February 11, 2022

Since ancient times, new knowledge and new inventions

Since ancient times, new knowledge and new inventions have periodically remade human societies. Today, however, knowledge is assuming greater importance than ever before. Now more essential to the wealth of nations than either capital or labor, Peter Drucker argues here, it has already created a "postcapitalist" society and promises further transformations on a global scale. How do interest rates and raising capital like bonds help the "postcapitalist" society? Use the Peter Drucker article in Week 4 to help aid your post.

1. Write a 300 word description in which you discuss and answer the above questions by Wednesday.

2. Leave two questions for peers in your answer. (50 words for each)

3. Cite and use APA format if you use outside sources.

Corporate Finance

Fifth Edition

Chapter 5

Interest Rates

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Chapter Outline

5.1 Interest Rate Quotes and Adjustments

5.2 Application: Discount Rates and Loans

5.3 The Determinants of Interest Rates

5.4 Risk and Taxes

5.5 The Opportunity Cost of Capital

Learning Objectives (1 of 3)

Define effective annual rate and annual percentage rate.

Given an effective annual rate, compute the n-period effective annual rate.

Convert an annual percentage rate into an effective annual rate, given the number of compounding periods.

Describe the relation between nominal and real rates of interest.

Learning Objectives (2 of 3)

Given two of the following, compute the third: nominal rate, real rate, and inflation rate.

Describe the effect of higher interest rates on net present values in the economy.

Explain how to choose the appropriate discount rate for a given stream of cash flows, according to the investment horizon.

Learning Objectives (3 of 3)

Discuss the determinants of the shape of the yield curve.

Explain why Treasury securities are considered risk-free, and describe the impact of default risk on interest rates.

Given the other two, compute the third: after-tax interest rate, tax rate, and before-tax interest rate.

5.1 Interest Rate Quotes and Adjustments (1 of 2)

The Effective Annual Rate

Indicates the total amount of interest that will be earned at the end of one year

Considers the effect of compounding

Also referred to as the effective annual yield (E A Y) or annual percentage yield (A P Y)

5.1 Interest Rate Quotes and Adjustments (2 of 2)

Adjusting the Discount Rate to Different Time Periods

Earning a 5% return annually is not the same as earning 2.5% every six months.

General Equation for Discount Rate Period Conversion

Note: n = 0.5 since we are solving for the six month (or half year) rate.

Textbook Example 5.1 (1 of 3)

Valuing Monthly Cash flows

Problem

Suppose your bank account pays interest monthly with the interest rate quoted as an effective annual rate (E A R) of 6%. What amount of interest will you earn each month? If you have no money in the bank today, how much will you need to save at the end of each month to accumulate $100,000 in 10 years?

Textbook Example 5.1 (2 of 3)

From Eq. 5.1, a 6% E A R is equivalent to earning

per month. We can write the timeline for

our savings plan using monthly periods as follows:

That is, we can view the savings plan as a monthly annuity with

monthly payments. We can calculate the total

amount saved as the future value of this annuity, using Eq. 4.10:

We can solve for the monthly payment C using the equivalent monthly interest rate r = 0.4868%, and n = 120 months:

Textbook Example 5.1 (3 of 3)

We can also compute this result using the annuity spreadsheet:

BLANK	NPER	RATE	PV	PMT	FV	Excel Formula
Given	120	0.4868%	0	blank	100,000	blank
Solve for PMT	blank	blank	blank	− 615.47	blank	= PMT(0.004868,120,0,100000)

Thus, if we save $615.47 per month and we earn interest monthly at an effective annual rate of 6%, we will have $100,000 in 10 years.

Alternative Example 5.1 (1 of 2)

Problem

Suppose an investment pays interest quarterly with the interest rate quoted as an effective annual rate (E A R) of 9%.

What amount of interest will you earn each quarter?

If you have no money in the bank today, how much will you need to save at the end of each quarter to accumulate $25,000 in five years?

Alternative Example 5.1 (2 of 2)

Solution

From Eq. 5.1, a 9% E A R is approximately

equivalent to earning

per quarter.

To determine the amount to save each quarter to reach the goal of $25,000 in five years, we must determine the quarterly payment, C:

Annual Percentage Rates (1 of 4)

The annual percentage rate (A P R), indicates the amount of simple interest earned in one year.

Simple interest is the amount of interest earned without the effect of compounding.

The A P R is typically less than the effective annual rate (E A R).

Annual Percentage Rates (2 of 4)

The A P R itself cannot be used as a discount rate.

The A P R with k compounding periods is a way of quoting the actual interest earned each compounding period:

Annual Percentage Rates (3 of 4)

Converting an A P R to an E A R

The E A R increases with the frequency of compounding.

Continuous compounding is compounding every instant.

Annual Percentage Rates (4 of 4)

Table 5.1 Effective Annual Rates for a 6% A P R with Different Compounding Periods

Compounding Interval	Effective Annual Rate
Annual	left parenthesis start fraction 1 + 0.06 over 1 end fraction right parenthesis to the first power minus 1 = 6%
Semiannual	left parenthesis start fraction 1 + 0.06 over 2 end fraction right parenthesis squared minus 1 = 6.09%
Monthly	left parenthesis start fraction 1 + 0.06 over 12 end fraction right parenthesis to the power of 12 minus 1 = 6.1678%
Daily	left parenthesis start fraction 1 + 0.06 over 365 end fraction right parenthesis to the power of 365 minus 1 = 6.1831%

A 6% A P R with continuous compounding results in an E A R of approximately 6.1837%.

Textbook Example 5.2 (1 of 4)

Converting the A P R to a Discount Rate

Problem

Your firm is purchasing a new telephone system, which will last for four years. You can purchase the system for an upfront cost of $150,000, or you can lease the system from the manufacturer for $4,000 paid at the end of each month. Your firm can borrow at an interest rate of 5% A P R with semiannual compounding. Should you purchase the system outright or pay $4,000 per month?

Textbook Example 5.2 (2 of 4)

Solution

The cost of leasing the system is a 48-month annuity of $4,000 per month:

We can compute the present value of the lease cash flows using the annuity formula, but first we need to compute the discount rate that corresponds to a period length of one month. To do so, we convert the borrowing cost of 5% A P R with semiannual compounding to a monthly discount rate. Using Eq. 5.2,

the A P R corresponds to a six-month discount rate of

To convert

a six-month discount a one-month discount rate, we compound the six-month

rate by

using Eq. 5.1:

Textbook Example 5.2 (3 of 4)

(Alternatively, we could first use Eq. 5.3 to convert the A P R to an E A R.

Then we can convert the E A R to

a monthly rate using Eq. 5.1: (1.050625)

month.) Given this discount rate, we can use the annuity formula (Eq. 4.9) to compute the present value of the 48 monthly payments:

We can also use the annuity spreadsheet:

Blank	N P E R	R A T E	P V	P M T	F V	Excel Formula
Given	48	0.4124%	blank	− 4,000	0	blank
Solve for P V	blank	blank	173,867	blank	blank	= PV(0.004124,48,4000,0)

Textbook Example 5.2 (4 of 4)

Thus, paying $4,000 per month for 48 months is equivalent to paying a present value of $173,867 today. This cost is

higher than the cost of

purchasing the system, so it is better to pay $150,000 for the system rather than lease it. We can interpret this result as meaning that at a 5% A P R with semiannual compounding, by promising to repay $4,000 per month, your firm can borrow $173,867 today. With this loan it could purchase the phone system and have an additional $23,867 to use for other purposes.

Alternative Example 5.2 (1 of 3)

Problem

A firm is considering purchasing or leasing a luxury automobile for the C E O. The vehicle is expected to last three years. You can buy the car for $65,000 up front , or you can lease it for $1,800 per month for 36 months. The firm can borrow at an interest rate of 8% A P R with quarterly compounding. Should you purchase the system outright or pay $1,800 per month?

Alternative Example 5.2 (2 of 3)

Solution

The first step is to compute the discount rate that corresponds to monthly compounding. To convert an 8% rate compounded quarterly to a monthly discount rate, compound the quarterly rate using Eqs. 5.3 and 5.1:

Alternative Example 5.2 (3 of 3)

Solution

Given a monthly discount rate of 0.66227%, the present value of the 36 monthly payments can be computed:

Paying $1,800 per month for 36 months is equivalent to

paying $57,486 today. This is

lower than the cost of purchasing the system, so it is better to lease the vehicle rather than buy it.

5.2 Application: Discount Rates and Loans (1 of 4)

Computing Loan Payments

Payments are made at a set interval, typically monthly.

Each payment made includes the interest on the loan plus some part of the loan balance.

All payments are equal and the loan is fully repaid with the final payment.

5.2 Application: Discount Rates and Loans (2 of 4)

Computing Loan Payments

Consider a $30,000 car loan with 60 equal monthly payments, computed using a 6.75% A P R with monthly compounding.

6.75% A P R with monthly compounding corresponds

to a one-month discount rate of

5.2 Application: Discount Rates and Loans (3 of 4)

Computing Loan Payments

Financial Calculator Solution

5.2 Application: Discount Rates and Loans (4 of 4)

Computing the Outstanding Loan Balance

One can compute the outstanding loan balance by calculating the present value of the remaining loan payments.

Textbook Example 5.3 (1 of 3)

Computing the Outstanding Loan Balance

Problem

Two years ago your firm took out a 30-year amortizing loan to purchase a small office building. The loan has a 4.80% A P R with monthly payments of $2,623.33. How much do you owe on the loan today? How much interest did the firm pay on the loan in the past year?

Textbook Example 5.3 (2 of 3)

Solution

After 2 years, the loan has 28 years, or 336 months, remaining:

The remaining balance on the loan is the present value of these

remaining payments, using the loan rate of

During the past year, your firm made total payments of

on the loan. To determine the amount that was interest, it is

easiest to first determine the amount that was used to repay the principal. Your loan balance one year ago, with 29 years (348 months) remaining, was

Textbook Example 5.3 (3 of 3)

Therefore, the balance declined by

in the past year. Of the total payments made, $8,022

was used to repay the principal and the remaining

was used to pay interest.

5.3 The Determinants of Interest Rates (1 of 2)

Inflation and Real Versus Nominal Rates

Nominal Interest Rate: The rates quoted by financial institutions and used for discounting or compounding cash flows

Real Interest Rate: The rate of growth of your purchasing power, after adjusting for inflation

5.3 The Determinants of Interest Rates (2 of 2)

The Real Interest Rate

Textbook Example 5.4 (1 of 3)

Calculating The Real Interest Rate

Problem

In May of 2014, one-year U.S. government bond rates were about 0.1%, while the rate of inflation over the

following year was around

At the

start of 2017, one-year interest rates were about 0.8%, and inflation over the following year was approximately 2.1%. What were the real interest rates in May 2014 and in 2017?

Textbook Example 5.4 (2 of 3)

Solution

Using Eq. 5.5, the real interest rate in May 2014 was

In 2017, the real interest rate was

Textbook Example 5.4 (3 of 3)

Solution

Note that the real interest rate was negative in 2017, indicating that interest rates were insufficient to keep up with inflation: Investors in U.S. government bonds were able to buy less at the end of the year than they could have purchased at the start of the year. On the other hand, because prices actually decreased (deflation) in the year following May 2014, the real interest rate briefly exceeded the nominal interest rate.

Alternative Example 5.4 (1 of 2)

Problem

On December 31, 2015, the average one-year Treasury Constant Maturity rate was about 0.65% and the 2015 annual inflation rate was about 0.70%.

What was the real interest rate in 2015?

Alternative Example 5.4 (2 of 2)

Solution

Using Eq. 5.5, the real interest rate in 2008 was

Which is equal to the difference between the nominal

rate and inflation:

Figure 5.1 U.S. Interest Rates and Inflation Rates, 1962–2017

Interest rates are one-year Treasury rates, and inflation rates are the increase in the U.S. Bureau of Labor Statistics’ consumer price index over the coming year, with both series computed on a monthly basis. The difference between them thus reflects the approximate real interest rate earned by holding Treasuries. Note that interest rates tend to be high when inflation is high.

Both plots rise and fall numerous times, and although there is not a perfect correlation between them, changes in inflation rate tend to be followed by similar changes in the nominal interest rate. The plot for inflation rate begins at (1962, 1.2) and has large upward spikes peaking at (1974, 12.5), (1980, 14.5), (1990, 6), and (2008, 5.5). Small peaks appear at (2011, 3.9) and (2016, 2.5). The plot for nominal interest rate begins at (1962, 3.5), and its corresponding peaks are at (1974, 9.5), (1982, 17), (1989, 9.5), and (2008, 5.75). After 2008, the interest rates drop almost near to 0. The gap between the plots was largest during the 1980s, with nominal interest rates 9% above inflation during 1985. All values estimated.

Investment and Interest Rate Policy (1 of 2)

An increase in interest rates will typically reduce the N P V of an investment.

Consider an investment that requires an initial investment of $10 million and generates a cash flow of $3 million per year for four years. If the interest rate is 5%, the investment has an N P V of

Investment and Interest Rate Policy(2 of 2)

If the interest rate rises to 9%, the N P V becomes negative and, the investment is no longer profitable:

Monetary Policy, Deflation, and the 2008 Financial Crisis

When the 2008 financial crisis struck, the Federal Reserve responded by cutting its short-term interest rate target to 0%.

While this use of monetary policy is generally quite effective, because consumer prices were falling in late 2008, the inflation rate was negative, and so even with a 0% nominal interest rate, the real interest rate remained positive.

The Yield Curve and Discount Rates (1 of 2)

Term Structure: The relationship between the investment term and the interest rate

Yield Curve: A graph of the term structure

Figure 5.2 Term Structure of Risk-Free U.S. Interest Rates, November 2006, 2007, and 2008

The graph in panel b shows interest rate, E A R, in percentage, versus term, in years. The plot for November 2006 is a curve that falls from (0.5, 5.23) to (4, 4.63), rises to (15, 4.89), then falls to (20, 4.87). The plot for November 2007 is a curve that falls from (0.5, 3.32) to (3, 3.12), then rises through (15, 4.44) and (20, 4.45). The plot for November 2008 is a curve that rises from (0.5, 0.47) through (2, 0.98), (15, 3.86), and (20, 3.87). All values from panel a.

The Yield Curve and Discount Rates (2 of 2)

The term ‘structure’ can be used to compute the present and future values of a risk-free cash flow over different investment horizons.

Present Value of a Cash Flow Stream Using a Term Structure of Discount Rates

Textbook Example 5.5 (1 of 2)

Using the Term ‘Structure’ to Compute Present Values

Problem

Compute the present value in November 2008 of a risk-free five-year annuity of $1,000 per year, given the yield curve for November 2008 in Figure 5.2

Textbook Example 5.5 (2 of 2)

Solution

To compute the present value, we discount each flow by the corresponding interest rate:

Note that we cannot use the annuity formula here because the discount rates differ for each cash flow.

Alternative Example 5.5 (1 of 2)

Problem

Compute the present value of a risk-free three-year annuity of $500 per year, given the following yield curve:

Treasury Rates	Blank
Term (Years)	Rate
1	0.261%
2	0.723%
3	1.244%

Alternative Example 5.5 (2 of 2)

Solution

Each cash flow must be discounted by the corresponding interest rate:

The Yield Curve and the Economy (1 of 2)

Interest Determination

The Federal Reserve determines very short-term interest rates through its influence on the federal funds rate, which is the rate at which banks can borrow cash reserves on an overnight basis.

All other interest rates on the yield curve are set in the market and are adjusted until the supply of lending matches the demand for borrowing at each loan term.

The Yield Curve and the Economy (2 of 2)

Interest Rate Expectations

The shape of the yield curve is influenced by interest rate expectations.

An inverted yield curve indicates that interest rates are expected to decline in the future.

Because interest rates tend to fall in response to an economic slowdown, an inverted yield curve is often interpreted as a negative forecast for economic growth.

Each of the last six recessions in the United States was preceded by a period in which the yield curve was inverted.

The yield curve tends to be sharply increasing as the economy comes out of a recession, and interest rates are expected to rise.

Figure 5.3 Short-Term Versus Long-Term U.S. Interest Rates and Recessions

One-year and ten-year U.S. Treasury rates are plotted, with the spread between them shaded in blue if the shape of the yield curve is increasing (the one-year rate is below the ten-year rate) and in red if the yield curve is inverted (the one year rate exceeds the ten-year rate). Gray bars show the dates of U.S. recessions as determined by the National Bureau of Economic Research. Note that inverted yield curves tend to precede recessions by 12–18 months. In recessions, interest rates tend to fall, with short-term rates dropping further. As a result, the yield curve tends to be steep coming out of a recession.

1969 to 1970, 1973 to 1957, 1980, 1981 to 1982, 1990 to 1991, 2001, and 2008 to 2009. With the exception of the recession in 1990, each recession was preceded, within 2 years, by a brief period wherein 1-year interest rates were higher than 10-year interest rates. These narrow, red areas showing inverted yield curve are followed by wider blue areas showing increasing yield curve.

Textbook Example 5.6 (1 of 4)

Comparing Short- and Long-Term Interest Rates

Problem

Suppose the current one-year interest rate is 1%. If it is known with certainty that the one-year interest rate will be 2% next year and 4% the following year, what will the interest rates r1, r2, and r3 of the yield curve be today? Is the yield curve flat, increasing, or inverted?

Textbook Example 5.6 (2 of 4)

Solution

We are told already that the one-year rate r1 = 1%. To find the two-year rate, note that if we invest $1 for one-year at the current one-year rate and then reinvest next year at the new one-year rate, after two-years we will earn

We should earn the same payoff if we invest for two-years at the current two-year rate r2:

Textbook Example 5.6 (3 of 4)

Otherwise, there would be an arbitrage opportunity: if investing at the two-year rate led to a higher payoff, investors could invest for two-years and borrow each year at the one-year rate. Investing at the two-year rate could led to a lower payoff. Investors could invest each year at the one-year rate and borrow at the two-year rate.

Solving for r2, we find that

Textbook Example 5.6 (4 of 4)

Similarly, investing for three years at the one-year rates should have the same payoff as investing at the current three-year rate:

We can solve for

Therefore, the current yield curve has r1 = 1%, r2 = 1.499%, and r3 = 2.326% The yield curve is increasing as a result of the anticipated higher interest rates in the future.

Alternative Example 5.6 (1 of 4)

Problem

Suppose the current one-year interest rate is 3%. If it is known with certainty that the one-year interest rate will be 2% next year and 1% the following year, what will the interest rates r1, r2, and r3 of the yield curve be today? Is the yield curve flat, increasing, or inverted?

Alternative Example 5.6 (2 of 4)

Solution

We are told already that the one-year rate r1 = 3%.

To find r2, we know that if we invest $1 for one year at the current one-year rate and then reinvest next year at the new one-year rate, after two years we will earn:

Alternative Example 5.6 (3 of 4)

Solution

We should earn the same payoff if we invest for two-years at the current two-year rate r2:

Otherwise, there would be an arbitrage opportunity.

Solving for r2, we find that:

Alternative Example 5.6 (4 of 4)

Solution

Similarly, investing for three years at the one-year rates should have the same payoff as investing at the current three-year rate:

We can solve for

Therefore, the current yield curve has r1 = 3%,

The yield curve is decreasing as a result of the anticipated lower interest rates in the future.

5.4 Risk and Taxes

Risk and Interest Rates

U.S. Treasury securities are considered “risk-free.” All other borrowers have some risk of default, so investors require a higher rate of return.

Figure 5.4 Interest Rates on Five-Year Loans for Various Borrowers, July 2018

Source: FINRA.org.

• U.S. Treasury: 2.7

• Microsoft: 3.1

• Carnival Corp.: 3.5

• Union Pacific: 4.0

• Kia Motors: 4.1

• Netflix: 5.1

• Rite Aid: 5.7

• Mattel: 6.1

• J.C. Penney: 7.3

• Sprint: 9.4

• Avon Products: 11.0

Textbook Example 5.7 (1 of 2)

Discounting Risky Cash Flows

Problem

Suppose the U.S. government owes your firm $1,000 to be paid in five years. Based on the interest rates in Figure 5.4, what is the present value of this cash flow? Suppose instead J CPenney owes your firm $1,000. Estimate the present value in this case.

Textbook Example 5.7 (2 of 2)

Solution

Assuming we can regard the government’s obligation as risk free (there is no change you won’t be paid), then we discount the cash flow using the risk-free Treasury interest rate of 2.7%:

The obligation from JCPenney is not risk-free. J C Penney may face financial difficulties and fail to pay the $1,000. Because the risk of this obligation is likely to be comparable to the five-year bond quoted in Figure 5.4, the 7.3% interest rate of the loan is a more appropriate discou

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Since ancient times, new knowledge and new inventions

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