Module 1: Time Value of Money Flashcards
what is a(0)
1
what is A(t)
k*a(t) or k/v(t)
how to calculate interest earned?
A(n)-A(n-1)
exact simple interest method
ignores leap years, assume every year has 365 days, t = # of days/365
ordinary simple interest method
approximates by rounding full months to 30 days each and each year to 12 x 30 = 360 days, t = approximate # of days/360
bankers rule
counts # of days exactly but uses 360 as # of days in a year, t = number of days/360
a(t) for simple interest
1 + it
v(t) for simple interest
1 - it
a(t) for compound interest
(1+i)^t, t>=0
v(t) for compound interest
(1-i)^t
what happens to a(t) between 0
a(t) for simple interest > a(t) for compound interest
what happens to a(t) at t = 0 and t = 1
a(t) for simple interest = a(t) for compound interest
Effective interest rate of period (t1, t2)
i[t1, t2] = interest earned during period/amount at beginning of period = A(t2) - A(t1)/A(t1) = a(t2)-a(t1)/a(t1)
what is the accumulation factor
(1+i). This is the growth factor
effective rate of discount in nth year
=discount (interest) earned in nth year / amount at the end of nth year
= A(n) - A(n-1) / A(n)
= a(n) - a(n-1) / a(n)
EIR
(1+i/n)^n - 1
EDR
(1-i/n)^-n - 1
a(t) when t < 0
a(t) < 1
a(t) when t > 0
a(t) > 1
v(t) for simple discount
v(t) = 1 - dt, 0 <= t < 1/d
are effective rates with simple interest increasing/decreasing with time?
decreasing
are effective rates with simple discount increasing/decreasing with time?
increasing
v(t) for compound discount
(1-d)^t for t>=0
effective rate formula for i
(1+(i/m))^m - 1
what is i^(m)
annual effective rate compounded m times
what is i^(m)/m
“quarterly” effective rate
effective rate formula for d
1-(1-d/m)^m
going from effective interest rate to discount rate
i = d / (1-d)
going from effective discount rate to interest rate
d = i / (1 + i)
how does I^m change as m increases
decreases at a decreasing rate
how does d^m change as m increases
increases at a decreasing rate
how does i^m and d^m change as m changes
they are both approaching the same limit, i^m as m is small and d^m as m is big
what is δ
constant force of interest; annual nominal rate of interest with continuous compounding
formula for δ
= i^(infinity) = ln(1+i)
what is δt
instantaneous relative change in accumulation function
what is a(t) for δ
a(t) = e^δt
what is a(t) for δt
a(t) = e^(integral from time 1 to time 2 of δt)
what is p(t)
price index function
what is ã
real accumulation function
what is p(0)
1
formula for real accumulation function ã
ã = a(t) / p(t)
what is p(t) formula
(1+r)^t
what is ã(t) formula
(1 + ĭ)^t
what is r
annual effective rate of inflation
what is ĭ
annual effective rate of “real” interest
what is ĭ formula
i - r / 1 + r
how to solve for ĭ^(m)
ĭ^(m)/m = i^(m)/m - r^(m)/m / 1 + r^m / m
what does payments of $100 at time t mean?
$100 / (1+i)^t
what is i^(infinity) = δ formula in general
derivative of a(t) / a(t) = d/dt of ln(a(t))
