## Application A Level Coupon Bond

Let us exercise your new found knowledge in a more elaborate example—this time with bonds. Bonds come in many different varieties, but one useful classification is into coupon bonds and zero bonds (short for zero coupon bonds). A coupon bond pays its holder cash at many different points in time, whereas a zero bond pays only a single lump sum at the maturity of the bond. Many coupon bonds promise to pay a regular coupon similar to the interest rate prevailing at the time of the bond's original sale, and then return a "principal amount" plus a final coupon at the end of the bond.

For example, think of a coupon bond that will pay $1,500 each half-year (semi-annual payment is very common) for five years, plus an additional $100,000 in 5 years. This payment pattern is so common that it has specially named features: A bond with coupon payments that remain the same for the life of the bond is called a level-coupon bond. The $100,000 here would be called the principal, in contrast to the $1,500 semi-annual coupon. Level bonds are commonly named by just adding up all the coupon payments over one year (here, $3,000), and dividing this sum of annual coupon payments by the principal. Thus, this particular bond would be called a "3% semi-annual coupon bond" ($3,000 coupon per year, divided by the principal of $100,000). Now, the "3% coupon bond" is just a naming convention for the bond with these specific cash flow patterns—it is not the interest rate that you would expect if you bought this bond. In Section 2 ■ 3.C, we called such name designations interest quotes, as distinct from interest rates. Of course, even if the bond were to cost $100,000 today (and you shall see below that it usually does not), the interest rate would not be 3% per annum, but (1 + 1.5%)2 - 1 « 3.02% per annum.

Par value is a vacuous concept, sometimes used to compute coupon payout schedules. Principal and par value, and/or interest and coupon payment need not be identical, not even at the time of issue, much less later. For the most part, par value is best ignored.

Coupon bonds pay not only at the final time.

Bonds are specified by their promised payout patterns.

SIDE NOTE

Step

: Write down the bond's payment stream.

Step 3: Compute the discount factor is

Common naming conventions for this type of bond: coupon rate is not interest rate!

### Using the annuity to make this faster.

Now solve for the value of the coupon bond. Incidentally, you may or may not find the annuity formula helpful—you can use it, but you do not need it. Your task is to find the value of a "3% coupon bond" today. First, you should write down the payment structure for a 3% semi-annual coupon bond. This comes from its defined promised patterns,

Due |
Bond |
Due |
Bond | ||

Year |
Date |
Payment |
Year |
Date |
Payment |

0.5 |
Nov 2002 |
$1,500 |
3.0 |
May 2005 |
$1,500 |

1.0 |
May 2003 |
$1,500 |
3.5 |
Nov 2005 |
$1,500 |

1.5 |
Nov 2003 |
$1,500 |
4.0 |
May 2006 |
$1,500 |

2.0 |
May 2004 |
$1,500 |
4.5 |
Nov 2006 |
$1,500 |

2.5 |
Nov 2004 |
$1,500 |
5.0 |
May 2007 |
$101,500 |

Step 2: find the appropriate costs of capital.

Second, you need to determine the appropriate expected rates of return to use for discounting. In this example, assume that the prevailing interest rate is 5% per annum, which translates into 2.47% for 6 months, 10.25% for two years, etc.

Maturity |
Yield |
Maturity |
Yield |

6 Months |
2.47% |
36 Months |
15.76% |

12 Months |
5.00% |
42 Months |
18.62% |

18 Months |
7.59% |
48 Months |
21.55% |

24 Months |
10.25% |
54 Months |
24.55% |

30 Months |
12.97% |
60 Months |
27.63% |

Your third step is to compute the discount factors, which are just 1/(1 + r0,t), and to multiply each future payment by its discount factor. This will give you the present value (PV) of each bond payment, and from there the bond overall value:

Due |
Bond |
Rate of |
Discount |
Present | |

Year |
Date |
Payment |
Return |
Factor |
Value |

0.5 |
Nov 2002 |
$1,500 |
2.47% |
0.976 |
$1,463.85 |

1.0 |
May 2003 |
$1,500 |
5.00% |
0.952 |
$1,428.57 |

1.5 |
Nov 2003 |
$1,500 |
7.59% |
0.929 |
$1,349.14 |

2.0 |
May 2004 |
$1,500 |
10.25% |
0.907 |
$1,360.54 |

2.5 |
Nov 2004 |
$1,500 |
12.97% |
0.885 |
$1,327.76 |

3.0 |
May 2005 |
$1,500 |
15.76% |
0.864 |
$1,295.76 |

3.5 |
Nov 2005 |
$1,500 |
18.62% |
0.843 |
$1,264.53 |

4.0 |
May 2006 |
$1,500 |
21.55% |
0.823 |
$1,234.05 |

4.5 |
Nov 2006 |
$1,500 |
24.55% |
0.803 |
$1,204.31 |

5.0 |
May 2007 |
$101,500 |
27.63% |
0.784 |
$79,527.91 |

Sum |
$91,501.42 |

You now know that you would expect this 3% level-coupon bond to be trading for $91,501.42 today in a perfect market. Because the current price of the bond is below the so-named final principal payment of $100,000, this bond would be said to trade at a discount. (The opposite would be a bond trading at a premium.)

The above computation is a bit tedious. Can you translate it into an annuity? Yes! Let's work in half-year periods. You thus have 10 coupon cash flows, each $1,500, at a per-period interest rate of 2.47%. According to the formula, the coupon payments are worth

CFt+1

$13,148.81

In addition, you have the $100,000 repayment of principal, which is worth $100, 000

Together, these present values of the bond's cash flows add up to $91, 501.42.

Prevailing Interest Rates and Bond Values: You already know that the value of one fixed The effect of a change future payment and the interest rate move in opposite directions. Given that you now have in interest rates. many payments, what would happen if the economy-wide interest rates were to suddenly move from 5% per annum to 6% per annum? The semi-annual interest rate would now increase from 2.47% to r = V1 + 6% - 1

To get the bond's new present value, reuse the formula

CFt+1

$12,823.89

This bond would have lost $3,951.72, or 4.3% of the original investment—which is the same inverse relation between bond values and prevailing economy-wide interest rates that you first saw on Page 26.

Important Repeat of Quotes vs. Returns: Never confuse a bond designation with the interest it pays. The "3%-coupon bond" is just a designation for the bond's payout pattern. The bond will not give you coupon payments equal to 1.5% of your $91,502.42 investment (which would be $1,372.52). The prevailing interest rate (cost of capital) has nothing to do with the quoted interest rate on the coupon bond. You could just as well determine the value of a 0%-coupon bond, or a 10% coupon bond, given the prevailing 5% economy-wide interest rate. Having said all this, in the real world, many corporations choose coupon rates similar to the prevailing interest rate, so that at the moment of inception, the bond will be trading at neither premium nor discount. At least for this one brief at-issue instant, the coupon rate and the economy-wide interest rate may actually be fairly close. However, soon after issuance, market interest rates will move around, while the bond's payments remain fixed, as designated by the bond's coupon name.

Q 3.16 If you can recall it, write down the annuity formula.

Q3.17 What is the PV of a 360 month annuity paying $5 per month, beginning at $5 next month, if the monthly interest rate is a constant 0.5%/month (6.2%/year)?

Q3.18 Mortgages are not much different from rental agreements. For example, what would your rate of return be if you rented your $500,000 warehouse for 10 years at a monthly lease payment of $5,000? If you can earn 5% elsewhere, would you rent out your warehouse?

Q3.19 What is the monthly payment on a 15-year mortgage for every $1,000 of mortgage at an effective interest rate of 6.168% per year (here, 0.5% per month)?

Q 3.20 Solve Fibonacci's annuity problem from Page 24: Compare the PV of a stream of quarterly cash flows of 75 bezants vs. the PV of a stream of annual cash flows of 300 bezants. Payments are always at period-end. The interest rate is 2 bezants per month. What is the relative value of the two streams? Compute the difference for a 1-year investment first.

Interest Rates vs. Coupon Rates.

Solve Now!

Q3.21 In L'Arithmetique, written in 1558, Jean Trenchant posed the following question: "In the year 1555, King Henry, to conduct the war, took money from bankers at the rate of 4% per fair [quarter]. That is better terms for them than 16% per year. In this same year before the fair of Toussaints, he received by the hands of certain bankers the sum of 3,945,941 ecus and more, which they called 'Le Grand Party' on the condition that he will pay interest at 5% per fair for 41 fairs after which he will be finished. Which of these conditions is better for the bankers?" Translated, the question is whether a perpetuity at 4% per quarter is better or worse than a 41-month annuity at 5%.

Q 3.22 Assume that a 3% level-coupon bond has not just 5 years with 10 payments, but 20 years with 40 payments. Also, assume that the interest rate is not 5% per annum, but 10.25% per annum. What are the bond payment patterns and the bond's value?

Q 3.23 Check that the rates of return in the coupon bond valuation example on Page 46 are correct.

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