Business Mathematics: Stocks and Bonds - Bonds

A bond is a long-term contract between a borrower (usually a corporation or the government) and a lender (bondholder). The bondholder is a creditor and is paid interest on the face value or par value of the bond at a specified rate of interest, called bond rate. The interest is usually payable semiannually on the dates specified on the bond certificate. On the redemption date (or maturity date) the bondholder receives the par value of the bond.

Table: Bond Quotations

Bonds may be bought and sold in bond exchanges. Current market prices, called market quotations, of $100 par value bonds are published daily in the financial pages of major newspapers.

Typical bond quotations are shown below.

Example
Find the market price of four $1,000 bonds D in Table 7.2 at the previous day’s closing price.


Example
Find the market price of four $1,000 bonds D in Table 7.2 at the previous day's closing price.


Example
Find the market price of a $5,000 bond C in Table 7.2 at that day’s low price.


Example
How much bond interest will be paid semiannually to an investor who owns six $1,000 bonds E in Table 7.2?

The market prices of bonds change to reflect changes in the economy and in the current interest rates. A bond priced below its face value is selling at a discount, and a bond priced above its face value is selling at a premium.

Example
Find the premium on a $1,000 bond A in Table 7.2 at the day’s high price.


Example
Find the discount on a $2,000 bond B in Table 7.2 at the day’s low price.

Bonds pay interest semiannually. For bonds purchased between interest dates, the purchaser pays the seller the interest earned (accrued) from the last bond interest payment date to the date of purchase. This so-called accrued bond interest is computed by the simple interest formula

Approximate time (each month = 30 days) and a 360-day year is used to determine time t.

Total purchase price of a bond purchased between bond interest dates is then the sum of the market price and the accrued bond interest. Any commission to the broker should also be added to the purchase price. Since these commissions are relatively small, they are ignored in this book.

Example
A $5,000, 8% bond paying interest on February 1 and August 1 is sold on April 8. Find the accrued bond interest.

Example
On March 26, an investor bought eight $1,000 bonds D at the day’s closing price, shown in Table 7.2. Calculate the total purchase price if the interest dates are June 1 and December 1.

To measure the rate of return on an investment in bonds, we may calculate either the rate of current yield or the rate of yield to maturity.

The formula for the rate of current yield is

The estimate for the rate of yield to maturity is based on the average annual interest divided by the average investment:


Example
A $1,000, 8% bond is quoted at 110 five years before maturity. Find the rate of current yield.

Example
Estimate the rate of yield to maturity for the bond in the above example:

Solved Problems:

7.13   Find the market price of each $5,000 bond in Table 7.2 at the day’s low price.
Solution

7.14   Find the semiannual bond interest payment on each $10,000 bond in the table above.
Solution



7.15   Find the premium or discount on each $2,000 bond in Table 7.2 at the day’s closing price.
Solution

7.16   Jessica Guiso bought three $1,000, % bonds on August 9 at market quotation . If the interest dates are March 1 and September 1, find the accrued bond interest and the total purchase price.
Solution

7.17   Find the total purchase price in Prob. 7.16 assuming that the market quotation was
Solution

7.18 Andrew owns two $1,000, % bonds paying interest on May 15 and November 15. What will be his proceeds from sale of the bonds on July 20 at market quotation 92?
Solution

7.19   How much would Andrew receive in Prob. 7.18 at market quotation 103?
 

Solution


7.20   Find the rate of current yield in Prob. 7.16.
Solution

7.21   In Prob. 7.16, at what price would Jessica have to buy a $1,000 bond for the rate of current yield to be 12%?
 

Solution
We rewrite the formula for current yield to solve for the market price:

The annual interest rate in Prob. 7.16 is 11.5%. The annual interest on a $1,000 bond is therefore
0.115 × $1,000 = $115

By substituting the known values into the formula for market price, we get

For the current yield to be 12%, the price of a $1,000 bond in Prob. 7.16 would have to be $958.33.

7.22   What market price would give the buyer of a $1,000 bond B in Table 7.2 a 10.5% rate of current yield?
Solution

7.23   Twelve years before maturity, a $5,000, 9% bond is quoted at 93. Find the rate of current yield.
 

Solution

7.24   Estimate the rate of yield to maturity for the bond in Prob. 7.23.
 

Solution


7.25 Estimate the rate of yield to maturity for a $2,000, 12% bond quoted at 107 eight and a half years before maturity.
 

Solution


7.26   Dan Thomas owns some 9 bonds from TTR, Inc. that are due in 5 years, 3 months. He is considering selling these and buying some Satco 11% bonds due in 17 years. If he can sell his TTR at 92 and buy Satco at 102, which bonds would give him a better rate of current yield?
 

Solution

The Satco bond gives a better rate of current yield.

7.27 In Prob. 7.26, which bond would give Dan Thomas a better rate of yield to maturity?
 

Solution
TTR $1,000 bond:

Satco $1,000 bond:

The TTR bond gives a better rate of yield to maturity.

More Problems:

7.28 Refer to the table above and find the total dividends on (a) 120 shares of stock A, (b) 200 shares of stock B, (c) 50 shares of stock C, (d) 90 shares of stock D, and (e) 250 shares of stock E.

7.29 Find the total purchase cost of the shares in Prob. 7.28 at the closing price of the day.

7.30 Find the total purchase cost of the shares in Prob. 7.28 at the previous day’s closing price.

7.31 Find the gross proceeds from the sale of the shares in Prob. 7.28 at the high price of that day.

7.32 Find the gross proceeds from the sale of the shares in Prob. 7.28 at the closing price of that day.

7.33 Fairway Electronics issued 20,000 shares of 7c) 1,500 shares, (b) 200 shares, and (c) 380 shares.

7.34 For each of the following, calculate the amount of dividends payable on $100 par value cumulative preferred stock.


7.35 Natco, Inc. has outstanding 4,000 shares of 6%, $100 par value cumulative preferred stock and 10,000 shares of common stock. The company declares dividends as follows from 1998 to 2001.
1998   $20,000
1999   $45,000
2000   $25,000
2001   $87,000
No dividends are in arrears as of the end of 1997. What amount of dividends is paid each year to (a) preferred and (b) common stock?

7.36  Assuming that the preferred stock is noncumulative, find the total dividends for the common stock in Prob. 7.35 for the 4 years.

7.37  Gemini, Inc. has outstanding 7,000 shares of 8%, $100 par value cumulative preferred stock and 20,000 shares of common stock. Dividends on the preferred stock are currently $45,000 in arrears. If a dividend of $184,000 is declared in the current year, find the dividend per share for (a) preferred stock and (b) common stock.

7.38  Delita Corporation declares a dividend of $72,750 to its 5,000 shares of 8a) preferred and (b) common stock.

7.39  Calculate the dividend per share for each class of stock in Prob. 7.38 assuming that the dividend declared is (a) $76,150, (b) $48,350.

7.40  Assuming that the preferred stock is cumulative participating preferred stock and $7,000 dividends are in arrears, calculate the dividend per share for (a) preferred and (b) common stock in Prob. 7.38.

7.41  Assuming that the preferred stock is cumulative nonparticipating preferred stock and $11,900 dividends are in arrears, calculate the dividend par value for (a) preferred and (b) common stock in Prob. 7.38.

7.42  Calculate the broker’s commission at the commission rate of 1.8% on the purchase of (a) 80 shares at 17b) 100 shares at 2lc) 280 shares at 3l.

7.43  What is the broker’s commission in Prob. 7.42 if the transactions were sales?

7.44  Find the total cost, including 2.1% commission, of a purchase of 320 shares of stock E in Table 7.1 at the day’s closing price.

7.45  Find the net proceeds, after a commission of 1.7% has been paid, on the sale of 190 shares of stock B in Table 7.1 at the day’s low price.

7.46  Find the total cost for each of the following purchases:


7.47  Find the net proceeds if the transactions in Prob. 7.46 were sales.

7.48  What is the investor’s annual yield on common stocks with
   (a)  Cost per share of $17.25 and quarterly dividend of $0.55
   (b)  Cost per share of $45 and semiannual dividend of $2.12
   (c)  Cost per share of $105.75 and annual dividend of $6.80

7.49  Determine the rate of annual yield for each stock in Table 7.1, if bought at the closing price of that day.

7.50  Determine the rate of annual yield for each stock in Table 7.1, if bought at the closing price of the previous day.

7.51  Arlene purchased 200 shares of stock A in Table 7.1 four years ago at 12a) her total gain and (b) the annual rate of gain based on cost.

7.52  Assuming that Arlene bought the stock in Prob. 7.51 at 33a) her total loss and (b) the annual rate of loss based on cost?

7.53  Find the market price of each $2,000 bond in Table 7.2 at that day’s high price.

7.54  At what market price would the $1,000 bonds in Table 7.2 give an 11% rate of current yield?

7.55  At what market price would the $1,000 bonds in Table 7.2 give a 7% rate of current yield?

7.56  Find the total semiannual interest payable to the owner of two $1,000 bonds A and three $1,000 bonds E in Table 7.2.

7.57  Find the total semiannual interest payable to the investor who owns one $2,000 bond of each type listed in Table 7.2.

7.58  Calculate the premium for the following bonds:


7.59  Calculate the discount for the following bonds:


7.60  Find the accrued bond interest on each of the following bond transactions:


7.61  Find the total proceeds of the bond transactions in Prob. 7.60.

7.62  Find the rate of current yield for the bonds in Prob. 7.60.

7.63  Referring to Table 7.2, find the accrued bond interest on each of the following bond transactions:


7.64  Find the total purchase price of the bonds in Prob. 7.63 if purchased at that day’s closing price.

7.65  Calculate the rate of current yield for the following bonds:


7.66  Estimate the rate of yield to maturity for the bonds in Prob. 7.65.

Answers to Above Problems:

7.28  (a) $180, (b) $120, (c) $100, (d) $495, (e) $195
7.29  (a) $2,942.50, (b) $1,925, (c) $5,437.50, (d) $5,793.75, (e) $5,193.75
7.30  (a) $2,837.50, (b) $1,950, (c) $5,456.25, (d) $5,771.25, (e) $5,256.25
7.31  (a) $2,937.50, (b) $1,950, (c) $5,512.50, (d) $5,771.25, (e) $5,181.25
7.32  (a) $2,937.50, (b) $1,925, (c) $5,425, (d) $5,771.25, (e) $5,181.25
7.33  (a) $11,250, (b) $1,500, (c) $2,850
7.34  (a) $750, (b) $42,000, (c) $2,560, (d) $937.50
7.35  (a) 1998: $20,000, 1999: $32,000, 2000: $25,000, 2001: $27,000; (b) 1998: 0, 1999: $13,000, 2000: 0, 2001: $60,000
7.36  $80,000
7.37  (a) $14.43, (b) $4.15
7.38  (a) $6, (b) $2.85
7.39  (a) Preferred: $6.17, common: $3.02; (b) preferred: $4.78, common: $1.63
7.40  (a) $7.05, (b) $2.50
7.41  (a) $6.63, (b) $2.64
7.42  (a) $25.92, (b) $38.25, (c) $157.05
7.43  (a) $25.56, (b) $38.25, (c) $156.69
7.44  $6,781.99
7.45  $1,786.60
7.46  (a) $9,622.86, (b) $2,774.05, (c) $2,368.95
7.47  (a) $9,352.21, (b) $2,669.81, (c) $2,271.15
7.48  (a) 12.75%, (b) 9.42%, (c) 6.43%
7.49  A, 6.12%; B, 6.23%; C, 1.84%; D, 8.56%; E, 3.76%
7.50  A, 6.35%; B, 6.15%; C, 1.83%; D, 8.59%; E, 3.71%
7.51  (a) $3,675, (b) 37.89%
7.52  (a) $650, (b) 2.41%
7.53  A, $2,047.50; B, $1,780; C, $2,202.50; D, $1,497.50; E, $2,060
7.54  A, $852.27; B, $647.73; C, $1,136.36; D, $431.82; E, $790.91
7.55  A, $1,250; B, $950; C, $1,666.67; D, $633.33; E, $1,160
7.56  $244.25
7.57  $424.50
7.58  (a) $71.25, (b) $93.75, (c) $87.50
7.59  (a) $525, (b) $900, (c) $12,750
7.60  (a) $67.32, (b) $170, (c) $96.25, (d) $429.33
7.61  (a) $1,917.32, (b) $4,520, (c) $5,146.25, (d) $8,899.33
7.62  (a) 10%, (b) 9.20%, (c) 10.40%, (d) 11.33%
7.63  (a) $73.44, (b) $37.60, (c) $72.92, (d) $172.85, (e) $285.17
7.64  (a) $3,077.19, (b) $1,817.60, (c) $3,372.92, (d) $7,647.85, (e) $10,585.17
7.65  (a) 11.48%, (b) 8.72%, (c) 11.49%, (d) 8.92%
7.66  (a) 11.14%, (b) 9.57%, (c) 10.80%, (d) 10.97%