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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