RMSC2001_Tutorial_4(3).pdf - RMSC2001 Introduction to Risk Management Tutorial 4 Interests Chen Sihan Oct 9 2019 1 Interest 1.1 Accumulation Time Value

# RMSC2001_Tutorial_4(3).pdf - RMSC2001 Introduction to Risk...

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RMSC2001 - Introduction to Risk Management Tutorial 4: Interests Chen, Sihan Oct 9, 2019 1 Interest 1.1 Accumulation Time Value of Money : the same amount of money will worth less in the future. 1. Principal : The initial amount of an investment, denoted by A (0). 2. Accumulated Value : The value of an investment after time t , denoted by A ( t ). 3. Accumulation Function : The accumulative value at time t for unit principal, de- noted by a ( t ) := A ( t ) /A (0). a (0) = A (0) /A (0) = 1, as no money can be earned at time 0. a ( t ) is increasing with t and continuous. 1.2 Interest 1.2.1 Definitions 1. Interest : The extra money you obtained at time t , I ( t ) = A ( t ) - A (0). 2. Interest Rate : The interest you obtained for unit principal, i ( t ) = I ( t ) /A (0). Nominal Rate : Suppose n - 1 and n are two consecutive interest-paying time, and it is payed m times annually, then nominal rate i m = m [ A ( n ) - A ( n - 1)] /A ( n - 1). Effective Rate : Suppose t - 1 and t are any two time points, then effective rate during the t th period i ( t ) = [ A ( t ) - A ( t - 1)] /A ( t - 1). 1
1.2.2 Types of Interest 1. Simple Interest : (Nominal) simple interest rate i s : t, a ( t ) = a (0)(1+ i s t ) = 1+ i s t . Effective rate: i ( t ) = (1+ i s t ) - (1+ i s ( t - 1)) 1+ i s ( t - 1) = i s 1+ i s ( t - 1) , which decreases as t increases. 2. Compound Interest : Suppose the interest is paid per unit time, then after t unit times, a ( t ) = (1 + i c ) t , where i c is the (nominal) compound interest rate. For each unit t , the effective rate i ( t ) = (1+ i c ) t - (1+ i c ) t - 1 (1+ i c ) t - 1 = i c is constant. Suppose the interest is paid m times per year and t is measured in years, then a ( t ) = (1 + i c /m ) mt .

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