Actuarial Mathematics Flashcards
Accumulation
A = P * (1+i)^n
Interest for one time unit
I = i * P
Interest after n time periods
I = n * i * P
Accumulation n times
An = ( 1 + i )^n
Principal P accumulates
( 1 + i )^n * P
Accumulation factor
Let A(t1,t2) to be the accumulated value at time t2 of $1 at time t1
Accumulation factor general properties
- A(t,t) = 1
- Accumulation of a Principal P invested at time t1 is:
P * A(t1,t2)
- Principal of consistency
t1 <= t2 <= t3
- A(t1,t3) = A(t1,t2) * A(t2, t3)
A(t1, t2) is equivalent to
( 1 + i )^(t2-t2)
Compounded p-thly
Interest paid p times in each unit time period, i.e ever period 1/p
Nominal interest rate
i^(p) where the p is a subscript not a power
Nominal interest rate per time unit
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
Effective interest rate i
Total interest paid on $1 over one time unit
Effective Annual Rate (EAR)
The actual interest rate over a year, accounting for compounding
1 + i = ( 1 + i(p)/p) )^p
Annual Equivalent Rate (AER)
the interest rate that would yield the same accumulation after one year if compounded annually, rather than at different intervals
Annual Percentage Rate (APR)
This is EAR including extra fees
Time dependent nominal rate
i(p)(t), is the nominal interest rate applied over a specific term 1/p starting at a given time t
Accumulation factor for interest converted p-thly
A(t, t+1/p) = 1 +i(p)(t)/p
Force of interest
We define force of interest per unit time at time t as the limit:
d(t) = lim i(p)(t)
p -> infinity
Constant force of interest
A(t1,t2) = e^(d * (t2-t1)
Effective Rate of Discount
the interest rate applied when the discount is taken at the beginning of a time period rather than the end
d = i/ (1+i)
Nominal rate of Discount
the interest discount compounded p-thly
d(p)
Relationship between nominal rates of discount and interest
d(p)/p * A(t, t+1/p) = i(p)/p
We have A(t, t+1/p) = 1 + i(p)/p
Relationship between effective and nominal rate of discount
(1 + i(p)/p)^p = 1+i
Arrears
The practice of paying interest at the end of a time period
Discounting factor
is used to calculate the present value of a future amount by discounting it back to the present
v = 1/(1+i) = 1- d
Discrete Cash flow
Two cash flows are equivalent if their P.Vs are equal
Equation of Cash flow values
P.V (Outgoing cash flow) = P.V (Incoming cash flow)
Annuity-certain
Number of payments is fixed
Level annuity
Payments are equal
Perpetuity
The limit n-> infinity corresponds to payments made “in perpetuity”
Immediate perpetuity
1/i where v= 1/1+i < 1
Perpetuity due
1/d
Deferred annuity
When payments begin after a specified delay or deferral period
Annuities payable p-thly
pays $1 per unit time over n time periods in instalments of $1/p at p-thly intervals
Annuities payable continuously
In the limit p->infinity payments are made continuously at the rate of $1 per time unit
Equation of value at time 0
P * annuities n
where p is the premium per time unit
Schedule of payments
details how much capital is repaid and how much interest is paid with each premium, and how much of the loan is outstanding
Discounted payback period (DPP)
the smallest time t such that the investor’s accumulation is positive
Yield
The yield of an investment is the effective rate of interest at which the outgoing cash flows are equal to the incoming ones.
Accumulation, simple interest
A = P(1 + ni)
Accumulation, compound interest
A = P(1 + i)^n
Discounted present value of C due in time t
Cv^t
P.V. of annuity-due
..an = (1-v^n)/(1-v)
P.V. of immediate-annuity
an = v..an
P.V.s of deferred annuities
m|..an =v^m..an
m|..an=v^man
P.V. of annuity-due payable p-thly
..an = 1/p((1-v^n)/(1-v^1/p))
P.V. of immediate-annuity payable p-thly
an = v^1/p..an
P.V. of continuous annuity
_an = (1-v^n)/delta
