Actuarial Mathematics

Question 1

Accumulation

Answer 1

A = P * (1+i)^n

Question 2

Interest for one time unit

Answer 2

I = i * P

Question 3

Interest after n time periods

Answer 3

I = n * i * P

Question 4

Accumulation n times

Answer 4

An = ( 1 + i )^n

Question 4

Principal P accumulates

Answer 4

( 1 + i )^n * P

Question 4

Accumulation factor

Answer 4

Let A(t1,t2) to be the accumulated value at time t2 of $1 at time t1

Question 5

Accumulation factor general properties

Answer 5

1. A(t,t) = 1
2. Accumulation of a Principal P invested at time t1 is:

P * A(t1,t2)

3. Principal of consistency

t1 <= t2 <= t3

4. A(t1,t3) = A(t1,t2) * A(t2, t3)

Question 5

A(t1, t2) is equivalent to

Answer 5

( 1 + i )^(t2-t2)

Question 5

Compounded p-thly

Answer 5

Interest paid p times in each unit time period, i.e ever period 1/p

Question 5

Nominal interest rate

Answer 5

i^(p) where the p is a subscript not a power

Question 5

Nominal interest rate per time unit

Answer 5

interest is compounded p-thly with an interes rate of i(p)/p for any 1/p interval

A(t, t+1/p) = 1 + i(p)/p

Question 6

Effective interest rate i

Answer 6

Total interest paid on $1 over one time unit

Question 7

Effective Annual Rate (EAR)

Answer 7

The actual interest rate over a year, accounting for compounding

1 + i = ( 1 + i(p)/p) )^p

Question 8

Annual Equivalent Rate (AER)

Answer 8

the interest rate that would yield the same accumulation after one year if compounded annually, rather than at different intervals

Question 9

Annual Percentage Rate (APR)

Answer 9

This is EAR including extra fees

Question 10

Time dependent nominal rate

Answer 10

i(p)(t), is the nominal interest rate applied over a specific term 1/p starting at a given time t

Question 11

Accumulation factor for interest converted p-thly

Answer 11

A(t, t+1/p) = 1 +i(p)(t)/p

Question 12

Force of interest

Answer 12

We define force of interest per unit time at time t as the limit:

d(t) = lim i(p)(t)
p -> infinity

Question 13

Constant force of interest

Answer 13

A(t1,t2) = e^(d * (t2-t1)

Question 14

Effective Rate of Discount

Answer 14

the interest rate applied when the discount is taken at the beginning of a time period rather than the end

d = i/ (1+i)

Question 15

Nominal rate of Discount

Answer 15

the interest discount compounded p-thly

d(p)

Question 16

Relationship between nominal rates of discount and interest

Answer 16

d(p)/p * A(t, t+1/p) = i(p)/p

We have A(t, t+1/p) = 1 + i(p)/p

Question 17

Relationship between effective and nominal rate of discount

Answer 17

(1 + i(p)/p)^p = 1+i

Question 18

Arrears

Answer 18

The practice of paying interest at the end of a time period

Question 19

Discounting factor

Answer 19

is used to calculate the present value of a future amount by discounting it back to the present

v = 1/(1+i) = 1- d

Question 20

Discrete Cash flow

Answer 20

Two cash flows are equivalent if their P.Vs are equal

Question 21

Equation of Cash flow values

Answer 21

P.V (Outgoing cash flow) = P.V (Incoming cash flow)

Question 22

Annuity-certain

Answer 22

Number of payments is fixed

Question 23

Level annuity

Answer 23

Payments are equal

Question 24

Perpetuity

Answer 24

The limit n-> infinity corresponds to payments made “in perpetuity”

Question 25

Immediate perpetuity

Answer 25

1/i where v= 1/1+i < 1

Question 26

Perpetuity due

Answer 26

1/d

Question 27

Deferred annuity

Answer 27

When payments begin after a specified delay or deferral period

Question 28

Annuities payable p-thly

Answer 28

pays $1 per unit time over n time periods in instalments of $1/p at p-thly intervals

Question 29

Annuities payable continuously

Answer 29

In the limit p->infinity payments are made continuously at the rate of $1 per time unit

Question 30

Equation of value at time 0

Answer 30

P * annuities n
where p is the premium per time unit

Question 31

Schedule of payments

Answer 31

details how much capital is repaid and how much interest is paid with each premium, and how much of the loan is outstanding

Question 32

Discounted payback period (DPP)

Answer 32

the smallest time t such that the investor’s accumulation is positive

Question 33

Yield

Answer 33

The yield of an investment is the effective rate of interest at which the outgoing cash flows are equal to the incoming ones.

Question 34

Accumulation, simple interest

Answer 34

A = P(1 + ni)

Question 35

Accumulation, compound interest

Answer 35

A = P(1 + i)^n

Question 36

Discounted present value of C due in time t

Answer 36

Cv^t

Question 37

P.V. of annuity-due

Answer 37

..an = (1-v^n)/(1-v)

Question 38

P.V. of immediate-annuity

Answer 38

an = v..an

Question 39

P.V.s of deferred annuities

Answer 39

m|..an =v^m..an

m|..an=v^man

Question 40

P.V. of annuity-due payable p-thly

Answer 40

..an = 1/p((1-v^n)/(1-v^1/p))

Question 41

P.V. of immediate-annuity payable p-thly

Answer 41

an = v^1/p..an

Question 42

P.V. of continuous annuity

Answer 42

_an = (1-v^n)/delta