Textbook Notes (368,795)
ACTSC 231 (11)
Javid Ali (1)
Chapter 1

# Chapter 1.docx

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School
Department
Actuarial Science
Course
ACTSC 231
Professor
Javid Ali
Semester
Fall

Description
Chapter 1 – The time Value of Money  Interest o Definition: Compensation one receives for lending asset, it doesn’t have to be monetary. o [t1, t2](2 ) – A1t ) t = time A = Amount Function o I = S – P (S = Accumulated value P = Principal) o n = accumulated amount at end of period n – accumulated amount at end of period n-1 o n = A(n) – A(n-1)  Capital o Definition: Asset one lent out, it doesn’t have to be monetary.  Principal o Definition: Asset one lent out (monetary).  Accumulation Function – a(t) o Value at time t of \$1 invested at time 0. o a(0) = 1  Amount Function – A(t) o Value at time t of an initial capital of P invested at time 0. o A(0) = P o A(t) = P * a(t)  Effective Rate of Interest – “i” o Definition Amount of money that one unit of capital, invested at beginning of period will earn during the period, where interest is paid at the end of the period. Caters compounding interest. Interest earned over [t ,t ] [t ,t ]A(t )-A(t ) a(t )-a(t ) o [1 2t ] 1 2 = 1 2 = 2 1 = 2 1 Capital Invested at t1 A(t 1 A(t1) a(t1) o i = Interest earned over n-th period = In = A(n)-A(n-1) = a(n)-a(n-1) n Capital Invested at beginning of n-th period A(n-1) A(n-1) a(n-1) 1. No interest. 2. Simple Interest/Nominal Interest – “i” o Definition: Interest computed based on original principal during the whole term of loan. P = No reinvestment of interest  a(t+s) [0, t+s](t)[0, t] *)[t, t+s] Principal I = Simple o Linear growth  linear approximation of compound interest. Interest o Use: Short term, less than 1 year. S = Amount r = Annual o I = Prt = Interest earned on initial principal rate of o S = P + I  S = P + Prt  S = P (1 + rt) or S = P (1 + it) simple interest o The effective interest rate of simple interest is decreasing over time. t = time (year) o Time Types: Exact # of Days o Exact simple interest: t = 365 o Ordinary simple interest: t = Exact # of Days 360 3. Step function: earns interest at defined period, rises by set amount and remains constant over time. 4. Compound Interest o Definition: Interest earned from principal and the reinvestments of past interests. Accumulation Function: a(t)=(1+i) t o o 2 = a(1) * i = (1 + i) * i o a(2) = a(1) + 2 = (1 + i) + (1 + i) * i o a(2) = 1 + i + i + i = 1 + 2i + i = (1 + i) t o => a(t) = (1 + i) Amount Function: t o A(t)= P(1+i) ,t ³ 0 o a(t+s)[0, t+s]a(t[0, t] *s)[t, t+s] y
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