QAs
Mr.Q
lecture.rarlecture/Bond-Stock Valuation.ppt
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BOND VALUATION
I. BOND VALUTION TERMINOLOGY
Bond
Par or Face Value: Maturity Value, M
Coupon Interest Rate: Coupon Payment
Maturity: Maturity Date
Call Provision; Call Protection; Call Premium
Issue Date
Default Risk
Yield to Maturity or Yield, kd
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II. GENERAL VALUTION MODEL
The value of any asset can be found as the present value of its expected future cash flows, discounted at the yield to maturity (ytm):
P = Principal value or face value or maturity value
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The discount rate depends on:
- The risk of the cash flows.
- The general level of interest rates, which reflects inflation, supply and demand for money, production opportunities and time preferences for consumption.
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III. VALUING A BOND
1.Suppose you are a CFO and want to issue a two-year $1000 face value note with 8% semiannual coupons. What is a the price at which you can sell the note? You consider notes of similar characteristics should yield 10%.
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2.You want to issue the note for around $965, but find there is not much demand for it. What should you do?
3.Suppose you offer the note for $955. What is the implied yield?
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4.Suppose when you wanted to issue the note for $965 there is a great deal of interest in the note. So you want to sell the note for $975. What is the implied yield?
5.What is the relationship between bond price and yield?
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6.Suppose an investor buys the bond today at $975 and sells it after six months (after the first coupon is paid) when the yield to maturity is 10%. (a) What would be the 6-month holding period return (HPR)?
(b) What would be the annualized return?
IV. INFLATION AND REAL VERSUS NOMINAL INTEREST RATES
- Quotes of interest rates in the financial press are commonly referred to as the nominal (or quoted) interest rates. Real rate of interest adjusts for the effects of inflation.
- Real Rate of Interest (approximation)
≈ Nominal interest rate – Inflation premium
Fisher Effect: The Nominal and Real Rate of Interest
- The relationship between the nominal rate of interest, rnominal , the anticipated rate of inflation, rinflation , and the real rate of interest is known as the Fisher effect.
V. INTEREST-RATE DETERMINANTS
The nominal return or interest rate on a note or bond can be thought of including five basic components:
INTEREST-RATE DETERMINANTS (CONT’D)
- The inflation premium
- Default–risk premium
- Maturity-risk premium
- Liquidity-risk premium
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VI. WHAT WE HAVE LEARNT
- Finding bond price
- Finding yield to maturity (YTM)
- Relationship between bond price and YTM
- Finding HPR
- Relationship between YTM and HPR
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STOCK VALUTION
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Time Pattern of Dt
$ g = Nonconst.
g = Constant
Do g = 0
g = Neg. const.
t
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ZERO GROWTH CASE
Applicable for firms with no growth prospects
Zero growth in earnings => Zero growth in dividends => Find value of a perpetuity
The valuation formula reduces to:
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Suppose HotJobs, Inc. sees bleak future and expects its earnings to remain flat at $2.00 per share indefinitely. Investors require a 16% yield from it.
- Vps = the value of a share of preferred stock
- Dps = the annual preferred stock dividend
- rps = the market yield or the rate of return on the preferred stock’s promised dividend
ZERO GROWTH CASE (Cont’d)
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CONSTANT GROWTH CASE
Applicable for a firm that expects to grow at the same (constant) rate indefinitely
For a constant growth stock, earnings, dividends and stock prices are all expected to grow at the constant rate ‘g’
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CONSTANT GROWTH CASE (Cont’d)
Under the assumption of constant growth in dividends indefinitely, the valuation model reduces to:
rCS the required return on the stock
Note (1) that ‘g’ must be constant forever, and (2) that rCS must be greater than ‘g’.
Determinants of the Investor’s Required Rate of Return
The investor’s required rate of return is determined by two key factors:
The level of interest rates in the economy;
The risk of the firm’s stock.
Determinants of the Investor’s Required Rate of Return (cont’d)
CAPM (SML relationship) suggests that if risk-free rate and/or systematic risk (beta) rises, the investor’s required rate of return will rise and the stock price will fall.
Determinants of the Growth Rate of Future Dividends
The growth rate of future dividends (g) can also change and lead to a change in the stock price. The two key determinants of a firm’s growth opportunities relate to:
the return on equity (ROE), and
the retention ratio (b)
Determinants of Growth Rate of Future Dividends (cont’d)
The growth rate is formally expressed as follows:
- g = the expected rate of growth of dividends
- D1/E1 = the dividend payout ratio
- b = the proportion of firm’s earnings that are retained and reinvested in the firm.
- ROE = the return on equity earned when the firm reinvests a portion of its earning back into the firm.
Determinants of the Investor’s Required Rate of Return (cont’d)
- Dividend payout ratio = 0.5
- Return on equity (ROE) = 12%
- g = (1 - 0.5)12% = 6%
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CONSTANT GROWTH CASE (Cont’d)
Assume HotJobs, Inc. foresees better times, and it expects to grow, but at a constant rate indefinitely. Data on the stock are:
b = 1.2 rrf = 10% rm = 15%
(a) Find rcs
SML: rcs = rrf +(rm + rrf)b
= 10%+(15% - 10%)1.2= 16%
(b) If Do = $2 and g = 6% = constant, find:
D1 =
D2 =
D3 =
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CONSTANT GROWTH CASE (Cont’d)
(c) Find
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CONSTANT GROWTH CASE (Cont’d)
Consider previous facts as given for HotJobs:
ks = 16%, D0 = $2, g = 6% (constant)
D1 = $2.12
D2 = $2.247
D3 = $2.382
Find
*
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SUPERNORMAL NORMAL GROWTH
HotJobs sees growth prospects of 30% over the next 3 years which is likely to fall to 6% level indefinitely thereafter.
(a) Find its intrinsic value today.
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SUPERNORMAL NORMAL GROWTH (Cont’d)
PV of Supernormal dividends
Stock price at t=3
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SUPERNORMAL NORMAL GROWTH (Cont’d)
(b) Find the intrinsic value expected at Yr. 1.
(c) Find the 1-year HPR.
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STOCK VALUATION MODELS
We can value a stock using the following approaches:
PV of all future dividends
PV of all future cash flows
PV of future earnings minus future investments
PV of current earnings + PV of growth opportunities
Defining the P/E Ratio Valuation Model
- Vcs = the value of common stock of the firm.
- P/E1 = the price earnings ratio for the firm based on the current price per share divided by earnings for end of year 1.
- E1 = estimated earnings per share of common stock for the end of year 1.
The Problem
After some careful analysis and reflection on the valuation of the Heals’ shares the company CFO suggested that the earnings projection are too conservative and earnings for the coming year could easily jump to $2.00. What does this do for your estimate of the value of Heals’ shares?
Solution
Suppose the PE ratio is 18.20.
Vcs = 18.20 × $2
= $36.40
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lecture/Chapter 10-Stock Valuation.ppt
Chapter 10
Stock
Valuation
10-*
Slide Contents
- Learning Objectives
- Principles Applied in This Chapter
Common Stock
The Comparables Approach to Valuing Common Stock
Preferred Stock
The Stock Market
- Key Terms
10-*
Learning Objectives
Identify the basic characteristics and features of common stock and use the discounted cash flow model to value common shares.
Use the price-to-earnings (P/E) ratio to value common stock.
Identify the basic characteristics and features of preferred stock and value preferred shares.
Use the secondary markets for common stock.
10-*
Principles Applied in This Chapter
- Principle 1: Money Has a Time Value.
- Principle 2: There is a Risk-Reward Tradeoff.
- Principle 3: Cash Flows are the Source of Value.
- Principle 4: Market Prices Reflect Information.
- Principle 5: Individuals Respond to Incentives.
10-*
Common Stock
Common stockholders are the owners of the firm. They elect the firm’s board of directors who in turn appoint the firm’s top management team. The firm’s management team then carries out the day-to-day management of the firm.
10-*
Common Stock Characteristics
Claim on Income Common stockholders have the right to the firm’s income after bondholders and preferred stockholders have been paid. The common stockholders either receive dividends or any increase in value that results from the reinvested earnings.
10-*
Common Stock Characteristics (cont.)
- Claim on Assets In case of liquidation, common stockholders have residual claim on assets.
- Voting Rights In general, common shareholders are the only security holders given the right to vote. Most shareholders vote by proxy.
10-*
Common Stock Characteristics (cont.)
Agency Costs and Common Stock Shareholders elect the board. In reality, board members are nominated by the management. As a result, management effectively elects the board. This may lead to agency problems.
10-*
Valuing Common Stock Using the Discounted Dividend Model
Like bonds, common stock’s value is equal to the present value of all future cash flows that the stockholder expects to receive from owning the shares of stock. However, unlike bonds, the future cash flows in the form of dividends are not fixed and there is no maturity date.
10-*
Three Step Procedure for Valuing Common Stock
Step 1: Estimate the amount and timing of the receipt of the future cash flows the common stock is expected to provide.
Step 2: Evaluate the riskiness of the common stock’s future dividends to determine the stock’s required rate of return.
10-*
Three Step Procedure for Valuing Common Stock (cont.)
Step 3: Calculate the present value of the expected dividends by discounting them back to the present at the stock’s required rate of return.
- The three steps show that the value of a common stock is equal to the present value of all future dividends.
10-*
The Constant Dividend Growth Rate Model
If a firm’s cash dividend grow by a constant rate, then the common stock can be valued as follows:
10-*
The Problem
What is the value of a share of common stock that paid $6 dividend at the end of last year and is expected to pay a cash dividend every year from now to infinity, with that dividend growing at a rate of 5 percent per year, if the investor’s required rate of return is 12% on that stock?
10-*
Step 1: Picture the Problem
With a perpetuity, a timeline goes on for ever with the growing cash flow occurring every period.
i=12%
Years
Cash flows $6 $6(1.05) $6(1.05)2
0
1
2 …
Value of common
stock = Present
Value of Expected
Dividends.
The growing
dividends go on
forever
…
10-*
Step 2: Decide on a Solution Strategy
- The value of a share of stock can be viewed as a the present value of a growing perpetuity.
- Here we know the expected dividends, the growth rate, and investor’s required rate of return.
- We can use equation 10-2 to determine the value of a share of common stock.
10-*
Step 3: Solve
- We need to first determine D1, the dividend next period.
- Since dividends at the end of last year was $6 and dividends are expected to grow at a rate of 5%, dividends for next period will be:
- D1 = D0 (1+g) = $6 (1.05) = $6.30
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Step 3: Solve (cont.)
- Vcs = $6.30 ÷ (0.12-0.05) = $6.30 ÷ 0.07
= $90
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Step 4: Analyze
Equation 10-2 is based on the assumption that dividends will grow at a constant rate for ever. While not a realistic assumption, it enables us to determine the value of common stock easily and also helps us to identify the factors that move the stock prices.
10-*
What Causes Stock Prices to Go Up and Down?
Equation 10-2 indicates that there are three variables that drive share value:
- The most recent dividend (D0),
- Investor’s required rate of return (rcs ), and
- Expected rate of growth in future dividends (g).
10-*
What Causes Stock Prices to Go Up and Down? (cont.)
Since most recent dividend (D0) has already been paid, it cannot affect price. Thus the other two variables, rcs and g, can vary and lead to changes in stock prices.
10-*
Determinants of the Investor’s Required Rate of Return
The investor’s required rate of return is determined by two key factors:
The level of interest rates in the economy;
The risk of the firm’s stock.
10-*
Determinants of the Investor’s Required Rate of Return (cont.)
CAPM suggests that if risk-free rate and/or systematic risk (beta) rises, the investor’s required rate of return will rise and the stock price will fall.
10-*
Determinants of the Growth Rate of Future Dividends
The growth rate of future dividends (g) can also change and lead to a change in the stock price. The two key determinants of a firm’s growth opportunities relate to:
- the return on equity (ROE), and
- the retention ratio (b)
10-*
Determinants of Growth Rate of Future Dividends (cont.)
The growth rate is formally expressed as follows:
- g = the expected rate of growth of dividends
- D1/E1 = the dividend payout ratio
- b = the proportion of firm’s earnings that are retained and reinvested in the firm.
- ROE = the return on equity earned when the firm reinvests a portion of its earning back into the firm.
10-*
The Comparables Approach to Valuing Common Stock
This method estimates the value of the firm’s stock as a multiple of some measure of firm’s performance. The most common metric is earnings per share. Thus values are determined from the price-to-earnings ratio of comparable firms.
10-*
Defining the P/E Ratio Valuation Model
- Vcs = the value of common stock of the firm.
- P/E1 = the price earnings ratio for the firm based on the current price per share divided by earnings for end of year 1.
- E1 = estimated earnings per share of common stock for the end of year 1.
10-*
CHECKPOINT 10.2:
CHECK YOURSELF
Valuing Common Stock
Using the P/E Ratio
10-*
The Problem
After some careful analysis and reflection on the valuation of the Heals’ shares the company CFO suggested that the earnings projection are too conservative and earnings for the coming year could easily jump to $2.00. What does this do for your estimate of the value of Heals’ shares?
10-*
Step 1: Picture the Problem
EPS
= $2.00
P/E
Multiple
Stock Price
10-*
Step 2: Decide on a Solution Strategy
- The common stock value can be computed by multiplying the firm’s estimated earnings per share for the coming year by what the analyst estimates to be an appropriate P/E ratio.
- We can use equation 10-4 to estimate the value of common stock.
10-*
Step 3: Solve
Vcs = 18.20 × $2
= $36.40
10-*
Step 4: Analyze
- We estimated the value of Heales’ shares based on the P/E ratios of three comparable firms. However, this estimate is contingent on the appropriateness of the comparable set of companies to the Heals Shoe Company.
- Furthermore, if the market conditions change by the time the shares are sold in the market, the price estimate will not be appropriate.
10-*
What Determines the P/E Ratio for a Stock?
Using Equation 10-5a and 10-5b, there are two fundamental determinants of a firm’s P/E ratio:
Growth Rate in Dividends (higher the growth rate, higher the P/E ratio), and
Investor-Required Rates of Return (higher the required rate, lower the P/E ratio)
10-*
Features of Preferred Stock
- Dividend: In general, size of preferred stock dividend is fixed, and it is either stated as a dollar amount or as a percentage of the preferred stock’s par value.
- Multiple Classes: A company can issue more than one class of preferred stock, and each class can have different characteristics.
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Features of Preferred Stock (cont.)
- Claims on Assets and Income: Preferred stockholders have priority over those of common stockholders for payment of dividends and in settlement of claims at bankruptcy. Most preferred stock carry a cumulative feature i.e. all past unpaid dividends must be paid before any common stock dividends can be declared.
10-*
Features of Preferred Stock (cont.)
Preferred Stock as a Hybrid Security:
- Like common stocks, preferred stocks do not have a fixed maturity date. Also, like common stocks, nonpayment of dividends does not bring on bankruptcy, and dividends are not deductible for tax purposes.
- Like debt, preferred stocks have a fixed dividend. Also, most preferred stocks are periodically retired even though there is no stated maturity date.
10-*
Valuing Preferred Stock
Because preferred stocks are perpetuities (non-maturing), and because the cash dividend is the same every period, they can be valued using the present value of perpetuity equation introduced in chapter 6 (equation 6-5).
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Valuing Preferred Stock (cont.)
- Vps = the value of a share of preferred stock
- Dps = the annual preferred stock dividend
- rps = the market yield or the rate of return on the preferred stock’s promised dividend
10-*
Estimating the Market’s Required Yield
Estimating the Market Yield: We can use equation 10-6 to solve for the market’s required yield.
rps = Dps ÷Vps
10-*
CHECKPOINT 10.3:
CHECK YOURSELF
Valuing Preferred Stock
What is the present value of a share of preferred stock that pays a dividend of $12 per share if the market’s yield on similar issues of preferred stock is 8%?
10-*
Step 1: Picture the Problem
Preferred stocks are constant for all years and form a level perpetuity.
rps=8%
Years
Dividends $12 $12 $12 $12
0
1
2
3 …
Value of Preferred
Stock = Present
Value of promised
dividends.
The annual
$12 dividends
go on
forever.
…
10-*
Step 2: Decide on a Solution Strategy
Step 3: Solve
We can determine the present value of share of preferred stock using equation 10-6.
Vps = $12 ÷ 0.08 = $150
10-*
Step 4: Analyze
Since preferred stock is a level perpetuity, its value on any future date will be the same as its present value today as long as the promised rate of return on the share remains the same.
10-*
The Stock Market
As discussed in chapter 2, new securities trade in the primary market while currently outstanding securities trade in the secondary market. There are two types of secondary markets: organized exchanges and over-the-counter markets.
10-*
Organized Exchanges
The New York Stock Exchange (NYSE), also called the “Big Board,” is the oldest of all organized exchanges. While the NYSE is considered an organized exchange because of its physical location, the majority of its trades are done electronically without a face-to-face meeting of traders.
10-*
Organized Exchanges (cont.)
- To be listed on the NYSE, a firm must meet strict requirements dealing with profitability and market value, and be widely owned.
- Much of the trading on the NYSE is made up of block trades i.e. transactions involving 10,000 shares or more by a single individual or institution.
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Organized Exchanges (cont.)
- The American Stock Exchange (AMEX) is the nation’s second largest, floor-based exchange. However, in terms of trading volume, the AMEX is a distant number two with less than 3% of that on the NYSE.
- Although AMEX merged with NASDAQ in 1998 it continues to operate as a separate entity.
10-*
Over-the-Counter (OTC) Market
The over-the-counter market is a network of dealers that has no listing or membership requirements. Today, the OTC market is electronic rather than personal, with Nasdaq leading the way. It is also the primary market for bonds.
10-*
Over-the-Counter (OTC) Market (cont.)
- Nasdaq debuted in 1971 and was the world’s first electronic stock market. While Nasdaq lists more companies than the NYSE, they are relatively smaller companies (with a few exceptions)
- There are about 1,000 market participants, in general trading firms that are linked electronically, with price and trading information broadcast to over 350,000 terminals worldwide.
10-*
Over-the-Counter (OTC) Market (cont.)
The Nasdaq stock market has two tiers of listed companies:
- Nasdaq National Markets, made up of around 4,000 companies like Dell (D), Intel (INTC); and
- Nasdaq Smallcap Market, which includes over 1,000 smaller emerging growth companies.
10-*
Key Terms
- American Stock Exchange
- Block holding
- Block trade
- Constant dividend growth rate model
- Cumulative preferred stock
- Cumulative voting
- Initial public offering
10-*
Key Terms (cont.)
- Majority voting
- Market’s required yield
- Nasdaq
- New York Stock Exchange (NYSE)
- Over-the-counter (OTC) market
- Price-earnings ratio
- Proxy
lecture/Chapter 4(1).ppt
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Chapter 4
Financial
Analysis—
Sizing up Firm Performance
4-*
Slide Contents
- Learning Objectives
- Principles Applied in this Chapter
Why Do We Analyze Financial Statements
Common Size Statements – Standardizing Financial Information
Using Financial Ratios
Selecting a Performance Benchmark
Limitations of Ratio Analysis
- Key Terms
4-*
Learning Objectives
Explain what we can learn by analyzing a firm’s financial statements.
Use common size financial statements as a tool of financial analysis.
Calculate and use a comprehensive set of financial ratios to evaluate a company’s performance.
4-*
Learning Objectives (cont.)
Select an appropriate benchmark for use in performing a financial ratio analysis.
Describe the limitations of financial ratio analysis.
4-*
Principles Used in this Chapter
- Principle 3: Cash Flows Are the Source of Value.
- Principle 4: Market Prices Reflect Information.
- Principle 5: Individuals Respond to Incentives.
4-*
Why Do We Analyze Financial Statements?
- An internal financial analysis might be done:
- To evaluate the performance of employees
- To compare the performance of different divisions
- To prepare financial projections
- To evaluate the firm’s financial performance in light of its competitors’ performance
4-*
Why Do We Analyze Financial Statements? (cont.)
- External financial analysis is done by:
- Banks and other lenders
- Suppliers
- Credit-rating agencies
- Professional analysts
- Individual investors
4-*
Common Size Statements: Standardizing Financial Information
- A common size financial statement is a standardized version of a financial statement in which all entries are presented in percentages.
- It helps to compare a firm’s financial statements with those of other firms, even if the other firms are not of equal size.
4-*
Common Size Statements: Standardizing Financial Information (cont.)
- How to prepare a common size financial statement?
- For a common size income statement, divide each entry in the income statement by sales.
- For a common size balance sheet, divide each entry in the balance sheet by total assets.
4-*
Table 4.1 H. J. Boswell, Inc.
4-*
Table 4.1 Observations
- Table 4-1 created by dividing each entry in the income statement of Table 3.1 by firm sales for 2013.
- Cost of goods sold make up 75% of the firm’s sales resulting in a gross profit of 25%.
- Selling expenses account for about 3% of sales.
- Income taxes account for 4.1% of the firm’s sales.
- After all expenses, the firm generates net income of 7.6% of firm’s sales.
4-*
Table 4.2 H. J. Boswell, Inc.
4-*
Table 4.2 Observations
- Table 4.2 created by dividing each entry in the balance sheet of Table 3.2 by total assets.
- Total current assets increased by 5.6% in 2013 while total current liabilities declined by 2%.
- Long-term debt account for 39.2% of firm’s assets, showing a decline of 1.7%.
- Retained earnings increased by 5.8% .
4-*
Using Financial Ratios
- Financial ratios provide a second method for standardizing the financial information on the income statement and balance sheet.
- A ratio by itself may have no meaning. Hence, a given ratio is generally compared to: (a) ratios from previous years; or (b) ratios of other firms in the same industry.
4-*
Using Financial Ratios (cont.)
4-*
Liquidity Ratios
- Liquidity ratios address a basic question: How liquid is the firm?
- A firm is financially liquid if it is able to pay its bills on time. We can analyze a firm’s liquidity from two perspectives (see next slide).
Overall liquidity - analyzed by comparing the firm’s current assets to the firm’s current liabilities.
Liquidity of specific assets - analyzed by examining the timeliness in which the firm’s liquid assets (accounts receivable and inventories) are converted into cash.
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Liquidity Ratios: Current Ratio
- The overall liquidity of a firm is analyzed by computing the current ratio and acid-test ratio. Current Ratio: Current Ratio compares a firm’s current (liquid) assets to its current (short-term) liabilities.
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Liquidity Ratios: Current Ratio (cont.)
- What is the current ratio for 2012 for Boswell?
Current Ratio = $477 ÷ 292.5 = 1.63 times
- The firm had $1.63 in current assets for every $1 it owed in current liability.
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Liquidity Ratios: Quick Ratio
- Acid-Test (Quick) Ratio excludes the inventory from current assets as inventory may not be very liquid.
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Liquidity Ratios: Quick Ratio
(cont.)
- What is the quick ratio for Boswell for 2012?
- Quick Ratio
= ($477-$229.50) ÷ ($292.50) = 0.84 times
- The firm has only $0.84 in current assets (less inventory) to cover $1 in current liabilities.
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Liquidity Ratios:
Individual Asset Categories
We can also measure the liquidity of the firm by examining the liquidity of accounts receivable and inventories to see how long it takes the firm to convert its accounts receivables and inventories into cash.
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Liquidity Ratios: Accounts Receivable
Average Collection Period measures the number of days it takes the firm to collects its receivables.
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Liquidity Ratios: Accounts Receivable (cont.)
- What will be the average collection period for Boswell, Inc. for 2012 if we assume that the annual credit sales were $2,500 million?
- Daily Credit Sales
= $2,500 ÷ 365 days = $6.85 million
- Average Collection Period
= $139.5m ÷ $6.85m = 20.37 days
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Liquidity Ratios: Accounts Receivable Turnover Ratio
Accounts Receivable Turnover Ratio measures how many times receivables are “rolled over” during a year.
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Liquidity Ratios: Accounts Receivable Turnover Ratio (cont.)
- What will be the accounts receivable turnover ratio for Boswell, Inc. for 2012 if we assume that the annual credit sales were $2,500 million?
- Accounts Receivable Turnover
= $2,500 million ÷ $139.50 = 17.92 times
- The firm’s accounts receivable were turning over at 17.92 times per year.
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Liquidity Ratios:
Inventory Turnover Ratio
Inventory turnover ratio measures how many times the company turns over its inventory during the year. Shorter inventory cycles lead to greater liquidity since the items in inventory are converted to cash more quickly.
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Liquidity Ratios:
Inventory Turnover Ratio (cont.)
- What will be the inventory turnover ratio for 2012 for Boswell, Inc. if we assume that the cost of goods sold were $1,980 million in 2012?
- Inventory Turnover Ratio
= $1,980 ÷ $229.50 = 8.63 times
- The firm turned over its inventory 8.63 times per year.
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Liquidity Ratios:
Days’ Sales in Inventory
- Days’ Sales in Inventory
= 365÷ inventory turnover ratio
= 365 ÷ 8.63 = 42.29 days
- The firm, on average, holds it inventory for about 42 days.
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Can a Firm Have Too Much Liquidity?
- A high investment in liquid assets will enable the firm to repay its current liabilities in a timely manner.
- However, an excessive investments in liquid assets can prove to be costly as liquid assets (such as cash) generate minimal return.
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CHECKPOINT 4.1:
CHECK YOURSELF
Evaluating Dell’s Liquidity
Why do you think HP’s inventory turnover ratio is so much lower than Dell’s inventory turnover ratio?
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Step 1: Picture the Problem
- The inventory turnover ratio will measure how many days items remain in inventory before being sold.
- Inventory turnover ratio is important as it has implications for cash flows and profitability of a firm.
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Step 2: Decide on a Solution Strategy
Step 3: Solve
- We will use the following equation to compute the Inventory Turnover (IT) ratio
IT ratio = Cost of Goods Sold ÷ Inventories
- Inventory Turnover Ratio for HP
= $97,529,000 ÷ 7,490,000 = 13.02
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Step 4: Analyze
- HP’s inventory turnover ratio indicates that the inventory at HP remains on shelf for (365 ÷ 13.02) days or 28.03 days. This is much higher than Dell that has an inventory turnover ratio of 34.37 or shelf life of only 10.61 days.
- The significant difference must be investigated further as the two firms are in the same industry.
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Step 4: Analyze (cont.)
There are two reasons why HP has a lower turnover of inventories relative to Dell:
- HP sells computers out of inventory of computers while Dell builds computers only when orders are received.
- HP carries more parts inventory on hand than does Dell.
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Capital Structure Ratios
Capital structure refers to the way a firm finances its assets. Capital structure ratios address the important question: How has the firm financed the purchase of its assets?
*
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Capital Structure Ratios (cont.)
Debt ratio measures the proportion of the firm’s assets that are financed by borrowing or debt financing.
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Capital Structure Ratios (cont.)
- What is the debt ratio for H.J. Boswell, Inc. for 2012?
- Debt Ratio
= $1,012.50 million ÷ $1,764 million = 57.40%
- The firm financed 57.39% of its assets with debt.
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Capital Structure Ratios (cont.)
- Times Interest Earned Ratio measures the ability of the firm to service its debt or repay the interest on debt.
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Capital Structure Ratios (cont.)
- What will be the times interest earned ratio for Boswell for 2012 if we assume interest expense of $65 million and EBIT of $350 million?
- Times Interest Earned
= $350m ÷ $65m = 5.38 times
- The firm can pay its interest expense 5.38 times or interest used 1/5.38th or 18.58% of its EBIT.
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CHECKPOINT 4.2:
CHECK YOURSELF
Comparing the Financing Decisions
of HD and LOW
What would be Home Depot’s times interest earned ratio if interest payments remained the same, but net operating income dropped by 80% to only $1.332 billion? Similarly if Lowes’ net operating income dropped by 80%, what would its times interest earned ratio be?
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Step 1: Picture the Problem
- Times interest earned ratio is an important ratio for firms that use debt financing. It measures the firm’s ability to service its debt.
- The ratio requires comparing net operating income or EBIT with Interest expense. Both items are found on the income statement.
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Step 1: Picture the Problem (cont.)
- Picture an Income Statement
- Sales
- Less: Cost of Good Sold
- Equals: Gross Profit
- Less: Operating Expenses
- Equals: Net Operating Income (EBIT)
- Less: Interest Expense
- Equals: Earnings before Taxes
- Less: Taxes
- Equals Net Income
EBIT
Interest
Expense
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Step 2: Decide on a Solution Strategy
- Here we are considering the impact of a drop in EBIT on the times interest earned ratio of Home Depot and Lowes. We will use the following ratio to measure the times interest earned (TIE) ratio.
- TIE = EBIT ÷ Interest Expense
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Step 3: Solve
- TIE (Home Depot)
= $1.332 billion ÷ $0.606 billion = 2.20 times
- TIE (Lowes)
= $0.655 billion ÷$0.371 billion = 1.77 times
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Step 4: Analyze
- We observe that a drop in net operating income leads to a significant drop in times interest earned ratio for both the firms. Should creditors be worried by this drop?
- The ratio is still reasonably safe. For example, for Home Depot, even if the EBIT shrank further by 55.55% (1-1/2.20 ), it can still pay its interest expense.
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Asset Management Efficiency Ratios
- Asset management efficiency ratios measure a firm’s effectiveness in utilizing its assets to generate sales.
- They are commonly referred to as turnover ratios as they reflect the number of times a particular asset account balance turns over during a year.
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Asset Management Efficiency Ratios (cont.)
- Total Asset Turnover Ratio represents the amount of sales generated per dollar invested in firm’s assets.
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Asset Management Efficiency Ratios (cont.)
What will be the total asset turnover ratio for Boswell, Inc. for 2012 if we assume total sales to be $2,500 million?
- Total Asset Turnover
= $2,500 million ÷ $1,764 million = 1.42 times
- The firm generated $1.42 in sales per dollar of assets in 2012.
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Asset Management Efficiency Ratios (cont.)
- Fixed asset turnover ratio measures firm’s efficiency in utilizing its fixed assets (such as property, plant and equipment).
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Asset Management Efficiency Ratios (cont.)
What will be the fixed asset turnover ratio for Boswell for 2012 if we assume sales of $2,500 million for 2012?
- Fixed Asset Turnover
= $2,500 million ÷ $1,287 million = 1.94 times
- The firm generated $1.94 in sales per dollar invested in plant and equipment.
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Asset Management Efficiency Ratios (cont.)
The following grid summarizes the efficiency of Boswell’s management in utilizing its assets to generate sales in 2013.
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Profitability Ratios
Profitability ratios address a very fundamental question: Has the firm earned adequate returns on its investments?
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Profitability Ratios (cont.)
Two fundamental determinants of firm’s profitability and returns on investments:
- Cost Control – How well has the firm controlled its costs relative to each dollar of firm sales?
- Efficiency of asset utilization – How effective is the firm in using the assets to generate sales?
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Cost Control: Is the Firm Earning Reasonable Profit Margins?
Gross profit margin shows how well the firm’s management controls its expenses to generate profits.
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Cost Control: Is the Firm Earning Reasonable Profit Margins? (cont.)
What will be the gross profit margin ratio for 2012 for Boswell if we assume sales of $2,500 million and gross profit of $650 million?
- Gross Profit Margin
= $650 million ÷ $2,500 million = 26%
- The firm spent $0.74 for cost of goods sold and thus $0.26 out of each dollar of sales went towards gross profits.
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Cost Control: Is the Firm Earning Reasonable Profit Margins? (cont.)
Operating Profit Margin measures how much profit is generated from each dollar of sales after accounting for both costs of goods sold and operating expenses. It also indicates how well the firm is managing its income statement.
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Cost Control: Is the Firm Earning Reasonable Profit Margins? (cont.)
What will be the operating profit margin ratio for Boswell for 2012 if we assume sales of $2,500 million and net operating income of $350 million?
- Operating Profit Margin
= $350 million ÷ $2,500 million = 14%
- The firm generates $0.14 in operating profit for each dollar of sales.
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Cost Control: Is the Firm Earning Reasonable Profit Margins? (cont.)
Net Profit Margin measures how much income is generated from each dollar of sales after adjusting for all expenses (including income taxes).
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Cost Control: Is the Firm Earning Reasonable Profit Margins? (cont.)
What will be the net profit margin ratio for 2012 if we assume sales of $2,500 million and net income of $217.75 million?
- Net Profit Margin
= $217.75 million ÷ $2,500 million = 8.71%
- The firm generated $0.087 for each dollar of sales after all expenses were accounted for.
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Return on Invested Capital
Operating Return on Assets ratio is the summary measure of operating profitability. It takes into account the management’s success in controlling expenses and its efficient use of assets.
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Profitability Ratios (cont.)
What will be the operating return on assets ratio for Boswell for 2012 if we assume EBIT or net operating income of $350 million for 2012?
- Operating Return on Assets
= $350 million ÷$1,764 million = 19.84%
- The firm generated $0.1984 of operating profits for every $1 of its invested assets.
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Decomposing the Operating Return on Assets Ratio
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Figure 4.1 Analyzing H. J. Boswell, Inc.’s Operating Return on Assets (OROA)
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Figure 4-1 Observations
- Firm’s OROA (operating return on assets) is better than its peers.
- Firm’s OPM (operating profit margin) is lower than its peers.
- Firm’s TATO (total asset turnover ratio) is higher than that of its peers.
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Figure 4-1 Recommendations
Reduce costs - The firm must investigate the cost of goods sold and operating expenses to see if there are opportunities to reduce costs.
Reduce inventories – The firm must investigate if it can reduce the size of its inventories.
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CHECKPOINT 4.3:
CHECK YOURSELF
Evaluating the Operating Return on Assets (OROA) for HD and LOW
If Home Depot were able to raise its total asset turnover ratio to 2.5 while maintaining its current operating profit margin, what would happen to its operating return on assets?
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Step 1: Picture the Problem
- The operating return on assets ratio for a firm is determined by two factors: cost control and asset utilization. Here the focus is on asset utilization.
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Step 2: Decide on a Solution Strategy
We will analyze the impact on operating return on assets of improvement on the total asset turnover ratio by using the following equation:
- Operating Return on Assets (OROA)
= Total Asset Turnover × Operating Profit Margin
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Step 3: Solve
- Operating Return on Assets (OROA)
- = Total Asset Turnover × Operating Profit Margin
- Before = 1.74 × 9.46% = 16.46%
- Now = 2.5 × 9.46% = 23.65%
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Step 4: Analyze
- An improvement in total asset turnover ratio has a favorable impact on Home Depot’s operating return on assets (OROA).
- If Home Depot wants to increase its OROA more, it should focus on cost control that will help improve the net operating profit.
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Is the Firm Providing a Reasonable Return on the Owner’s Investment?
Return on Equity (ROE) ratio measures the accounting return on the common stockholders’ investment.
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Is the Firm Providing a Reasonable Return on the Owner’s Investment (cont.)
What will be the ROE ratio for Boswell for 2012 if we assume net income of $217.75 million?
- ROE = $217.75m ÷ $751.50 mi = 28.98%
- Thus the shareholders earned 28.97% on their investments.
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Using the DuPont Method for Decomposing the ROE ratio
- DuPont method analyzes the firm’s ROE by decomposing it into three parts.
- ROE = Profitability × Efficiency × Equity Multiplier
- Equity multiplier captures the effect of the firm’s use of debt financing on its return on equity. The equity multiplier increases in value as the firm uses more debt.
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Using the DuPont Method for Decomposing the ROE ratio (cont.)
ROE = Profitability × Efficiency × Equity Multiplier
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Using the DuPont Method for Decomposing the ROE ratio (cont.)
The following table shows why Boswell’s return on equity was higher than its peers.
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Using the DuPont Method for Decomposing the ROE ratio (cont.)
Figure 4.2
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Market Value Ratios
Market value ratios address the question, how are the firm’s shares valued in the stock market?
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Price-Earnings Ratio
Price-Earnings (PE) Ratio indicates how much investors are currently willing to pay for $1 of reported earnings.
Peer–group average PE ratio = 12.0 times
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Price-Earnings Ratio (cont.)
What will be the PE ratio for 2012 for Boswell, Inc. if we assume the firm’s stock was selling for $22 per share at a time when the firm reported a net income of $217.75 million, and the total number of common shares outstanding are 90 million?
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Market Value Ratios (cont.)
- Earnings per share
= $217.75 million ÷ 90 million = $2.42
- PE ratio = $22 ÷ $2.42 = 9.09
- The investors were willing to pay $9.09 for every dollar of earnings per share that the firm generated.
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Market Value Ratios (cont.)
Market-to-Book Ratio measures the relationship between the market value and the accumulated investment in the firm’s equity.
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Market Value Ratios (cont.)
What will be the market-to-book ratio for 2012 for Boswell if the market price of the stock is $22 and the firm has 90 million shares outstanding?
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Market Value Ratios (cont.)
- Book Value per Share
- = 751.50 million ÷ 90 million = $8.35 per share
- Market-to-Book Ratio
= Market price per share ÷ Book value per share
= $22 ÷ $8.35
= 2.63 times
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CHECKPOINT 4.4:
CHECK YOURSELF
Comparing the Valuation of DELL to APPL Using Market Value Ratios
What price per share for Dell would it take to increase the firm’s price-to-earnings ratio to the level of Apple?
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Step 1: Picture the Problem
Price-to-earnings (PE) ratio depends on earnings per share and price per share, pictured as follows:
Price per share standardized by
EPS =
Net income ÷ number
of shares outstanding
PE Ratio =
Price per share ÷
Earnings per share
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Step 2: Decide on a Solution Strategy
We need to determine the price per share that will make PE ratio of Dell (4.83) equal to the PE ratio of Apple (13.22).
- PE ratio = Price per share ÷ Earnings per share
==> 13.22 = ? ÷ 2.01
- Price per share = 13.22 × 2.01 = $26.57
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Step 4: Analyze
- PE ratio allows us to compare two stocks with different prices by standardizing the stock prices by earnings.
- Apple has a much higher PE ratio. To reach the same PE valuation, the stock price of Dell will have to increase from $9.70 to $26.57.
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Selecting a Performance Benchmark
- There are two types of benchmarks that are commonly used:
- Trend Analysis – compares a firm’s financial statements over time (time-series comparisons).
- Peer Group Comparisons – compares the subject firm’s financial statements with “peer” firms.
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Trend Analysis
- Comparing a firm’s recent financial ratios with the past financial ratios provides insight into whether the firm is improving or deteriorating over time. This type of financial analysis is referred to as trend analysis.
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Figure 4-3 A Time-Series (Trend) Analysis: Dell’s Inventory Turnover Ratio Versus Hewlett Packard’s: 1995–2011
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Peer Firm Comparisons
Peer groups often consist of firms from the same industry. Industry average financial ratios can be obtained from a number of financial databases and internet sources (such as yahoo finance and google finance).
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Figure 4-4 Financial Analysis of the Gap, Inc., June 2009
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The Limitations of Ratio Analysis
Picking an industry benchmark can sometimes be difficult.
Published peer-group or industry averages are not always representative of the firm being analyzed.
An industry average is not necessarily a desirable target or norm.
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The Limitations of Ratio Analysis (cont.)
Accounting practices differ widely among firms.
Many firms experience seasonal changes in their operations.
Financial ratios offer only clues.
The results of financial analysis are no better than the quality of the financial statements.
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Key Terms
- Accounts receivable turnover ratio
- Acid-test (quick) ratio
- Average collection period
- Book value per share
- Capital structure
- Current ratio
- Days’ sales in inventory
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Key Terms (cont.)
- Debt ratio
- DuPont method
- Equity Multiplier
- Earnings per share (EPS)
- Financial leverage
- Financial ratios
- Fixed asset turnover ratio
- Inventory turnover ratio
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Key terms (cont.)
- Liquidity ratios
- Market-to-book ratio
- Market value ratios
- Notes payable
- Operating return on assets (OROA)
- Price-earnings (PE) ratio
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Key terms (cont.)
- Return on assets (ROA)
- Return on equity (ROE)
- Times interest earned
- Total asset turnover ratio (TATO)
- Trend analysis
lecture/Chapter 5-TVM-1.ppt
Chapter 5
The Time Value
of Money—
The Basics
*
Copyright ©2014 Pearson Education, Inc. All rights reserved.
5-*
Slide Contents
- Learning Objectives
- Principles Applied in this Chapter
- 5.1 Using Timelines to Visualize Cash Flows
- 5.2 Compounding and Future Value
- 5.3 Discounting and Present Value
- 5.4 Making Interest Rates Comparable
- Key Terms
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Learning Objectives
Construct cash flow timelines to organize your analysis of problems involving the time value of money.
Understand compounding and calculate the future value of cash flows using mathematical formulas, a financial calculator, and an Excel spreadsheet.
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Learning Objectives (cont.)
Understand discounting and calculate the present value of cash flows using mathematical formulas, a financial calculator and an Excel spreadsheet.
Understand how interest rates are quoted and know how to make them comparable.
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Principles Applied in this Chapter
Principle 1: Money Has a Time Value.
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Using Timelines to Visualize Cashflows
- A timeline identifies the timing and amount of a stream of payments – both cash received and cash spent - along with the interest rate earned.
- A timeline is typically expressed in years, but it could also be expressed as months, days or any other unit of time.
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Time Line Example
i=10%
Years
Cash flow -$100 $30 $20 -$10 $50
The 4-year timeline illustrates the following:
- The interest rate is 10%.
- A cash outflow of $100 occurs at the beginning of the first year (at time 0), followed by cash inflows of $30 and $20 in years 1 and 2, a cash outflow of $10 in year 3 and cash inflow of $50 in year 4.
0
1
2
3
4
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Compounding and Future Value
Time value of money calculations involve Present value (what a cash flow would be worth to you today) and Future value (what a cash flow will be worth in the future).
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Compound Interest and Time
Example: Suppose that you deposited $500 in your savings account that earns 5% annual interest. How much will you have in your account after two years? After five years?
- FV2 = PV(1+i)n = 500(1.05)2 = $551.25
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Compound Interest and Time
Using Equation 5-1a: FV = PV(1+i)n
= 500(1.05)5 = $638.14
| YEAR | PV or Beginning Value | Interest Earned (5%) | FV or Ending Value |
| 1 | $500.00 | $500*.05 = $25 | $525 |
| 2 | $525.00 | $525*.05 = $26.25 | $551.25 |
| 3 | $551.25 | $551.25*.05 =$27.56 | $578.81 |
| 4 | $578.81 | $578.81*.05=$28.94 | $607.75 |
| 5 | $607.75 | $607.75*.05=$30.39 | $638.14 |
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Figure 5.1 Future Value and Compound Interest Illustrated
(Panel A) Calculating Compound Interest
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Figure 5.1 Future Value and Compound Interest Illustrated (cont.)
(Panel B) The Power of Time
Copyright ©2014 Pearson Education, Inc. All rights reserved.
5-*
Figure 5.1 Future Value and Compound Interest Illustrated (cont.)
(Panel C) The Power of the Rate of Interest
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Applying Compounding to Things Other Than Money
Example A DVD rental firm is currently renting 8,000 DVDs per year. How many DVDs will the firm be renting in 10 years if the demand for DVD rentals is expected to increase by 7% per year?
- Using Equation 5-1a,
- FV = 8000(1.07)10 = 15,737.21 DVDs
Copyright ©2014 Pearson Education, Inc. All rights reserved.
5-*
CHECKPOINT 5.2:
CHECK YOURSELF
Calculating the FV of a Cash Flow
What is the FV of $10,000 compounded at 12% annually for 20 years?
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Step 1: Picture the Problem
i=12%
Years
Cash flow -$10,000
0
1
2 …
20
Future
Value=?
*
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Step 2: Decide on a Solution Strategy
This is a simple future value problem. We can find the future value using Equation 5-1a.
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Step 3: Solve
Solve Using a Mathematical Formula
FV = $10,000(1.12)20
= $10,000(9.6463)
= $96,462.93
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Step 3: Solve (cont.)
Solve Using a
Financial Calculator
N = 20
I/Y = 12%
PV = -10,000
PMT = 0
FV = $96,462.93
Solve Using an Excel Spreadsheet
=FV(rate,nper,pmt, pv)
=FV(0.12,20, 0,-10000)
= $96,462.93
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Step 4: Analyze
If you invest $10,000 at 12%, it will grow to$96,462.93 in 20 years.
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Compound Interest with Shorter Compounding Periods
Banks frequently offer savings account that compound interest every day, month, or quarter.
More frequent compounding will generate higher interest income and lead to higher future values.
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Table 5-2 The Value of $100 Compounded at Various Non-Annual Periods and Various Rates
Copyright ©2014 Pearson Education, Inc. All rights reserved.
5-*
CHECKPOINT 5.3:
CHECK YOURSELF
Calculating Future Values Using
Non-Annual Compounding Periods
If you deposit $50,000 in an account that pays an annual interest rate of 10% compounded monthly, what will your account balance be in 10 years?
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Step 1: Picture the Problem
i=10%
Months
Cash flow -$50,000
0
1
2 …
120
FV of $50,000
Compounded for
120 months
@ 10%/12
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Step 2: Decide on a Solution Strategy
This involves solving for future value of $50,000. Since the interest is compounded monthly, we will use equation 5-1b.
Copyright ©2014 Pearson Education, Inc. All rights reserved.
5-*
Step 3: Solve
Using a Mathematical Formula
FV = PV (1+i/12)m*12
= $50,000 (1+0.10/12)10*12
= $50,000 (2.7070)
= $135,352.07
Using a Financial Calculator
N = 120
I/Y = .833%
PV = -50,000
PMT = 0
FV = $135,352
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Step 3: Solve (cont.)
Using an Excel Spreadsheet
=FV(rate,nper,pmt, pv)
=FV(0.00833,120, 0,-50000)
= $135,346.71
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Step 4: Analyze
- More frequent compounding leads to a higher FV as you are earning interest more often on interest you have previously earned.
- If the interest was compounded annually, the FV would have been equal to only $129,687.12
- $50,000 (1.10)10 = $129,687.12
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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The Key Question
- What is value today of cash flow to be received in the future?
- The answer to this question requires computing the present value (PV) i.e. the value today of a future cash flow, and the process of discounting, determining the present value of an expected future cash flow.
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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The Mechanics of Discounting Future Cash Flows
- The term in the bracket is known as the Present Value Interest Factor (PVIF).
- PV = FVn × PVIF
Copyright ©2014 Pearson Education, Inc. All rights reserved.
5-*
Figure 5.2 The Present Value of $100 Compounded at Different Rates and for Different
Time Periods
Copyright ©2014 Pearson Education, Inc. All rights reserved.
5-*
CHECKPOINT 5.4:
CHECK YOURSELF
Solving for the PV of a Future Cash Flow
What is the present value of $100,000 to be received at the end of 25 years given a 5% discount rate?
Copyright ©2014 Pearson Education, Inc. All rights reserved.
5-*
Step 1: Picture the Problem
i=5%
Years
Cash flow $100,000
0
1
2 …
25
Present
Value =?
Copyright ©2014 Pearson Education, Inc. All rights reserved.
5-*
Step 2: Decide on a Solution Strategy
Here we are solving for the present value (PV) of $100,000 to be received at the end of 25 years using a 5% interest rate. We can solve using equation 5-2.
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Step 3: Solve
Using a Financial Calculator
N = 25
I/Y = 5
PMT = 0
FV = 100,000
PV = -$29,530
Using a Mathematical Formula
PV
= $100,000 [1/(1.05)25)
= $100,000 [0.2953]
= $29,530
Copyright ©2014 Pearson Education, Inc. All rights reserved.
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Step 4: Analyze
Once you’ve found the present value, it can be compared to other present values. Present value computation makes cash flows that occur in different time periods comparable so that we can make good decisions.
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Two Additional Types of Discounting Problems
Solving for: (1) Number of Periods; and
(2) Rate of Interest
(1): How long will it take to accumulate a specific amount in the future?
- It is easier to solve for “n” using the financial calculator or Excel rather than mathematical formula. (See checkpoint 5.5)
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The Rule of 72
- It determine the number of years it will take to double the value of your investment.
N = 72/interest rate
For example, if you are able to generate an annual return of 9%, it will take 8 years (=72/9) to double the value of investment.
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CHECKPOINT 5.5:
CHECK YOURSELF
Solving for Number of Periods, n
How many years will it take for $10,000 to grow to $200,000 given a 15% compound growth rate?
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Step 1: Picture the Problem
i=15%
Years
Cash flow -$10,000 $200,000
0
1
2 …
N =?
We know FV,
PV, and i and
are solving for
N
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Step 2: Decide on a Solution Strategy
In this problem, we are solving for “n”. We know the interest rate, the present value and the future value. We can calculate “n” using a financial calculator or an Excel spreadsheet.
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Step 3: Solve
- Using a Financial Calculator
I/Y = 15
PMT = 0
PV = -10,000
FV = 200,000
N = 21.4 years
- Using an Excel
Spreadsheet
N = NPER(rate,pmt,pv,fv)
= NPER(.15,0,-10000,200000)
= 21.4 years
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Step 4: Analyze
It will take 21.4 years for $10,000 to grow to $200,000 at an annual interest rate of 15%.
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Solving for the Rate of Interest
(2): What rate of interest will allow your investment to grow to a desired future value?
We can determine the rate of interest using mathematical equation, the financial calculator or the Excel spread sheet.
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CHECKPOINT 5.6:
CHECK YOURSELF
Solving for the Interest Rate, i
At what rate will $50,000 have to grow to reach $1,000,000 in 30 years?
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Step 1: Picture the Problem
i=?%
Years
Cash flow -$50,000 $1,000,000
0
1
2 …
30
We know FV, PV
and N and are Solving
for “interest rate”
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Step 2: Decide on a Solution Strategy
Here we are solving for the interest rate. The number of years, the present value, the future value are known. We can compute the interest rate using mathematical formula, a financial calculator or an Excel spreadsheet.
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Step 3: Solve
Using a Mathematical Formula
I = (FV/PV)1/n - 1
= (1000000/50000)1/30 - 1
= (20)0.0333 - 1
= 1.1050 - 1
= .1050 or 10.50%
Using an Excel Spreadsheet
=Rate (nper, pmt, pv, fv)
=Rate(30,0,-50000,1000000)
=10.50%
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Step 4: Analyze
You will have to earn an annual interest rate of 10.50 percent for 30 years to increase the value of investment from $50,000 to $1,000,000.
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Annual Percentage Rate (APR)
The annual percentage rate (APR) indicates the interest rate paid or earned in one year without compounding. APR is also known as the nominal or quoted (stated) interest rate.
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Calculating the Interest Rate and Converting it to an EAR
We cannot compare two loans based on APR if they do not have the same compounding period.
To make them comparable, we calculate their equivalent rate using an annual compounding period. We do this by calculating the effective annual rate (EAR)
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CHECKPOINT 5.7:
CHECK YOURSELF
Calculating an EAR
What is the EAR on a quoted or stated rate of 13 percent that is compounded monthly?
*
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Step 1: Picture the Problem
i= an annual rate of 13% that is compounded monthly
Months
0
1
2 …
12
Compounding periods
are expressed in months
(i.e. m=12) and we are
Solving for EAR
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Step 2: Decide on a Solution Strategy
Here we need to solve for Effective Annual Rate (EAR). We can compute the EAR by using equation 5-4
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Step 3: Solve
EAR = [1+.13/12]12 - 1
= 1.1380 – 1
= .1380 or 13.80%
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Step 4: Analyze
- There is a significant difference between APR and EAR (13.00% versus 13.80%).
- If the interest rate is not compounded annually, we should compute the EAR to determine the actual interest earned on an investment or the actual interest paid on a loan.
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To the Extreme:
Continuous Compounding
- As m (number of compounding period) increases, so does the EAR. When the time intervals between when interest is paid are infinitely small, we can use the following mathematical formula to compute the EAR.
- EAR = (e quoted rate ) – 1
- Where “e” is the number 2.71828
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Continuous Compounding (cont.)
- Example What is the EAR on your credit card with continuous compounding if the APR is 18%?
- EAR = e.18 - 1
= 1.1972 – 1
= .1972 or 19.72%
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Key Terms
- Annual Percentage Rate (APR)
- Compounding
- Compound Interest
- Discounting
- Discount Rate
- Effective Annual Rate (EAR)
- Future Value
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Key Terms (cont.)
- Future Value Interest Factor
- Nominal or Stated Interest Rate
- Present Value
- Present Value Interest Factor
- Simple Interest
- Timeline
