Actuarial Mathematics
Q&A
Discount function (definition) - answer-A positive valued function v, of two nonnegative
variables, which satisfy the following relationship: v(s,t)v(t,u) = v(s,u) for all s, t, u.
Other desirable features:
v(s,s) = 1 for all s: v(s,t)v(t,s) = v(s,s) = 1 -> v(s,t) = v(t,s)^-1.
For simplicities sake: v(0,t) = v(t).
v(s,t) = v(t)/v(s).
Definition: The rate of interest for the time interval k to k+1 is the quantity: - answer-i_k = v(k+1,
k) - 1
Note: an investment of 1 unit at time k will produce v(k+1, k) = 1 + i_k units at time k+1.
Definition: The rate of discount for the time interval k to k+1 is the quantity: - answer-d_k = 1 -
v(k, k+1).
An investment of 1 - d_k units at time k will accumulate to 1 unit at time k+1.
Given i_k = v(k+1, k) - 1 and d_k = 1 - v(k, k+1), we have the following: - answer-d_k = i_kv(k,
k+1) = i_k / (1+i_k)
i+k = d_kv(k+1, k) = d_k / (1 - d_k)
Definition: For any time n = 0,1,...,N, the value at time n of the cash flow vector c with respect to
the discount function v is given by: - answer-Val_n(c;n) = sum(c_k * v(n,k))
Represents the single amount that we would accept at time n in place of all the other cash
flows, assuming that everything accumulates in accordance to the discount function v.
Annuity - answer-Sequence of periodic payments
Definition: Two cash flow vectors c and e are said to be actuarially equivalent with respect to the
discount function v, if, for some nonnegative integer n, - answer-Val_n(c;v) = Val_n(e;v)
Replacement principle - answer-Suppose that we are given a cash-flow vector and some subset
of entries (0,1,..,N). Take the value at time k of just those cash flows in the subset and then
replace all entries in the subset by a single payment at time k, equal to that value. This leaves a
vector that is actuarially equivalent to the original.
Assume that v(n) = v^n, for some constant v. Consider vectors (1_n) and j^n = (1,2,...,n-1,n).
Then: - answer-ä(1_n) = 1 + v + v^2 + ... + v^n-1 = (1-v^n) / (1-v) [proof below]
, ä(j^n) = 1 + 2v + 3v^2 + ... + nv^(n-1) = (ä(1_n) - nv^n) / (1 - v)
Multiply ä(1_n) by v we get: vä(1_n) = v + v^2 + ... + v^n-1 + v^n
Then subtract (2) from (1) and divide by 1-v leads to:
(ä(1_n) - vä(1_n)) / (1-v) = (1-v^n) / (1-v)
Now ä(0,1_n) (deferred annuity) = v + v^2 + ... + v^n-1 + v^n = (ä(1_n) - 1 + v^n) = (1-v^n) / i ,
where i = (1-v)/v (constant interest rate).
Definition 2.6: For k = 0,1,...,N, the balance at time k with respect to c and v, is defined by -
answer-B_k(c;v) = Val_k(_kc;v) = sum (j=0, k-1) (c_j * v(k,j)).
The balance at time k is simply the accumulated amount at time k resulting from all the
payments received up to that time, and answers the question of how much we will have.
Definition 2.7: For k = 0,1,...,N, the reserve at time k with respect to c and v is defined by -
answer-_kV(c;v) = -Val_k(^kc;v) = - sum(j=k,N) (c_j * v(k,j)).
The reserve at time k is the negative of the value at time k of the future payments, and is equal
to the amount we will need in order to meet future obligations.
Calculating the outstanding balance by means of the reserve is often called ...
And by means of balance if often called... - answer-The prospective method since it looks to the
future.
The retrospective method since it looks to the past.
Let l_0 be an arbitrary number. Suppose we start with a group of l_0 newly born individuals.
What does l_x denote, and what does d_x denote? - answer-l_x: the number of those aged 0
who will still be alive at age x. (number of people alive at age x)
d_x: the number of those who were orignially aged 0, who die between the ages of x and x+1.
(number of people who die at age x).
The basic relationship between l_x and d_x is: l_x+1 = l_x - d_x
The basic relationship between l_x and d_x is: - answer-l_x+1 = l_x - d_x
For nonnegative integers n and x let
_np_x =
_nq_x =
(chapter 3) - answer-_np_x = l_x+n / l_x
Q&A
Discount function (definition) - answer-A positive valued function v, of two nonnegative
variables, which satisfy the following relationship: v(s,t)v(t,u) = v(s,u) for all s, t, u.
Other desirable features:
v(s,s) = 1 for all s: v(s,t)v(t,s) = v(s,s) = 1 -> v(s,t) = v(t,s)^-1.
For simplicities sake: v(0,t) = v(t).
v(s,t) = v(t)/v(s).
Definition: The rate of interest for the time interval k to k+1 is the quantity: - answer-i_k = v(k+1,
k) - 1
Note: an investment of 1 unit at time k will produce v(k+1, k) = 1 + i_k units at time k+1.
Definition: The rate of discount for the time interval k to k+1 is the quantity: - answer-d_k = 1 -
v(k, k+1).
An investment of 1 - d_k units at time k will accumulate to 1 unit at time k+1.
Given i_k = v(k+1, k) - 1 and d_k = 1 - v(k, k+1), we have the following: - answer-d_k = i_kv(k,
k+1) = i_k / (1+i_k)
i+k = d_kv(k+1, k) = d_k / (1 - d_k)
Definition: For any time n = 0,1,...,N, the value at time n of the cash flow vector c with respect to
the discount function v is given by: - answer-Val_n(c;n) = sum(c_k * v(n,k))
Represents the single amount that we would accept at time n in place of all the other cash
flows, assuming that everything accumulates in accordance to the discount function v.
Annuity - answer-Sequence of periodic payments
Definition: Two cash flow vectors c and e are said to be actuarially equivalent with respect to the
discount function v, if, for some nonnegative integer n, - answer-Val_n(c;v) = Val_n(e;v)
Replacement principle - answer-Suppose that we are given a cash-flow vector and some subset
of entries (0,1,..,N). Take the value at time k of just those cash flows in the subset and then
replace all entries in the subset by a single payment at time k, equal to that value. This leaves a
vector that is actuarially equivalent to the original.
Assume that v(n) = v^n, for some constant v. Consider vectors (1_n) and j^n = (1,2,...,n-1,n).
Then: - answer-ä(1_n) = 1 + v + v^2 + ... + v^n-1 = (1-v^n) / (1-v) [proof below]
, ä(j^n) = 1 + 2v + 3v^2 + ... + nv^(n-1) = (ä(1_n) - nv^n) / (1 - v)
Multiply ä(1_n) by v we get: vä(1_n) = v + v^2 + ... + v^n-1 + v^n
Then subtract (2) from (1) and divide by 1-v leads to:
(ä(1_n) - vä(1_n)) / (1-v) = (1-v^n) / (1-v)
Now ä(0,1_n) (deferred annuity) = v + v^2 + ... + v^n-1 + v^n = (ä(1_n) - 1 + v^n) = (1-v^n) / i ,
where i = (1-v)/v (constant interest rate).
Definition 2.6: For k = 0,1,...,N, the balance at time k with respect to c and v, is defined by -
answer-B_k(c;v) = Val_k(_kc;v) = sum (j=0, k-1) (c_j * v(k,j)).
The balance at time k is simply the accumulated amount at time k resulting from all the
payments received up to that time, and answers the question of how much we will have.
Definition 2.7: For k = 0,1,...,N, the reserve at time k with respect to c and v is defined by -
answer-_kV(c;v) = -Val_k(^kc;v) = - sum(j=k,N) (c_j * v(k,j)).
The reserve at time k is the negative of the value at time k of the future payments, and is equal
to the amount we will need in order to meet future obligations.
Calculating the outstanding balance by means of the reserve is often called ...
And by means of balance if often called... - answer-The prospective method since it looks to the
future.
The retrospective method since it looks to the past.
Let l_0 be an arbitrary number. Suppose we start with a group of l_0 newly born individuals.
What does l_x denote, and what does d_x denote? - answer-l_x: the number of those aged 0
who will still be alive at age x. (number of people alive at age x)
d_x: the number of those who were orignially aged 0, who die between the ages of x and x+1.
(number of people who die at age x).
The basic relationship between l_x and d_x is: l_x+1 = l_x - d_x
The basic relationship between l_x and d_x is: - answer-l_x+1 = l_x - d_x
For nonnegative integers n and x let
_np_x =
_nq_x =
(chapter 3) - answer-_np_x = l_x+n / l_x
