The Foundation
4.10 Other Time Value Applications
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Solving for unknown interest rates is among the most common time value of money applications. Examples include solving for the expected yield-to-maturity for a corporate bond, discussed in Chapter 5, and the internal rate of return for a capital investment project, discussed in Chapter 8. Investors often determine the interest rate that equates the present value of expected future amount with the market price of a security to identify the expected return on that investment.
Interest portion of loan payment
Beginning of month loan balance
= ×r = 0.01 × $20,000 = $200.00
Since the total monthly payment of $444.89 consists of both interest and principal, the principal portion of the loan payment can be computed by subtracting the interest portion of the payment from the total payment.
Principal portion
of loan payment = Loan payment − Interest = $444.89 − $200.00 = $244.89 Now compute the end-of-month balance by subtracting the principal portion of the payment from the beginning-of-month balance:
End of month loan balance
Beginning of month loan balance
Principal
= − paid = $20,000 − $244.89 = $19,755.11 Repeat this same process for months 2–60 to complete the amortization table. Table 4.4 contains partial results.
Several important patterns are illustrated in Table 4.4. First, note that in the first few months, a substantial portion of the monthly payment is interest, about 45 percent. As time elapses, the portion of the payment representing interest declines, with more of the payment going to reduce the principal balance. Toward the end of the life of the loan, nearly all of the monthly payment represents principal repayment.
Leasing
Leasing of assets has become an increasingly popular means for acquiring assets by both corporate and personal users. Basic time value of money principles can be used to determine the periodic lease payment over the term of the lease. We illustrate the calculation of the monthly payment for an auto lease in Example 4.17.
Table 4.4 Amortization table for a $20,000 loan at 12 percent for 60 months
Month Beginning of Loan payment Interest Principal End of month
month balance ($) ($) ($) ($) balance ($)
1 20,000.00 444.89 200.00 244.89 19,755.11
2 19,755.11 444.89 197.55 247.34 19,507.77
3 19,507.77 444.89 195.08 249.81 19,257.96
58 1,308.41 444.89 13.08 431.81 876.60
59 876.61 444.89 8.77 436.12 440.49
60 440.49 444.88 4.40 440.49 0.00
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Example 4.17 Leasing
Vitel Incorporated has agreed to lease a fleet of Leopard luxury cars from Sports Motors Inc. (SMI) for its executive management. The Leopard model sells at its MSRP of $49,900. SMI estimates the residual market value of a Leopard to be $28,500 after three years. Assuming a market interest rate of 9 percent per year (compounded monthly) on auto loans and leases, what is the monthly lease payment?
Solution: In a leasing payment structure, the estimated residual payment at the end of the leasing term is essentially a final balloon payment. By returning the car, the lessee is giving an asset worth the estimated residual amount to the lessor. Thus, the time diagram for the lease can be depicted as follows:
Year
Cash flow
The present value of the lease ($49,900) must be equal to the present value of the lease payments and the present value of the terminal residual lease amount of $28,500 at the monthly discount rate of 0.75 percent a month:
49 900
1 1
1 0075
0 0075 28 500 1
1 0075
36
, 36
( . )
. ,
( . )
= ⎡ −
⎣
⎢⎢
⎢
⎤
⎦
⎥⎥
⎥+ ⎡
⎣⎢ ⎤
⎦⎥ PMT
The keystrokes on the BA II PLUS® calculator would be:
1 2nd CLR TVM 2 36 N
3 0.75 I/YR 4 49,900 PV 5 28,500 +/− FV 6 CPT PMT
These steps will look as follows on the BA II PLUS® calculator:
36 N
0.75 I/Y
49,900 CPT
PV PMT
28,500 +/−
FV (last step)
The lease payment should be set at $894.26.
. . .
0 1 2 3 4 5 6 33 34 35 36
PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT +
$28,500
Solving for Unknown Time Periods
Suppose a financial manager wants to know the amount of time needed for an amount of money (present value) to grow to a specified (future value) amount. We illustrate how to solve this type of problem in Example 4.18.
Example 4.18 Solving for an Unknown Time Period
A financial manager wants to know the approximate time needed to double a $1 million investment if the investment earns an annual interest rate of 5 percent.
Solution: Given a future value that is twice the size of the present value and an annual interest rate of 5 percent, what is n? Substituting PV = 1, FV = 2 and r = 0.05 into the future value of a present amount formula (Equation 4.1):
FV = PV (1 + r)n 2 = 1(1.05)n
Using the BA II PLUS® calculator enter:
1 2nd CLR TVM 2 5 I/YR 3 1 PV 4 2 +/− FV 5 CPT N
N
5 I/Y
1 CPT
PV PMT
2 +/−
FV (last step)
These keystrokes should give N = 14.21 years. Be careful in interpreting this number.
Recall that with annual compounding, interest is paid annually. After years 14 and 15, the future value will be about $1.98 million and $2.08 million, respectively. The financial manager will not have $2 million as an intermediate amount any time during year 14 because interest is not paid during the year. Thus, the best answer is 15 years.
The financial manager will have more than doubled the investment at that time.
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Summary
This chapter discusses major principles and applications of time value of money. The key concepts covered are summarized below.
1 Because cash can be invested to earn a positive rate of interest, most individuals would prefer to have $1 today rather than $1 at some time in the future. Thus, money is said to have time value.
2 The most basic formulas involving time value of money include determining the future value of a present amount or the present value of a future amount.
3 Compounding is the process used to determine the future value of a current amount.
4 Discounting is the process used to determine the present value of some future amount.
5 The interest rate used to compute a present value is called a discount rate. Present values are inversely related to the discount rate.
6 An annuity is a series of payments that occur at evenly spaced intervals over time.
With an ordinary annuity, payments occur at the end of each period. With an annuity due, payments occur at the beginning of each period.
7 A perpetuity is an annuity with an infinite life. Computing the present value of a perpetuity involves dividing the perpetuity payment amount by the discount rate.
8 A nominal interest rate is a quoted annual interest rate. It ignores the compounding effect when interest is compounded more frequently than once per year. An effec- tive annual interest rate is the annual rate that takes into account the compounding frequency in the nominal rate.
FURTHERREADING
Brigham, Eugene F. and Michael C. Ehrhardt. Financial Management: Theory and Practice, 11th edn, Thomson, 2005.
Keown, Arthur J., John D. Martin, J. William Petty, and David F. Scott, Jr. Financial Management:
Principles and Applications, 10th edn, Pearson-Prentice Hall, 2005.
SELECTED WEBSITES
http://financenter.com is where calculators may be obtained for solving a variety of problems in finance.
http://invest-faq.com/articles/analy-fut-prs-val.html explains basic time value of money principles.
Chapter 5
Valuation
The problem in valuation is not that there are not enough models to value an asset, it is that there are too many. Choosing the right model to use in valuation is as critical to arriving at a reasonable value as understanding how to use the model. (Aswath Damodaran, Investment Valuation, John Wiley &
Sons, 1996, p. 501.)
Overview
Major decisions of a company are all interrelated in their effect on the valuation of the firm and its securities. Corporate managers, especially financial managers, need to understand how their decisions affect the value of their firm’s common stock and shareholder wealth. They also need to understand how the market values the financial instruments of a company. Finally, to achieve the goal of maximizing shareholder wealth, managers must comprehend what creates it. Thus, valuation is a fundamental issue in finance.
In the previous chapter, we discussed the basic procedures used to value future cash flows. In this chapter, we show how to use time value of money techniques to value a firm’s long-term securities – bonds, preferred stock, and common stock. Although the concept of value has several meanings, we focus on determining a security’s intrinsic value, which is what the security ought to be worth. We discuss two major approaches to valuation – discounted cash flow valuation and relative valuation. We focus on discounted cash flow models to estimate the value of bonds, preferred stock, and common stock. We also discuss relative valuation, which uses multiples to estimate stock values. Along the way, we discuss some terminology and features associated with bonds, preferred stock, and common stock.
Learning Objectives
After completing this chapter, you should be able to:
• explain the concept of valuation and five different types of value;
• differentiate between discounted cash flow valuation and relative valuation;
• describe the key inputs to the basic model used in the valuation process;
• estimate the intrinsic value of annual-pay, semiannual-pay, and zero coupon bonds;
• understand six fundamentals of bond pricing that relate bond prices to coupon rates, the required return, and the remaining term to maturity;
• explain the importance of interest rate risk for bonds;
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• explain and calculate the current yield, yield to maturity, and yield to call of a bond;
• calculate the intrinsic value of preferred stock;
• calculate the intrinsic value of common stock using dividend discount models with different growth rate assumptions;
• discuss when using multiples is appropriate to value a stock;
• describe the major characteristics and features of bonds, preferred stock, and common stock.