## What is a call feature

Many bonds have a contractual feature that allows the issuer to repay all or part of the bond issue before maturity. This feature is called a call feature, and the bond is said to be callable. For example, the United States Treasury has a bond outstanding that has a 10 coupon and matures on May 15, 2010. It is callable at par (100) on May 15, 2005. This means that on May 15, 2005 (or any interest payment date thereafter), the United States Treasury may repay those bonds still outstanding. These...

## Web Sites

The Chicago Board of Trade maintains a Web site at www.cbot.com. This site contains a variety of recent information on various CBOT activities, both current and historical. The CBOT also offers a conversion factor calculation program and a table of conversion factors. The Chicago Mercantile Exchange maintains a Web site, at www.cme.com. This site contains a variety of information. The Chicago Mercantile Exchange offers a number of interest rate products. The Board of Trade Clearing Corporation...

## Solving polynomial equations

The equations we have been looking at have the general form as described earlier. The fundamental theorem of algebra assures us that any polynomial equation of degree n has n solutions, counting repetitions. However, it makes no statement about whether the solutions are real or complex, or about the signs of the solutions if they are real. If an equation has a complex solution a bi, then the complex conjugate a bi is also a solution to the equation that is, complex solutions occur in pairs, in...

## A note on accrued days in the settlement period a in equation 61 and dated date

The settlement period starts on the date of the last interest payment. But what happens if the upcoming interest payment is the first interest payment This will happen in the case of new issues. The date used is the dated date of the bond. The dated date is, almost always, the date the bond starts to accrue interest. Suppose a new issue of bonds is sold in mid-June 2002, has interest payments on June 1 and December 1, and matures on June 1, 2012. Its dated date would probably be June 1, 2002....

## Calculations for Other Securities

Many other types of fixed-income securities exist. Almost all of them use the equations and methods we have discussed earlier in this book, either the present value equation of the sort used to price bonds, or a discount yield of the sort used to price Treasury bills. When you examine the security, you should be able to figure out the mathematical equation for pricing it. Here are two examples of other types of fixed-income securities, using pricing methods discussed earlier. When you finish...

## What is a bond

A bond consists of a principal amount, usually approximately the amount borrowed, together with periodic interest payments until the principal amount is repaid, together with the last interest payment. The date on which the principal amount is repaid is called the maturity date of the bond, and the bond is said to mature on that date. The periodic interest payments are also called the coupon payments, or coupons. The income from the coupon payments is called the coupon income. The frequency of...

## Why annuities certain are important

Most people like to receive a regular income, and most people, both borrowers and lenders, like to have debts paid off in equal installments over the life of the original loan. There are good business reasons for this preference. It is good loan management policy to pay off the loan in installments over the original life of the loan. For example, before the Great Depression of the 1930s, many homeowners borrowed money to buy their homes, just as they do now. In those...

## Convexity Calculation

Table 16.1 is an example of computing the convexity of a bond. We use the same bond we used in computing the duration a 4 bond, due in 3 years, at a 4 yield. Note that in this case, k 2, the number of compoundings per year. Each semiannual period is counted as 1, so the answer was in half years. To change to years, we divided by k2 4. If we had measured the periods in half years, going from t .5 to t 3, then the amount in the fourth column in Table 16.1 would have been t t 1 2 , and the result...

## Method 2 Calculating i Given S Snt t and n a Numerical Analysis Approach

Suppose you are given S, Snt, n, and t, and you wish to know the value of i, which gave Snt, starting with S. How would you do it Many calculators have a function that will compute this for you. Here are two other methods you can use. The first, the bisection method, can be used in many approximation problems presented in this book. You should have some idea how to use these methods. The Bisection Method for Solving Equations A Practical Approach This method is not nearly so formidable as its...

## Portfolio Duration

A portfolio is a collection of bonds, considered as a group. Any group of bonds could be considered a portfolio, but most portfolios have some feature or features that distinguish them from other portfolios. These features describe the characteristics of the portfolio. Usually, these include the purpose of the portfolio, which would in turn determine the types of bonds bought by the portfolio. Here are some examples of portfolios frequently encountered 1. You have invested your 200,000...

## Weighted Average Cash Flow

Weighted average cash flow expands from average life to include the coupon payments. The equation is as follows Weighted average cash flow - Equation 15.2 where c,, M, are the coupons and maturities payable at time t,. Weighted average cash flow is considered superior to average life because it includes coupon payments. However, this also requires that the coupons either be already known or estimated. Like average life, weighted average cash flow also does not contain a present value factor. In...

## How To Analyze This Problem

To analyze this problem properly, you must make some decisions about the fund you are setting up. You must choose 2. The interest rate earned on the reinvested interest 3. The time when the fund will be dissolved, called the investment horizon 4. The value of the investments in the fund when the fund is dissolved Each of these decisions will partly determine the total amount in the fund when the fund is dissolved. The interest rate nominal annual rate compounded semiannually that equates the...

## Rule 2 Exact Days over Exact Days

In this case, the exact number of days since the last interest payment period is divided by the exact number of days in the settlement period. United States Treasury bonds and notes and some municipal notes, use this rule. If you are trading some municipal notes, how will you know which rule is followed The description of the notes or the municipal note trader will tell you. In this case, the numerator for calculating t is the total number of days since the last interest payment, or the day the...

## What is a put option

A put option is the opposite of a call option. The put option allows the owner of the bond, under certain conditions, to return the bond to the issuer and receive his or her money back the owner is said to put the bond back to the issuer. This must be done under the terms of the original bond contract. Usually, this can only be done on certain dates, and usually the owner must give the issuer advance notice. The most common bonds with a put feature are United States savings bonds. These allow...

## The general equation for computing a bond price given the yield

You can see from the previous description and examples that the usual bond consists of a series of coupon payments, together with a final maturity amount. The coupon payments are a set of equal payments, equally spaced in time. They form an annuity certain. We studied annuities certain in Chapter 5. We can evaluate the final maturity amount by using a present value calculation. We studied present value calculations in Chapter 4. We therefore have all the ingredients needed to compute the bond...

## Rule 1 30Day Month 360Day Year

In this rule, each month is assumed to have 30 days, and so a year will have 360 days. Thus, the periods from February 1 to March 1, from March 1 to April 1, and from April 1 to May 1 each have 30 days. The equation for the day count is as follows Number of days Y 2 - Y1 360 M2 - M1 30 D2 - D1 In this calculation, Y1, M1, and D1 are the year, month, and day, respectively, that the computation period begins, and Y2, M2, and D2 and the year, month, and day, respectively, that the computation...

## Using Modified Duration To Predict Price Change

We now use the modified duration computed above to calculate the expected new price for changes in the yield. We will compute the new price for yield changes of 10 basis points 0.10 and 300 basis points 3.0 for our 4 bond due in 3 years. EXAMPLE 15.1. Use the following duration calculations to compute the 10 basis point change from 4.00 to 3.90 and 4.10 2.801 x .0010 .002801 .2801 Old price New price Actual Delta estimate Decrease yield 100.000 100.280 100.28 0.00 Increase yield 100.000 99.720...

## Topics For Class Discussion

You are retiring at age 60 with a pension of 1,500 monthly. You can receive a Social Security pension of 1,100 monthly, starting in 2 years. Your human resources department has offered to pay you an increased pension for 2 years, until your Social Security payments start, and then a reduced pension. They will pay you 2,300 for 2 years, and then a lifetime pension of 1,200. This 1,200, combined with Social Security's 1,100, will give you 2,300 monthly. What factors should you consider in...

## Percentage rate and time period

The interest rate is often expressed as a percentage rather than as a decimal. In this case, to do the calculation, divide the percentage rate by 100 convert it to a decimal form . EXAMPLE 2.6. In Example 2.1, we converted 6 to .06, as follows The interest rate description must contain both a rate and the time period to which that rate applies. The problem does not make sense, and cannot be solved, without both the rate and the time period. The time period is not necessarily the same as the...

## How to compute interest

The traditional equation for interest computation is Interest Principal X Rate X Time I PRT This is called simple interest. Compound interest is the usual basis for extended calculation, and it is discussed in the next chapter. EXAMPLE 2.4. In the case of Example 2.1, P 100, the amount you lent to your neighbor R .06, the 6 per year rate you are charging T .25, one - quarter year, or 3 months If your neighbor borrowed for 1 year, the annual interest he owes you would However, he is borrowing...

## The perpetuity

An annuity that will continue without time limit forever is called a perpetuity. They rarely occur, although sometimes bonds have long maturity periods, such as the recent Walt Disney 100-year bonds. The equation for a perpetuity of 1 can be expressed as follows where 1 the size of the annuity, i the interest rate, P the value of the perpetuity Then P 1 i, by simply dividing through by i Not many perpetuities exist. The most famous were the old British Consols consolidated loan , which...

## Why we care about bond equivalent yield bey

We use BEY to compare the yields of discount securities and securities traded on a bond yield basis to determine which offers the higher yield. EXAMPLE 6.3. In the previous case, with the 1-year 10 discount security, suppose you are offered a 1-year T-bill at a 10 yield and a 1-year Treasury note at a 10.40 yield. Your broker urges you to buy the 1-year note because it offers you a better yield. You calculate the BEY for the bill, just as we did earlier, and discover that the BET is 10.81 . You...

## Reasons for using continuous functions in financial models

Why would anyone want to use continuous functions They are used because they represent some business activities more accurately and are much easier to use in these cases. Business and other operations that can be represented by a mathematical model, and that have continuous flows of income and expenses, may find it useful to use continuous functions in modeling their activities' functions. A business that has investment income flowing in every day, makes investments every day, has business...

## A general introduction to interest

Interest is payment for the use of capital, usually in the form of money. It can also be thought of as rent for money borrowed for a period, which must be returned at the end of the period. Interest is stated as a percentage per time period, frequently in the form of an annual rate. Most loans have a date on which the amount of the loan becomes due and payable to the lender. This is called the maturity date, and the loan is said to mature on that date. Some loans, such as most home mortgages,...

## Using the bisection method to find real solutions

Usually, the analyst will only want the real solutions to the problem, and even those must be within some reasonable range. For example, if the polynomial has a solution of 10, the analyst probably won't be interested, at least in that solution. The analyst probably will only have an interest in positive real solutions. One way of finding possible real solutions, if they exist, is to simply graph the function for all real values of x. The computer programs listed earlier will do this. If the...

## Suggestions for further study

For differential equations, the following is a well-known and widely used text Boyce, William E., and DiPrima, Richard C., Elementary Differential Equations and boundary Value Problems, 6th ed., John Wiley amp Sons, New York, 1997. For numerical analysis, the author has used Conte, S. D., and de Boor, Carl, Elementary Numerical Analysis An Algorithmic Approach, 3rd ed., McGraw-Hill, New York, 1980. Hamming, R. W., Numerical Methods for Scientists and Engineers, 2nd ed., Dover, New York, 1986....

## Convexity

Modified Prices for Convexity 208 A Misconception about Convexity 209 17 The Mathematical Development of Duration, Convexity, and the Equation to Predict New Bond Prices, Taylor's Series Expansion 215 Reasons for the Equations and Additional Factors Introduced in 18 Probability and Some Applications to Finance Elementary Concepts in Probability A Review 219 Probability as a Mathematical Model 221 Use of the Word Population 221 Probability Distribution Functions 226 Continuous Probability...

## Real and nominal rates

Suppose you lend some money at 6 , and in one year you receive your 6 interest payment. You have received the stated rate on your loan. This is also called the nominal rate of interest. But suppose during the year the money has decreased in value, according to any reasonable measure of value. In the United States, this could be the consumer price index CPI . In terms of real purchasing power, you have not earned 6 , but somewhat less. For example, suppose the CPI shows an inflation of 5 during...

## Return on investment

The discussion and examples shown so far have all involved an actual payment of money and a stated return of money. The money return is due on specific dates, and these terms are all stated in the loan contract at the beginning of the loan. This can be expanded to a general idea of an investment of some sort and a return of some sort. The investment need not be entirely money, and the return need not be entirely money either. However, the same mathematics applies in these cases. What sort of...

## Some famous mathematical models

One of the most famous developments of mathematical models is the development of equations for the movements of the planets. Copernicus is generally credited with first developing the idea, in the 16th century, that the planets moved in circular orbits around the sun. The mathematical model of this behavior would be a circle, with the sun at the center. In the early 17th century, Kepler improved the model. He showed that the planets moved around the sun in elliptical orbits a circle is a...

## Historical backgroundthe big change in investment loan and money management

Any person who has been active in finance and investments during the last 45 years is surely aware of the enormous changes that have occurred in the area of personal financial management during that time. A truly immense range of opportunities for both borrowing and investment has opened up for most people in this country. But these opportunities have also presented financial management problems. This book tries to give the reader the mathematical tools to solve these problems. In the 1950s,...

## An imprint of Elsevier Science

Amsterdam Boston London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo This book is printed on acid-free paper. ' S1 Copyright 2003, Bob Zipf All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier's Science amp...

## Introduction Who this Book is for and What it Hopes to Accomplish

Historical Background The Big Change in Investment, Loan, What this Book Hopes to Accomplish 3 What Sort of Problems Might this Book Help You to Solve 4 Who this Book is Meant to Address 4 The Mathematical Knowledge Required for this Book 5 The Role of Examples and Problems in this Book 6 2 Interest, Its Calculation, and Return on Investment A General Introduction to Interest 7 Percentage Rate and Time Period 10 Return on Investment 11 Analysis of Investments or Returns without Explicit Money...