PDF Exam FM/2 Interest Theory Formulas - Kent State University

Exam FM/2 Interest Theory Formulas

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This is a collaboration of formulas for the interest theory section of the SOA Exam FM / CAS Exam 2. This study sheet is a free non-copyrighted document for students taking Exam FM/2. The author of this study sheet is using some notation that is unique so that no designation will repeat. Each designation has only one meaning throughout the sheet.

Fundamentals of Interest Theory and Time Value of Money

FV = PV (1 + i)n

PV

=

FV

(1 + i)n

d

=

i

(1 +

i)

v

=

(1

1 +

i)

d =1-v v =1-d

i - d = id d = iv

a(t) A(t)

The amount an initial investment of 1 grows to by time t

The amount an initial investment of A(0) grows to by time t

a(t ) = (1 + i)t = etln(1+i)

A(t) = A(0) (1 + i)t = ( ) A 0 etln(1+i)

= ln(1+ i)

a(t ) = et

( ) vn = 1 + i -n = e-n

(t

)

=

a(t ) a(t )

( ) e

t

0

(u

)

du

=

a

t

( ) A 0

e

t

0

(u

)

du

=

A(t )

Effective interest rate with nominal rate i(m) convertible m-thly

i

=

1 +

i (m ) m

m

-1

Effective discount rate with nominal rate d ( p) convertible p-thly

1 -

d

=

1 -

d (p) p

p

Nominal Rate Equivalence

1+ i

=

e

=

1 v

=1 1- d

= 1 +

i(m) m

m

= 1 -

d

(p)

p

-

p

Effective annual rate it during the t-th year is given by:

it

=

amount earned beginning amount

=

a(t)- a(t -1) a(t -1)

=

A(t)- A(t -1) A(t -1)

Note that the t-th year is given by the time period [t -1,t]

Therefore, the interest earned during the t-th year is given by:

A(t -1)i = A(t)- A(t -1)

For equivalent measures of interest we have the following relationship: d < d (2) < d (3) < < < < i(3) < i(2) < i

Annuities

Annuity Immediate-- payments are made at the end of the period Annuity Due-- payments are made at the beginning of the period

Annuity Immediate

a = v + v2 + + vn = 1- vn

n|i

i

a = vn s

n|i

n|i

s = (1 + )i n-1 + (1 + )i n-2 + + 1 = (1 + i)n -1

n|i

i

s = (1 + i)n a

n|i

n|i

Annuity Due a =1+v +

n|i

+ vn-1 = 1 - vn d

a = vn s

n|i

n|i

s = (1+ i)n + (1+ )i n-1 + + (1+ i) = (1+ i)n -1

n|i

d

s = (1 + i)n a

n|i

n|i

Identities for Annuity Immediate and Annuity Due

a = i a = (1+ i) a

d n |

n|

n|

s = i s = (1+ i) s

d n |

n|

n|

a =1+ a

n|

n -1|

s n|

=

sn+1|

-1

Perpetuity

a | i

=

lim a

n n|i

=

v + v2

+ v3

+

=1 i

a | i

=

lim a

n n|i

=

1 d

Continuous Annuities

a = 1- vn = i a

n|i

n|i

( ) PV

=

n

- t (u ) du

e 0

p

t

dt

0

s = (1 + i)n -1 = i s

n|i

n|i

( ) FV

=

n

e

n t

(

u

)

du

p

t

dt

0

n

a = vt dt n|i 0

where p(t) = payment function

Increasing Annuities-- Payments are 1, 2, ... , n

(Ia)n|i

=

a n|

- i

nv n

(Ia)n|i

=

i d

(Ia)n|i

=

(1 +

i)(Ia)n|i

=

a n|

- nvn d

(Is)n|i

=

(1 + i)n (Ia)n|i

=

s- n| i

n

(Is )n|i

=

i d

(Is)n|i

=

(1 + i)n (Ia)n|i

=

s- n| d

n

(Ia)|i

=

( ) lim

n

Ia

n|i

=

1 di

=

1 i

+

1 i2

(Ia)|i

=

( ) lim

n

Ia

n|i

=

1 d2

Decreasing Annuities-- Payments are n, n-1,..., 2, 1

( ) n-a

Da n|i =

n|i

i

(Da )n | i

=

i d

(Da)n|i

=

(1 + i)(Da)n|i

=

n

-a n|i d

(Ds )n | i

=

(1 +

i)n (Da)n|i

=

n(1 + i)n

i

-

s n|i

(Ds )n|i = (1 + i)n (Da)n|i

Present Value of the annuity with terms X , X + Y , X + 2Y ,..., X + (n -1)Y

X

a n|i

+

Y

a n|

- i

nv n

Present Value of the perpetuity with terms X , X + Y , X + 2Y ,...

X i

+

Y i2

Annuities with Terms in Geometric Progression-- 1, (1 + q), (1 + q)2 ,..., (1 + q)n-1 Present Value is V (0) = 1 v + (1 + q) v2 + (1 + q)2 v3 + + (1 + )q n-1 vn = 1 - (1 + q)n vn

i-q

Useful Identities

an+k |

=

a n|

+

vna k|

( ) vn - vm = i a - a m| n|

(Da )n |

+

(Ia)n|

=

(n

+

1)

a n|

1= vn + i a n|

a 2n|

a

=

a 2n |

a

= 1- v2n 1- vn

=1+ vn

n|

n|

s 2n |

s n|

=

s 2n |

s n|

=

(1+ )i 2n -1 (1+ i)n -1

= (1+ i)n

+1

If the interest rate varies:

a n|

=

1

a(1)

+

1

a(2)

+

+

1

a(n)

s n|

=

a(n) a(1)

+

a(n) a(2)

+

+

a(n) a(n)

If the compounding frequency of the interest exceeds the payment frequency of k years--

Use an equivalent interest rate over k years: j = (1 + i)k -1

If the payment frequency exceeds the compounding frequency of the interest--

(1) Use an m-thly annuity

a (m) n|

=

i i(m)

a n|

s(m) n|

=

i i(m)

s n|

a(m) n|

=

d d (m)

a n|

s (m) n|

=

d d (m)

s n|

(2) Use an equivalent interest rate effective over the payment period: j = (1 + i)1 m -1

a(m) = a

n|i

n| j

s(m) = s

n|i

n| j

a(m) = a

n|i

n| j

s(m) = s

n|i

n| j

If the payments are

1 , 2 ,..., n , then the present value is mm m

( )Ia

(m)

n|i

=

a - nvn n|i i(m)

( ) If the payments are

1 m2

2 , m2

,...

,

n m2

, then the present value is

I (m)a

(m)

n|i

=

a(m) - nvn n|i i(m)

Loan Repayment-- Amortization

Amortization Method-- when a payment is made, it must be first applied to pay interest due and then any remaining part of the payment is applied to pay principle

Notation

L amount of the loan

n number of payment periods

P A

P(k )

amount of level payment at the end of the period (amortized payment) loan payment at time k

i

effective interest rate per payment period

B k

balance at time k, balance after k-th payment.

Note that B0 = L

P k

principle paid in payment P(k )

I k

interest paid in payment P(k )

Useful Equations for Level Payments

L

=

PA

a n|i

P A

=

L a

n|i

Prospective Method Retrospective Method

Bk = PA an-k |i

B k

=

L(1+ i)k

-

PA

s k |i

Bk+t = Bk (1 + i)t and Pk+t = Pk (1+ i)t

PA = Pk + Ik

( ) Ik = i Bk-1 = PA 1 - vn-k+1

Pk = PA - I k = PA vn-k+1

Useful Equations for Non-Level Payments

L = P(1)v + P(2)v2 + + P(n)vn

( ) Bk = P(k+1)v + P(k+2)v2 + + P(n)vn-k = Bk-1 1 + i - P(k )

Ik = i Bk-1

Pk = P(k ) - I k = Bk-1 - Bk

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