$3,170 is the same answer for the present value of the annuity as that reached by the first method. The present value of the five-year annuity may now be added to the present value of the inflows over the first three years to arrive at the total value. 

*MAIN SOURCE: BLOCK & HIRT, 2005, FOUNDATIONS OF FINANCIAL MANAGEMENT, 11TH ED., PP. 256-258*

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Foundations of Financial Management: An Analysis (part 29)

“Rule No.1: Never lose money. Rule No.2: Never forget rule No.1.” 

– Warren Buffett

The Time Value of Money (Part D)

by

Charles Lamson

Determining the Yield on an Investment

In our discussion thus far, we have considered the following time value of money problems.

In each case we knew three out of the four variables and solved for the fourth. We will follow the same procedure again, but now the unknown variable will be i, the interest rate, or yield on the investment.

Yield---Present Value of a Single Amount

An investment producing $1,464 after 4 years has a present value of $1,000. What is the interest rate, or yield, on the investment?

We take the basic formula for the present value of a single amount and rearrange the terms.

We next express this value (0.003) as a fraction of the preceding value (0.024) and multiply by the difference between the two interest rates (6 percent minus 5 percent). The value is added to the lower interest rate (5 percent) to get a more exact answer of 5.125 percent rather than the estimated 5 percent.

Yield---Present Value of an Annuity

We may also find the yield related to any other problem. Let's look at the present value of an annuity. Take the basic formula for the present value of an annuity, and rearrange the terms.

The appropriate table is Table 4 from last post (the present value of an annuity of $1). Assuming a $10,000 investment will produce $1,490 a year for the next 10 years, what is the yield on the investment?

Is the student will flip back two tables for and read across the columns 4 in equals 10 periods, he or she will see that the yield is 8%.

The same type of approximated or interpolated yield that applied to a single amount can also be applied to an annuity when necessary. 

*MAIN SOURCE: BLOCK & HIRT, 2005, FOUNDATIONS OF FINANCIAL MANAGEMENT, 11TH ED., PP. 254-256*

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Foundations of Financial Management: An Analysis (part 28)

“Friendship is like money, easier made than kept.” 

– Samuel Butler

The Time Value of Money (Part C)

by

Charles Lamson

 

Determining the Annuity Value

In our prior discussion of annuities from last post, we assumed the unknown variable was the future value or the present value---with specific information available on the annuity value (A), the interest rate, and the number of periods of years. In certain cases our emphasis may shift to solving for one of these other values on the assumption that future value or present value is given. For now, we will concentrate on determining an unknown annuity value.

Annuity Equaling a Future Value

Assuming we wish to accumulate $4,641 after 4 years at a 10% interest rate, how much must be set aside at the end of each of the four periods? We take the previously developed statement from last post for the future value of an annuity and solve for A.

The solution is the exact reverse of that previously presented under the discussion of the future value of an annuity. As a second example, assume the director of the Women's Tennis Association must set aside an equal amount for each of the next 10 years to accumulate $1,000,000 in retirement funds and the return on deposited funds is 6%. Solve for the annual contribution, A, using Table 3 from last post.

Annuity Equaling a Present Value

In this instance, we assume you know the present value and you wish to determine what size annuity can be equated to that amount. Suppose your wealthy uncle presents you with $10,000 now to help you get through the next four years of college, or whatever. If you are able to earn 6 percent on deposited funds, how many equal payments can you withdraw at the end of each year for 4 years? We need to know the value of an annuity equal to a given present value. We take the previously developed statement for the present value of an annuity and we reverse it to solve for A.

 The appropriate table is Table 4 from last post (present value of an annuity). We determine an answer of $2,886.

The flow of funds follow the pattern in Table 5. Annual interest is based on the beginning balance for each year.

Table 5 Relationship of present value to annuity

The same process can be used to indicate necessary repayments on a loan. Suppose a homeowner signs a $40,000 mortgage to be repaid over 20 years at 8 percent interest. How much must he or she pay annually to eventually liquidate the loan? In other words, what annuity paid over 20 years is the equivalent of a $40,000 present value with an 8 percent interest rate?

Part of the payments to the mortgage company will go toward the payment of interest, with the remainder applied to debt reduction, as indicated in Table 6.

Table 6 Payoff table for loan (amortization table)

If this same process is followed over 20 years, the balance will be reduced to zero. You might note that the homeowner will pay over $41,000 of interest during the term of the loan, as indicated below.

*MAIN SOURCE: BLOCK & HIRT, 2005, FOUNDATIONS OF FINANCIAL MANAGEMENT, 11TH ED., PP. 252-254*

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Foundations of Financial Management: An Analysis (part 27)

“Money never made a man happy yet, nor will it. The more a man has, the more he wants. Instead of filling a vacuum, it makes one.”

- Benjamin Franklin

The Time Value of Money (Part B)

by

Charles Lamson

Future Value---Annuity

Our calculations up to now have dealt with single amounts rather than an annuity, which may be defined as a series of consecutive payments or receipts of equal amount. The annuity values are generally assumed to occur at the end of each period. If we invest $1,000 at the end of each year for four years and our funds grow at 10%, what is the future value of this annuity? We may find the future value for each payment and then total them to find the future value of an annuity (Figure 2).

Figure 2 (Assumption for annuity is that 1st payment doesn't come till end of first year.)

The future value for the annuity in Figure 2 is $4,641. Although this is a four period annuity, the first $1,000 comes at the end of the first period and has but three periods to run, the second $1,000 at the end of the second period, with two periods remaining---and so on down to the last $1,000 at the end of the fourth period. The final payment (period 4) is not compounded at all.

Because the process of compounding the individual values is tedious, special tables are also available for annuity computations. We shall refer to Table 3 above, the future value of an annuity of $1. Let us define A as the annuity value and use Formula 3 for the future value of an annuity. Note that the A part of the subscript on both the left- and right-hand side of the formula below indicates we are dealing with tables for an annuity rather than a single amount. Lamont. Using Table 3 for A = $1,000, and n = 4, and i = 10%:

If a wealthy relative offered to set aside $2,500 a year for you for the next 20 years, how much would you have in your account after 20 years if the funds grew at 8 percent? The answer is as follows:

A rather tidy sum considering that only a total of $50,000 has been invested over the 20 years.

Present Value---Annuity

To find the present value of an annuity, the process is reversed. In theory each individual payment is discounted back to the present and then all of the discounted payments are added up, yielding the present value of the annuity.

Table Four allows us to eliminate extensive calculations and to find our answer directly.

*MAIN SOURCE: BLOCK & HIRT, 2005, FOUNDATIONS OF FINANCIAL MANAGEMENT, 11TH ED., PP. 242-245*

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