Formulas

Question 1

Amount function

Answer 1

A(t), A(0) = Principal

Question 2

Accumulation Function

Answer 2

a(t) = A(t)/A(0), a(0) = 1

Question 3

Interest (accounting)

Answer 3

A(t) - A(0)

Question 4

Compounded Interest

Answer 4

a(t) = (1 + i)^t = e^&t

Question 5

Simple Interest

Answer 5

a(t) = 1 + it, t = days/365

Question 6

Nu

Answer 6

v = 1/1+i

Question 7

Nu with discount rate

Answer 7

v = 1 - d, iv = d, v + d = 1, v + iv = 1

Question 8

Force of interest function

Answer 8

d/dt (ln (A(t))) or a(t)

Question 9

Force of Interest relationships

Answer 9

1+i = 1/v = e^& , 1/a(t) = v^t = e^-&t

Question 10

v + d = 1

Answer 10

d is interest

Question 11

a(3)/a(1)

Answer 11

Bringing time 0 to time 3, and then time 1 to time 0. Overall bringing time 1 to time 3.

Question 12

Nominal annual interest rate

Answer 12

i^(m)

Question 13

Effective periodic interest rate

Answer 13

j = i^(m)/m

Question 14

J with accumulation function

Answer 14

(1 + j) = a(1/m)

Question 15

Effective annual interest rate

Answer 15

(i^(m)/m + 1)^m - 1 or (j + 1)^m - 1

Question 16

Force of interest

Answer 16

1 + i = e^& , nominal interest compounded continually

Question 17

&

Answer 17

i^(inf)

Question 18

Real Rate of return (baskets)

Answer 18

i - r / 1 + r

Question 19

Level certain immediate annuity

Answer 19

an|i = 1 - v^n / i = v + … v^n = v - v^n+ 1 / 1 - v

Question 20

Cash flow annuity interpretation

Answer 20

1 = ia[n] + v ^n , payments of i with on payment of 1 pulled back n periods.

Question 21

Geometric sum formula

Answer 21

(First - PastLast)/(1- mult)

Question 22

Nominal annual discount rate

Answer 22

d^(m)

Question 23

Effective Periodic discount rate

Answer 23

d^(m)/m

Question 24

Nominal v and d relationship

Answer 24

v = 1 - d^(m)/m

Question 25

Annual discount factor

Answer 25

1/(1+i) = ( 1 - d^(m)/m)^m = 1 - d

Question 26

Sinking fund

Answer 26

S = a + ar + ... ar^n --> S=( a - ar^n+1)/(1-r) --> (1- (1+i)^n)/(1-(1+i)) = ((1+i)^n - 1)/i = a[n] (1+i) ^n

Question 27

m year deferred annuity

Answer 27

m|a[n] = a[n]v^m = (v^m+n)s[n]

Question 28

Relationship of perpetuity to annuity

Answer 28

a[inf] - n|a[inf] = a[n] -> 1/i - v^n/i

Question 29

Perpetuity

Answer 29

a[inf] = 1/i -> ia[inf] = 1

Question 30

Accumulation function backwards in time

Answer 30

a(-t) = 1/(1+it)

Question 31

Sinking fund due

Answer 31

s..[n] = ((1+i)^n - 1)/d = s[n] (1+i)

Question 32

Annuity due

Answer 32

a..[n] = (1+i)a[n] = (1 - v^n)/d = 1 + a[n-1]

Question 33

Increasing Annuity

Answer 33

(Ia)[n]i = v + 2v^2 + ... + n v^n ->i(Ia)[n]i = a..[n] + nv^n -> (Ia)[n]i = (a..[n] - nv^n)/i

Question 34

Increasing Sinking Fund

Answer 34

(Is)[n] = (s..[n] - 1)/i

Question 35

Geometric annuity

Answer 35

(1 - (1+r)^nv^n)/((1+i) - (1+r)) = (first - pastlast)/(1-mult)

Question 36

Geometric Perpetuity

Answer 36

1/(i-r)

Question 37

d/di(a[n])

Answer 37

-v(Ia)[n]

Question 38

Decreasing annuity

Answer 38

(Da)[n] = (n - a[n])/i

Question 39

Decreasing Sinking fund

Answer 39

(Ds)[n] = (n(1+i)^n - s[n])/i

Question 40

a[n]^(m)

Answer 40

(1/m)a[mn]j

Question 41

a_[n]

Answer 41

= integral from 0 to n of (v^t) dt = e^tln(v) dt = e^ln(v^t) dt = (1 - v^n)/&

Question 42

v^-k s[n]

Answer 42

s[n+k] - s[k]

Question 43

Increasing perpetuity

Answer 43

(1/i)a..[inf]

Question 44

Interest and Principal paid as nu

Answer 44

INT = 1 - v^n+t-1 PRN = v^n+t-1

Question 45

v^n / (1-v^n)

Answer 45

1/(is[n])

Question 46

PQEV of Bond

Answer 46

Fra[n]j + Cv^n

Question 47

(Da)[n] + (Ia)[n]

Answer 47

(n+1)a[n]

Question 48

PQEV of Bond redeemable at par

Answer 48

F + F(r-j)a[n]j

Question 49

Force of Interest with integration

Answer 49

e^(integral from 0 to n of &t dt)

Question 50

Dollar-weighted return

Answer 50

i = net cash out / avg daily balance