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IMFC2 - Chapter 2 Fixed-Income Securities and Interest...

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Chapter 2 Fixed-Income Securities and Interest Rates We now begin a systematic study of fixed-income securities and interest rates. By a fixed-income security we mean a financial instrument that promises fixed (or definite) payments at prescribed future dates. In some cases only one payment is made at maturity; in other cases, periodic payments are made. The payments are usually characterized by an interest rate which, in some sense, is simply a rental rate for money. In practice, there is sometimes ambiguity regarding what is or is not a fixed- income security because the payments for some securities that are classified as being fixed-income are actually tied to quantities such as (variable) interest rates and credit ratings that can fluctuate, and consequently the payments are not known exactly at the time the security is purchased. In this chapter we focus primarily on situations where payments and interest rates are described precisely in advance and we assume that there is no risk of default. We assume first that interest rates are constant; in particular that they are independent of both the date on which the investment is made and the length (or term) of the investment. 2.1 Basic Interest Rate Mechanics Interest may be either simple or compound . With simple interest, only the original principal accrues interest and no interest is paid on the interest. On the other hand, with compound interest, interest is paid on the previously accrued interest as well as on the principal. Unless stated otherwise, the time t is measured in years and interest rates are given as annualized rates. In order to describe the basic mechanics of interest calculations, we shall assume that an initial amount A > 0 is invested at time 0, that no additional investments (or withdrawals) are made, and that the annual interest rate is r 0. We are interested in the value V t of the investment at future times t . Analogous formulas apply to the case of a loan. (Indeed, a loan can be considered as an investment of an amount A < 0.) 33
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2.1.1 Simple Interest The interest earned up to time t is simply equal to rtA . (This is just the familiar formula “Interest = (Principal) × (rate) × (time)”.) Therefore, we have V t = A + rtA = A (1 + rt ) . Notice that with simple interest the value of the account grows linearly with time. It is sometimes convenient to express the simple-interest rule as a differential equation together with an initial condition, namely, d dt V t = rA ; V 0 = A. In practice, the simple interest convention is used only for investments or loans of relatively short maturity (one year or less). Example 2.1. Suppose that you invest $100 in an account that pays 8% simple interest. (a) At the end of three months, you will have V . 25 = $100(1 + ( . 08) × . 25) = $102 in the account. (b) At the end of one year you will have V 1 = $100(1 + ( . 08) × 1) = $108 in the account.
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