Money, Banking, and International Finance ( PDFDrive )

Money, Banking, and International Finance

= $20,000[1.09 + 1.09 + 1.09 + 1.09 + 1.09 ] = $119,694.21

Do you notice anything strange about the exponents? We raise the first term in Equation 18 to the fourth power because your initial payment occurred at the end of period 1, and has earned four years of interest. Finally, the last term has a zero exponent, and the final $20,000 does not earn interest. Moreover, mathematicians derived a formula to calculate an annuity without calculating a long series of numbers. They derived a formula in Equation 19, and c is the periodic payment into an annuity. Using the previous example, the value of the annuity still equals $119,694.21.

= ( ) = $20,000 ( . ) = $119,694.21 (19)

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We also have the other side of an ordinary annuity. For example, you saved a $60,000 annuity that earns 4% APR. You plan to withdraw equal annual payments over 10 years. How much do you receive annually? Remember, you receive your first payment at the end of the first period, which is the beginning of the second period. That $60,000 earns interest for the first period. We compute an annual withdrawal payment of $7,397.46 in Equation 20.

Financial analysts use the present value formula to calculate mortgage payments, which is vital to building an amortization table. An amortization table itemizes every payment for a mortgage loan and decomposes every payment into interest and the amount that reduces the principal. A mortgage is a bank loan for a property, and the property becomes the collateral. For instance, if a person has a mortgage for a house and defaults on the loan, the bank can legally take possession of the house. We use the present value formula to build an amortization table.

Mathematical notation for a mortgage is:

All future mortgage payments (FV) are equal and are usually monthly.

Interest rate (i) is loan rate and becomes fixed throughout life of the loan.

Bank loan is amount recorded for PV0 because the bank loaned you money at time 0. We show a mortgage as a stream of cash flows to the bank in Equation 21.

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All loan payments are equal, so we set FV = FV1 = FV2 = FV3 = ... = FVT. Thus, we can factor the FV terms from all interest terms in Equation 22.

We solve for FV, which becomes the loan payment, yielding Equation 23.

For example, a bank granted a mortgage for $60,000 at an interest rate of 12% APR. Mortgage is a six-year loan and paid annually. We solve for FV and calculate your annual payment of $14,594 in Equation 24.

= $14,594

We can use the mortgage loan information to build an amortization table. We show an amortization table in Table 9. For Year 0, you have $60,000 outstanding because you did not make a payment yet. Then you make your first payment in Year 1. Your interest is 12% multiplied by $60,000, equaling $7,200. If your payment is $14,594, then $7,200 is the interest while the remainder reduces the principal. Thus, you subtract $7,394 from the loan balance. For Year 2, and beyond, you repeat the sequence until you pay the loan in full in Year 6.

Table 9. An Amortization Table

All amortization tables have one feature. First payment has the highest interest while the lowest principal applied to the loan balance. Then the interest amount declines over the life of the loan until it becomes the smallest for the last payment.

If a mortgage is monthly, then you divide the interest rate by 12 and multiply the number of years by 12. For instance, a 20-year mortgage will have 240 payments, 12 × 20. As you can see,

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Equation 24 would have 240 terms. Consequently, mathematicians devised a formula to calculate a mortgage with many payments.

For example, compute your monthly payment if you bought a $150,000 home at 6% APR with a 30-year mortgage. We calculated your monthly payment of $899.33 in Equation 25. If you notice, Equation 25 is the same formula for an annuity payout with compounding frequency

Amortization table can also handle balloon payments and variable interest rate mortgages. However, these topics go beyond the textbook’s scope. A balloon payment is a person pays a low monthly payment every month. For the last payment, the person would pay the remaining balance, which could be large. Moreover, variable-interest rate loan is the bank can adjust the loan’s interest rate as market interest rates change.

Foreign Investments

We can use the net present value (NPV) to calculate the monetary return to an investment in Equation 26. This equation is almost identical to the present value formula, except the PV0 is negative and located on the right-hand side while we add a new variable, NPV. If the net present value (NPV) equals zero, then this equation reduces to the present value formula. With the NPV formula, we could invest the amount PV0 today that generates the future cash flows, FVi, that ends at Time T. Market interest rate is i, and it automatically compares out investment to the market interest rate.

If we calculate a positive, net present value, then our investment is paying off. Consequently, the investment is increasing the investor’s wealth because more money flows in than out. Furthermore, investors would use the net present value formula to evaluate several investment projects. Then they select the project with the highest NPV, as long as the NPV is positive. An investor would never choose a project with a negative NPV because the project’s return would be negative. Over time, more money flows out than in, creating a net loss.

For example, your brother wants you to invest $10,000 into his business. He promises to repay you $12,000 in two years. If you invested your money into financial securities, you believe you would earn an annual 10% APR. Is it profitable to invest in your brother’s business? We calculated a net present value of -$82.64 in Equation 27. Unfortunately, you could earn more on the financial securities than your brother’s business because the NPV is negative.

Money, Banking, and International Finance

= −$30,000 + $9,523.81 + $7,256.24 + $3,455.35 = −$9,764.60

Chapter Questions

1.Compute the net income if a company sold $50,000 of goods in cash, sold $60,000 of goods on accounts receivable, paid $100,000 in costs and operating expenses, paid taxes of $30,000, and paid $30,000 in administrative expenses.

2.Calculate the retained earnings of a corporation at the end of the year if retained earnings were $20,000 at the beginning of the year, net income was $50,000, declared $60,000 in dividends, and sold $50,000 in additional stock.

3.Compute a business’s cash balance at the end of the year if the company starts with a cash balance of $10,000, paid salaries of $70,000, received $100,000 in cash sales from customers and $30,000 for accounts receivable, and paid taxes of $10,000.

4.You will receive $1,000,000 in one hundred years exactly. How much is this worth to you today if the market interest rate equals 5% APR?

5.You deposit $5,000 into a bank that earns 10% APR. How much will your balance grow in 50 years?

6.You deposit $1,000 in a bank for two years. Which interest rate in APR must you earn for your ending balance to be $1,200?

7.You deposit your savings into a money market that earns 3% APR. How many years would it double?

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7. Valuation of Stocks and Bonds

This chapter provides an overview of stocks and bonds, and the methods financial analysts use to calculate the market price using the present value formula. Furthermore, corporations issue a variety of bonds and stocks, and use them to expand business operations. Corporations sell their bonds to investors who buy these bonds and stock for investment. They either hold the bonds until maturity or sell the bonds and stock for a capital gain or loss. Consequently, investors must know the difference between yield to maturity and the rate of return. This chapter expands on Chapter 6, and we expand the present value formula to value a variety of bonds and stocks.

Overview of Bonds

Corporations often borrow money by issuing bonds. A bond is similar to notes payable because they are written promises to pay interest and principal. We show a picture of a bond in Figure 1. Face value of this bond equals $1,000, and this bond matures on February 1, 2020. Consequently, whoever holds this bond will receive $1,000 on this date, and the bondholder also earns $100 ( 0.1 × $1,000) per year in interest. Most bonds pay interest twice annually or $50 every six months for this example.

Bond

$1,000

10%

February 1, 2020

Figure 1. A picture of a bond

Bonds, however, differ from notes payable. A notes payable is a loan from a single creditor such as a bank, while a bond is a loan that corporations issue in denominations of $1,000, $2,000, etc. Finally, bonds are standardized, and thus, investors can purchase them. Moreover, investors can buy and sell these bonds on the financial markets before the bonds mature.

Bonds differ from corporate stock. A share of stock represents ownership in a corporation. For instance, if a shareholder owns 1,000 shares out of 10,000, then he or she owns 10% of the corporation’s equity. Moreover, the shareholder also receives 10% of the corporation’s earnings, when the board of directors declares dividends. On the other hand, a bond represents a debt or a liability to the corporation. For example, if a person owns a bond with a face value of $1,000

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with an 11% coupon interest rate and 20-year maturity, then the bondholder has two legal rights. Bondholder has a legal right to receive 11% or $110 interest each year while the bond is outstanding. Furthermore, the bondholder has a legal right to receive $1,000 when the bond matures in 20 years.

A corporation needing long-term funds may consider issuing additional shares of stock or issuing new bonds. However, if the corporation issues new stock, then the existing stockholders share control with new stockholders. Consequently, the stockholders lose part of control of the corporation. On the other hand, the bondholders do not share in the management or earnings of the corporation. Although the corporation must pay the bond interest, whether it earns profits or losses, bonds reduce net income, thus lowering a corporation’s taxes. U.S. corporations pay between 15 and 35% of their net income in taxes. Nevertheless, bond interest payments are an expense, which lowers the corporation’s net income. If a corporation issues new bonds, then the common stockholders could increase their dividend earnings.

We show an example of a corporation expanding operations in Table 1. This corporation had sold 300,000 shares of outstanding common stock to investors, and it needs $2 million to expand its operations. After the expansion, the management estimates the company can earn $1,000,000 annually. Consequently, the corporation has two plans. For Plan A, the corporation issues 200,000 new shares of the corporation’s stock at $10 per share. For Plan B, the corporation issues $2 million of bonds with a 10% interest rate. Hence, the interest expense equals $200,000 per year.

Examining these two plans, Plan B results in a greater income per share for the shareholders because the bond’s interest lowered the tax burden by $60,000. Thus, the stockholders retained control of the corporation, and they potentially earn higher dividends per share by using bond financing.

Table 1. A Corporation Finance an Expansion through Bonds or Stocks

The Valuation of Bonds

Governments and corporations issue a variety of bonds with different characteristics and cash flows. Consequently, we explain the main bonds, and the methods investors and analysts use to value them. We show a discount bond in Figure 2, and it is the simplest to calculate. This

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Money, Banking, and International Finance

discount bond is a Treasury Bill issued by the U.S. government. For this example, the Treasury Bill has a face value of $10,000, which we call T-bills for short. Discount bonds do not have an interest rate listed on them. Thus, U.S. government sells T-bills at a discount or lower price. Lower price reflects the market interest rate.

Treasury Bill

U.S. Government

$10,000

August 10, 2013

Figure 2. A picture of a Treasury Bill

For example, if the U.S. federal government sold this T-bill to you for $9,500, then the present value, PV0, becomes the market price. Subsequently, the federal government will repay you $10,000 for this instrument on August 10, 2013. The $500 difference reflects the interest on this loan.

Investors and analysts calculate the yield to maturity (YTM), the return to an investment. Yield to maturity reflects an investor’s profit from a security that is similar to an interest rate. We state both interest rates and yield to maturity in annual percentage terms. For our example, we calculated the yield to maturity of 5.26% in Equation 1 if the bond’s face value equals $10,000 while the market value is $9,500 with a maturity of one year.

1+YTM = $10,000 $9,500

YTM = 0.0526

If a discount bond has a maturity less than one year, then the time subscript remains one year. However, we adjust the yield to maturity to annual terms. For example, if the T-bill in the previous example matured in 180 days, then we calculate it in the same way. Nevertheless, we multiply the rate of return by two because 365 days divided by 180 is approximately two. In our case, the return would equal 10.52%.

Discount bonds usually have a maturity of one year or less. However, we can adjust the present value formula to calculate bonds with longer maturities. For example, you purchased a

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