Financial Mathematics for Actuaries

The following sections are included:

• Dedication
• About the Authors
• Preface to the Second Edition
• Contents
• List of Mathematical Symbols

From time to time we are faced with problems of making financial decisions. These may involve anything from borrowing a loan from a bank to purchase a house or a car; or investing money in bonds, stocks or other securities. To a large extent, intelligent wealth management means borrowing and investing wisely…

An annuity is a series of payments made at equal intervals. There are many practical examples of financial transactions involving annuities, such as

• a car loan being repaid with equal monthly installments
• a retiree purchasing an annuity from an insurance company upon his retirement
• a life insurance policy being purchased with monthly premiums
• a bondholder receiving an annuity in the form of semiannual coupon payments…

The annuity formulas in Chapter 2 are derived based on the assumption that the rates of interest are constant through time and do not vary with the periods of the payments due. In practice, however, we would expect interest rates to change over time. Furthermore, the rates of interest may vary according to the time to maturity (the due date) of the payments. In this chapter we introduce the spot rate of interest and discuss the relationship between the rates of interest and the time to maturity, called the term structure of interest rates. We also define the notion of the forward rate of interest, and describe the relationship between the spot and forward rates of interest…

The performance of investment projects, funds and portfolios of assets is typically measured by their rates of return. The most important measure of the return of a project or fund is the internal rate of return, also called the yield rate. In this chapter we define the internal rate of return and explain how it can be applied to various investment projects. We discuss measures of 1-period rate of return, including methods such as the dollar-weighted rate of return, time-weighted rate of return, as well as some approximation methods. Performance of funds over an extended period of time can be measured by the geometric mean rate of return and arithmetic mean rate of return. We also show how returns of a portfolio can be calculated from the returns of its component assets…

Loans are usually paid back by periodic installments. Upon payment of the installments, the balance of the loan reduces. We introduce two methods of calculating the balance of the loan, namely, the prospective method and the retrospective method. The two methods are mathematically equivalent…

A bond is a contract/certificate of debt for which the issuer promises to pay the holder a sequence of interest payments over a specified period of time, and to repay the principal at a specified terminal date. There are various types of bonds with different features: coupon-paying bonds, zero-coupon bonds, inflation-indexed bonds, callable bonds, etc. Different risks are involved in investing in different types of bonds…

We discuss the use of yield to maturity in measuring bond returns, which properly takes account of the time value of money and reflects the cash-flow pattern. When this method is applied to a callable bond assuming a known call date, the result is the yield to call. In addition, the par yield is often used as a summary of the spot-rate curve. To measure the return of a bond ex post, we can use the realized compound yield, which is also called the holding-period yield. Applying the realized compound yield over specific investment horizons with different interest-rate scenarios will provide useful information for bond management…

The rate of interest affects the prices of bonds differently, depending on the coupon rate of interest and the time to maturity of the bond. We introduce the Macaulay duration and the modified duration as measures of the price sensitivity of a bond with respect to the rate of interest. Duration can also be used to compute the approximate change in the bond price when interest rate changes. If better approximation is desired, convexity may be used to enhance the accuracy…

In this chapter we outline the determination of rates of interest from an economic view point. We discuss the yields on various financial securities, including Treasury bonds. We examine the role of the government in setting interest rate targets and policies. Our discussion will focus mainly on the U.S. economy, which is the largest in the world and whose policies have major impact on other economies…

In previous chapters we have assumed the rate of interest to be constant through time, or varying through time in a deterministic manner. A well-known deterministic set of interest rate scenarios is the collection of seven prescribed interest rate movements required by the U.S. 1990 Standard Valuation Law for cash-flow testing. These scenarios are similar to those used in the New York Regulation 126 and they are now commonly known as the NY7 model…

The following sections are included:

• Appendix A. Review of Mathematics and Statistics
  • Exponential Function
  • Logarithmic Function
  • Roots of a Quadratic Equation
  • Arithmetic Progression
  • Geometric Progression
  • Some Derivatives
  • Integration by Part
  • Taylor Series Expansion
  • Binomial Expansion
  • Expected Value and Variance of a Random Variable
  • Mean and Variance of Sum of Random Variables
  • Uniform Distribution
  • Normal Distribution
  • Lognormal Distribution
• Appendix B. Answers to Selected Exercises

The following section is included:

• Index