If an amount is loaned and then repaid in installments, we say that the loan is amortized. Therefore, loan amortization is the process of calculating

the loan payments that amortize the loaned amount. We can determine the amount of the loan payments once we know the frequency of payments, the interest rate, and the number of payments.

Consider a loan of $100,000. If the loan is repaid in four annual installments (at the end of each year) and the interest rate is 6% per year. The first thing we need to do is to calculate the amount of each payment. In other words, we need to solve for CF:

4

$100,000 =

CF

t=1 (1 + 0.06)t

We want to solve for the loan payment, that is, the amount of the annuity. The calculator and spreadsheet inputs for this calculation are:

PV = 100,000 i = 6%

N = 4

and then solve for PMT. This is the CF, the loan payment.

The loan payment is $28,859.15. We can calculate the amount of interest and principal repayment associated with each loan payment using a loan amortization schedule, as we show in Panel A of Exhibit 10.10.

The loan payments are determined such that after the last payment is made there is no loan balance outstanding. Thus, the loan is referred to as a fully amortizing loan. You can see this in Panel B of Exhibit 10.3. Even though the loan payment each year is the same, the proportion of interest and principal differs with each payment: the interest is 5% of the principal amount of the loan that remains at the beginning of the period, whereas the principal repaid with each payment is the difference between the payment and the interest. As the payments are made, the remainder is applied to repayment of the principal. This is the scheduled principal repayment or the amortization. As the principal remaining on the loan declines, less interest is paid with each payment.

Loan amortization works the same whether this is a mortgage loan to purchase a home, a term loan, or any other loan such as an automobile loan in which the interest paid is determined on the basis of the remaining amount of the loan. You can modify the calculation of the loan amortization to suit different principal repayments, such as additional lump-sum payments, known as balloon payments. See Exhibit 10.11.

arrangements. But this law does not apply beyond savings accounts. One investment may pay 10% interest compounded semiannually, whereas another investment may pay 9% interest compounded daily. One financing arrangement may require interest compounding quarterly, whereas another may require interest compounding monthly.

Want to compare investments or financing with different frequencies of compounding? We must first translate the stated interest rates into a common basis. There are two ways to convert interest rates stated over different time intervals so that they have a common basis: the annual percentage rate and the effective annual interest rate.

One obvious way to represent rates stated in various time intervals on a common basis is to express them in the same unit of time—so we annualize them. The annualized rate is the product of the stated rate of interest per compound period and the number of compounding periods in a year. Let i be the rate of interest per period and n be the number of compounding periods in a year. The annualized rate, which is as we indicated earlier in this chapter also referred to as the nominal interest rate or the annual percentage rate (APR), is

Another way of converting stated interest rates to a common basis is the effective rate of interest. The effective annual rate (EAR) is the true economic return for a given time period because it takes into account the compounding of interest. We also refer to this rate as the effective rate of interest. The formula is

Let’s look how the EAR is affected by the compounding. Suppose that the Safe Savings and Loan promises to pay 2% interest on accounts, compounded annually. Because interest is paid once, at the end of the year, the effective annual return, EAR, is 2%. If the 2% interest is paid on a semiannual basis—1% every six months—the effective annual return is larger than 2% since interest is earned on the 1% interest earned at the end of the first six months. In this case, to calculate the EAR, the interest rate per compounding period—six months—is 0.01 (that is, 0.02 ÷ 2) and the number of compounding periods in an annual period is 2:

EAR = (1 + 0.01)2 − 1 = 1.0201 − 1 = 0.0201 or 2.01%

In the case of continuous compounding, the EAR is simply:

Extending this example to the case of quarterly compounding and continuous compounding with a nominal interest rate of 2%, we first calculate the interest rate per period, i, and the number of compounding periods in a year, n:

Figuring out the effective annual rate is useful when comparing interest rates for different investments. It doesn’t make sense to compare the APRs for different investments having a different frequency of compounding within a year. But since many investments have returns stated in terms of APRs, we need to understand how to work with them.

To illustrate how to calculate effective annual rates, consider the rates offered by two banks, Bank A and Bank B. Bank A offers 4.2% compounded semiannually and Bank B other offers 4.158% compounded continuously. We can compare these rates using the EARs. Which bank offers the highest interest rate? The effective annual rate for Bank A is (1 + 0.021)2 − 1 = 4.2441%. The effective annual rate for Bank B is e0.04158 – 1 = 4.2457%. Therefore, Bank B offers a slightly higher interest rate.

Y i e l d s o n I n v e s t m e n t s

Suppose an investment opportunity requires an investor to put up $10,000 million and offers cash inflows of $4,000 after one year and $7,000 after two years. The return on this investment, or yield, is the interest rate that equates the present values of the $4,000 and $7,000 cash inflows to equal the present value of the $1 million cash outflow. This yield is also referred to as the internal rate of return (IRR) and is calculated as the rate that solves the following:

$4,000 $7,000 $10,000 = (1 + IRR)1 + (1 + IRR)2

Unfortunately, there is no direct mathematical solution (that is, closedform solution) for the IRR, but rather we must use an iterative procedure. Fortunately, financial calculators and financial software ease our burden

S o l u t i o n

The IRR for this investment is the interest rate that solves:

$200,000 $300,000 $900,000 $1,000,000 = (1 + IRR)2 + (1 + IRR)3 + (1 + IRR)4

The return is 10.172%.

We can use this approach to calculate the yield on any type of investment, as long as we know the cash flows—both positive and negative—and the timing of these flows. Consider the case of the yield to maturity on a bond. Most bonds pay interest semiannually—that is, every six months. Therefore, when calculating the yield on a bond, we must consider the timing of the cash flows to be such that the discount period is six months.

T R Y I T ! T H E Y I E L D O N A N I N V E S T M E N T

Suppose you invest $1,000 today in an investment that promises you $1,000 in two years and $10,000 in three years. What is the IRR on this investment?

E X A M P L E 1 0 . 1 1 : C A L C U L A T I N G T H E Y I E L D

O N A B O N D

Consider a bond that has a current price of 90; that is, if the par value of the bond is $1,000, the bond’s price is 90% of $1,000 or $900. And

suppose that this bond has five years remaining to maturity and an 8% coupon rate. With five years remaining to maturity, the bond has 10 six-month periods remaining.

(continued )

(Continued)

S o l u t i o n

With a coupon rate of 8%, this means that the cash flows for interest is $40 every six months. For a given bond, we therefore have the following information:

Present value = $900

Number of periods to maturity = 10

Cash flow every six months = $40

Additional cash flow at maturity = $1,000

The six-month yield, rd, is the discount rate that solves:

Using a calculator or spreadsheet, we calculate the six-month yield as 5.315% [PV = $900; N = ’10; PMT = $40; FV = $1,000]. Bond yields are generally stated on the basis of an annualized yield, referred to as the yield to maturity on a bond-equivalent basis. This measure is analogous to the APR with semiannual compounding. Therefore, yield to maturity is 10.63%.

T H E B O T T O M L I N E

The time value of money is one of the foundation concepts and tools in financial and investment management.

Using compound interest, we can estimate a value of in the future; using discounting, we can translate a future value into a value today—a present value.

It is important to consider the type of interest—compounding vs. simple—and the frequency of compounding in determining a present value of a future value.

The time value of money mathematics can be used to determine the present value or future value of a lump-sum amount or of a series of cash flows, the growth rate of values, the number of periods of interest to meet a goal, or to simply amortize a loan.

Given the cost of an investment and its cash flows, we can calculate the yield or implied interest rate. By calculating the yield or implied interest rate, we can then compare investment or financing opportunities. The yield or internal rate of return on an investment is the interest rate at which the present value of the cash flows equals the initial investment outlay.

S O L U T I O N S T O T R Y I T ! P R O B L E M S

Future Value

a.FV = $100 (1+ 0.02)5 = $110.41

b.FV = $100 (1+ 0.02)10 = $121.90

c.FV = $100 (1+ 0.02)20 = $148.59

Growth Rates

PV = $2,000; FV = $4,000; N = 5 Solve for i. i = 14.87%

Frequency of Compounding

a.$100 (1 + 0.02)10 = $121.899

b.$100 (1 + 0.005)40 = $122.079

c.$100 e 0.2 = $122.140

Present Value

PV = $1,000 ÷ (1 + 0.06)10 = $558.39

The Yield on an Investment

Cash flows are −$10,000, $0, $1,000 and $10,000. The yield is 3.332%

Q U E S T I O N S

1.What is the relationship between compounding and discounting of a lump-sum?

2.Complete the following: “The larger the interest rate, the (larger/smaller) the future value of a value today.”

3.Holding everything else the same, what is the effect of using a higher discount rate to discount a future value to the present?

4.If you invest the same amount in each of three accounts today, which account produces the highest future value if the annual percentage rate is the same? Account A: annual compounding, Account B: quarterly compounding, Account C: continuous compounding.

5.What distinguishes an ordinary annuity from an annuity due?

6.What distinguishes an ordinary annuity from a deferred annuity?

7.If a cash flow is the same amount each period, ad infinitum, how do we value the present value of this series of cash flows?

8.Which is most appropriate to use in describing the annual growth of the value of an investment: the arithmetic average growth rate or the geometric average growth rate? Why?

9.How can we break down the valuation of a deferred annuity into manageable parts for computation purposes?

10.Which has the highest present value if the payments and number of payments are identical, an ordinary annuity or an annuity due?

11.If you are offered two investments, one that pays 5% simple interest per year and one that pays 5% compound interest per year, which would you choose? Why?

12.Consider a borrowing arrangement in which the annual percentage rate (APR) is 8%.

a.Under what conditions does the effective annual rate of interest (EAR) differ from the APR of 8%?

b.As the frequency of compounding increases within the annual period, what happens to the relationship between the EAR and the APR?

13.Suppose you deposit $1,000 in an account with an APR of 4%, with compounding quarterly.

a.After 10 years, what is the balance in the account if you make no withdrawals?

b.After 10 years, how much interest on interest did you earn?

14.Suppose you are promised $10,000 five years from today. If the appropriate discount rate is 6%, what is this $10,000 worth to you today?

15.Suppose you buy a car today and finance $10,000 of its cost at an APR of 3%, with payments made monthly.

a.If you finance the car for 24 months, what is the amount of your monthly car payment?

b.If you finance the car for 36 months, what is the amount of your monthly car payment?

The operating performance of a company is a measure of how well a company has used its resources to produce a return on its investment. The financial condition of a company is a measure of its ability to satisfy its obligations, such as the payment of interest on its debt in a timely manner. An investor has many tools available in the analysis of financial information. These tools include financial ratio analysis and cash flow analysis. Cash flows provide a way of transforming net income based on an accrual system to a more comparable basis. Additionally, cash flows are essential ingredients in valuation because the value of a company today is the present value of its expected future cash flows. Therefore, understanding past and current cash flows may help in forecasting future cash flows and, hence, determine the value of the company. Moreover, understanding cash flow allows the assessment of the ability of a company to maintain current dividends and its current capital expenditure policy without relying on external financing.

In this chapter and the next, we describe and illustrate the basic tools of financial analysis. In this chapter, our focus is on financial ratio analysis. In the next chapter, we cover cash flow analysis.

C L A S S I F Y I N G F I N A N C I A L R A T I O S

A financial ratio is a comparison between one bit of financial information and another. Consider the ratio of current assets to current liabilities, which we refer to as the current ratio. This ratio is a comparison between assets that can be readily turned into cash—current assets—and the obligations that are due in the near future—current liabilities. A current ratio of 2, or 2:1, means that we have twice as much in current assets as we need to satisfy obligations due in the near future.

We can classify ratios according to the way they are constructed and the financial characteristic they are describing. For example, we will see that the current ratio is constructed as a coverage ratio (i.e., the ratio of current assets—available funds—to current liabilities, i.e., the obligation) that we use to describe a company’s liquidity (its ability to meet its immediate needs). We can also classify ratios according to the dimension of the company’s performance or condition. For example, a current ratio provides information on a company’s liquidity, whereas a turnover ratio provides information on the effectiveness to which the company puts its asset to use.

There are as many different financial ratios as there are possible combinations of items appearing on the income statement, balance sheet, and statement of cash flows. We can classify ratios according to the financial characteristic that they capture.

When we assess a company’s operating performance, a concern is whether the company is applying its assets in an efficient and profitable

manner. When an investor assesses a company’s financial condition, a concern is whether the company is able to meet its financial obligations. The investor can use financial ratios to evaluate five aspects of operating performance and financial condition:

1.Liquidity

2.Profitability

3.Activity

4.Financial leverage

5.Return on investment

There are several ratios reflecting each of the five aspects of a company’s operating performance and financial condition. We apply these ratios to the Exemplar Corporation, whose balance sheets, income statements, and statement of cash flows for two years we show in Exhibits 11.1, 11.2, and

E X H I B I T 1 1 . 1 Exemplar Corporation’s Balance Sheets

11.3, respectively. We refer to the most recent fiscal year for which financial statements are available, FY20X2, as the “current year.” The “prior year” is the fiscal year prior to the current year.

The ratios we introduce here are by no means the only ones that can be formed using financial data, though they are some of the more commonly

E X H I B I T 1 1 . 3 Exemplar Corporation’s Statement of Cash Flows