SOA EXAM FM QUESTIONS WITH
CORRECT ANSWERS
Value of $1.00 in a fund at a given time
Given by accumulation function a(t)
Note that a(0) = 1
Amount function: A_k(t) = k * a(t); represents value of investment of k dollars - Correct
Answers -Accumulated Value, Accumulation Function, Amount Function
i_t = [A(t) - A(t - 1)]/A(t - 1)
Note that A(t) = (1 + i_t)A(t - 1) and
A(t) = k prod_{n = 1}^t (1 + i_n) - Correct Answers -Effective Rate of Interest
A(t) = k(1 + it)
i_t = i/[1 + i(t - 1)] - Correct Answers -Simple interest
A(t) = k(1 + i)^t
i_t = i - Correct Answers -Compound Interest
If amount to be paid is not paid of in an integral number of payments, there are 3
options.
1: Small payment at exact time
2: Small payment on following period: drop payment
3: Large payment on previous period: balloon payment - Correct Answers -Balloon and
Drop Payments
Define a^(m)_n = (1 - v^n)/i^(n) = [i/i^(m)]a_n = s^(m)_1 a_n
Similarly,
a**^(m)_n = (1 - v^n)/d^(n) = [d/d^(m)]a_n = s**^(m)_1 a_n
Note a^(m)_\infty = 1/i^(m), a**^(m)_\infty = 1/d^(m), and
a**^(m)_\infty - a^(m)_\infty = 1/d^(m) - 1/i^(m) = 1/m - Correct Answers -Notation for
Fusion Method
a^(infty)_n = (1 - v^n)/del = i/del a_n = \int_0^n v^t dt - Correct Answers -Continuous
Annuities
Present Value: PV_k(t) = 1/A_k(t); represents present value of k dollars after t
years/periods. Discount rate is v_t = 1/(1 + i_t) - Correct Answers -Present Value
, d_t = [A(t) - A(t - 1)]/A(t)
Note: v = 1 - d, i - d = id - Correct Answers -Effective Rate of Discount
Given nominal interest rate i^(n), (1 + i^(n)/n)^n = 1 + i
Given nominal discount rate d^(n), (1 - d^(n)/n)^n = 1 - d
Note: n is number of compound periods per year - Correct Answers -Nominal Interest
Rate and Nominal Discount Rate
del = D a(t) / a(t) = D [ ln(a(t)) ] - Correct Answers -Force of Interest
del = ln(1 + i) = - ln(1 - d); del = i^(infty) = lim i^(n) = d^(infty) - Correct Answers -
Constant Force of Interest
e^x = sum_0 x^n/n!
ln(1 + x) = sum_1 (-1)^(n+1) x^n/n - Correct Answers -Useful Taylor Series
PV: (1 - v^n)/i; special notation is a_n | i
AV: [(1 + i)^n - 1]/i; special notation is s_n | i
Relationship: s_n = (1 + i)^n a_n - Correct Answers -Present and Accumulated Value of
an Annuity Immediate
PV: (1 - v^n)/d; special notation is a**_n | i
AV: [(1 + i)^n - 1]/d; special notation is s**_n | i
Relationships: a**_n = (1 + i)a_n = a_{n - 1} + 1
s**_n = (1 + i)s_n = s_{n + 1} - 1 - Correct Answers -Present and Accumulated Value of
an Annuity Due
m|a_n means advance 4 years and then do annuity immediate for n years; similar for s,
a**, s**
m|a_n = m+1|a**_n = a_{m + n} - a_m
Similar for s, s** - Correct Answers -Deferred Annuity Notation
Annuity with infinite payments; can be immediate and due
PV for immediate = a_\infty = 1/i
PV for due = a**_\infty = 1/d - Correct Answers -Perpetuity
a_2n / a_n = 1 + v^n
a_3n / a_n = 1 + v^n + v^{2n}
a_{mn} / a_n = sum{k = 0}^{m - 1} v^{kn}
Can be used to find i - Correct Answers -Quotients of Annuity Present Values
a**^(m)_p / a**^(m)_q = a**_p / a**_q = a^(m)_p / a^(m)_q =
a_p / a_q - Correct Answers -Cancellation Formulas
CORRECT ANSWERS
Value of $1.00 in a fund at a given time
Given by accumulation function a(t)
Note that a(0) = 1
Amount function: A_k(t) = k * a(t); represents value of investment of k dollars - Correct
Answers -Accumulated Value, Accumulation Function, Amount Function
i_t = [A(t) - A(t - 1)]/A(t - 1)
Note that A(t) = (1 + i_t)A(t - 1) and
A(t) = k prod_{n = 1}^t (1 + i_n) - Correct Answers -Effective Rate of Interest
A(t) = k(1 + it)
i_t = i/[1 + i(t - 1)] - Correct Answers -Simple interest
A(t) = k(1 + i)^t
i_t = i - Correct Answers -Compound Interest
If amount to be paid is not paid of in an integral number of payments, there are 3
options.
1: Small payment at exact time
2: Small payment on following period: drop payment
3: Large payment on previous period: balloon payment - Correct Answers -Balloon and
Drop Payments
Define a^(m)_n = (1 - v^n)/i^(n) = [i/i^(m)]a_n = s^(m)_1 a_n
Similarly,
a**^(m)_n = (1 - v^n)/d^(n) = [d/d^(m)]a_n = s**^(m)_1 a_n
Note a^(m)_\infty = 1/i^(m), a**^(m)_\infty = 1/d^(m), and
a**^(m)_\infty - a^(m)_\infty = 1/d^(m) - 1/i^(m) = 1/m - Correct Answers -Notation for
Fusion Method
a^(infty)_n = (1 - v^n)/del = i/del a_n = \int_0^n v^t dt - Correct Answers -Continuous
Annuities
Present Value: PV_k(t) = 1/A_k(t); represents present value of k dollars after t
years/periods. Discount rate is v_t = 1/(1 + i_t) - Correct Answers -Present Value
, d_t = [A(t) - A(t - 1)]/A(t)
Note: v = 1 - d, i - d = id - Correct Answers -Effective Rate of Discount
Given nominal interest rate i^(n), (1 + i^(n)/n)^n = 1 + i
Given nominal discount rate d^(n), (1 - d^(n)/n)^n = 1 - d
Note: n is number of compound periods per year - Correct Answers -Nominal Interest
Rate and Nominal Discount Rate
del = D a(t) / a(t) = D [ ln(a(t)) ] - Correct Answers -Force of Interest
del = ln(1 + i) = - ln(1 - d); del = i^(infty) = lim i^(n) = d^(infty) - Correct Answers -
Constant Force of Interest
e^x = sum_0 x^n/n!
ln(1 + x) = sum_1 (-1)^(n+1) x^n/n - Correct Answers -Useful Taylor Series
PV: (1 - v^n)/i; special notation is a_n | i
AV: [(1 + i)^n - 1]/i; special notation is s_n | i
Relationship: s_n = (1 + i)^n a_n - Correct Answers -Present and Accumulated Value of
an Annuity Immediate
PV: (1 - v^n)/d; special notation is a**_n | i
AV: [(1 + i)^n - 1]/d; special notation is s**_n | i
Relationships: a**_n = (1 + i)a_n = a_{n - 1} + 1
s**_n = (1 + i)s_n = s_{n + 1} - 1 - Correct Answers -Present and Accumulated Value of
an Annuity Due
m|a_n means advance 4 years and then do annuity immediate for n years; similar for s,
a**, s**
m|a_n = m+1|a**_n = a_{m + n} - a_m
Similar for s, s** - Correct Answers -Deferred Annuity Notation
Annuity with infinite payments; can be immediate and due
PV for immediate = a_\infty = 1/i
PV for due = a**_\infty = 1/d - Correct Answers -Perpetuity
a_2n / a_n = 1 + v^n
a_3n / a_n = 1 + v^n + v^{2n}
a_{mn} / a_n = sum{k = 0}^{m - 1} v^{kn}
Can be used to find i - Correct Answers -Quotients of Annuity Present Values
a**^(m)_p / a**^(m)_q = a**_p / a**_q = a^(m)_p / a^(m)_q =
a_p / a_q - Correct Answers -Cancellation Formulas
