Other equations

Question 1

Price of security (ignoring tax)

Answer 1

P = Da(p):<n> + Rv^n</n>

Question 2

Price of security (allowing for income tax)

Answer 2

P’ = D(1 - t1)a(p):<n> + Rv^n</n>

Question 3

Price of security (allowing for income and capital gains tax)

Answer 3

P’ = D(1 - t1)a(p):<n> + Rv^n - t2*(R - P'')*v^n</n>

Question 4

Capital gains test

Answer 4

i(p) > (D / R) * (1 - t1) capital gain
< capital loss
= price is equal to the redemption payment

Question 5

Price of equity immediately after dividend of D has been paid

Answer 5

P = D(1 + g) / (i - g) where g is the constant annual rate the annual dividends are expected to increase by

Question 6

Equation of value for property

Answer 6

P = sum(k=1, infinity)[(1/m) * D:(k/m) * v^(k/m)] where (1/m)*D:(k/m) is the rental income at time k/m

Question 7

Real rate of interest after allowing for inflation

Answer 7

i’ = (i - j) / (1 + j) ~= i - j
where i’ is the real rate, i the money rate, and j the effect of inflation

Question 8

Equation of value for an index-linked bond

Answer 8

P = sum(k=1, 2n)[(D/2) * (Q(k/2) / Q(0)) * (v:i)^(k/2) + R * (Q(n) /Q(0)) * (v:i)^n
where Q(t) is the index value at time t, D the nominal annual coupon payable half-yearly in arrears, and R the nominal redemption payment

Question 9

Relationship between forward rates of interest and spot rates

Answer 9

(1 + f:t,r)^r = [(1 + y:t+r)^(t+r)] / [(1 + y:t)^t] = P:t / P:t+r
where the discrete-time forward rate is agreed at time 0 for an investment made at time t > 0 for a period of r years, and P:t denotes the price at issue of a unit zero-coupon bond with term t years.

Question 10

Volatility

Answer 10

nu(i) = - A’ / A where A = sum(k=1, n)[C:k * v^t:k]

Question 11

DMT

Answer 11

tau(i) = (1 + i) * volatility

Question 12

Convexity

Answer 12

c(i) = A’’ / A

Question 13

Redington’s conditions

Answer 13

1. V(i0):A = V(i0):L i.e. PV(assets) = PV(liabilities)
2. V’(i0):A = V’(i0):L i.e. volatility(assets) = volatility(liabilities) or DMT(assets) = DMT(liabilities)
3. V’‘(i0):A > V’‘(i0):L i.e. convexity(assets) > convexity(liabilities)

Question 14

Force of mortality: tqx

Answer 14

tqx = int(0,t)[spx * mu:x+s ds]

Question 15

Force of mortality: tpx

Answer 15

tpx = exp{-int(0,t)[mu:x+s ds]}

Question 16

Gross premiums

Answer 16

EPV of premiums = EPV of benefits + EPV of expenses (+ EPV profit)

Question 17

Gross premium prospective reserves

Answer 17

EPV of future benefits + EPV of future expenses - EPV of future premiums

Question 18

Retrospective accumulation of a t-year payment stream

Answer 18

(AV):t = {EPV of the payments as at the start of the t years} * [(1 + i)^t / tpx]

Question 19

Gross premium retrospective reserves

Answer 19

Retrospective accumulation of (past premiums - past benefits and expenses)

Question 20

Recursive formula for reserves

Answer 20

(tV + G - e)(1 + i) - q:(x+t) * (S + f) = p:(x+t) * (t+1)V’

Question 20

Profit between policy durations

Answer 20

PRO:t = (tV + G - e)(1 + i) - q:(x+t) * (S + f) - p:(x+t) * (t+1)V’

Question 21

Net premium reserves for without-profit contracts

Answer 21

tV:x:<n> = A:(x+t):<n-t> - P:x:<n> * adue:(x+t):<n-t> = 1 - [adue:(x+t):<n-t> / adue:x:<n>]</n></n-t></n-t></n></n-t></n>

Question 22

DSAR

Answer 22

S - R - (t+1)V
where S is the sum assured paid on death during the year, R is the sum paid on survival to end of year, (t+1)V is reserve at the end of year and x + t is age at start of year

Question 23

Total EDS

Answer 23

sum(all policies)[q:x+t * DSAR]
where S is the sum assured paid on death during the year, R is the sum paid on survival to end of year, (t+1)V is reserve at the end of year and x + t is age at start of year

Question 24

Total ADS

Answer 24

sum(all deaths)[S - R - (t+1)V]

Question 25

Mortality profit

Answer 25

Total EDS - Total ADS

Question 26

Last survivor policies

Answer 26

Death event | DSAR Both lives die | S - (t+1)V^(both alive) x dies, y survives | (t+1)V^(y alive) - (t+1)V^(both alive) y dies, x survives | (t+1)V^(x alive) - (t+1)V^(both alive)

Question 27

Multiple state models

Answer 27

(tpx)^bar(ii) = probability of staying continuously in state i = exp{-int(0, t)[(mu:x+s)^ij ds]}

Question 28

Valuing lump sum benefit of state transitions

Answer 28

EPV = S * int(0, infinity)[v^t * (tpx)^ai * (mu:x+t)^ij dt]

Question 29

Valuing income benefit of state transitions

Answer 29

EPV = B * int(0, infinity)[v^t * (tpx)^aj dt]

Question 30

2 decrement model independent probabilities

Answer 30

(qx)^d = 1 - e^-mu (qx)^s = 1 - e^-sigma

Question 31

2 decrement model dependent probabilities

Answer 31

[(aq)x]^d = (px)^AD = [mu / (mu + sigma)] * [1 - e^-(mu + sigma)] [(aq)x]^s = (px)^AS = [sigma / (mu + sigma)] * [1 - e^-(mu + sigma)]

Question 32

2 decrement model probability of remaining in active state

Answer 32

(ap):x = 1 - [(aq):x]^d - [(aq):x]^s = e^-(mu + sigma)

Question 33

2 decrement model forces of transition

Answer 33

mu = {[(aq):x]^d / (aq):x} * [-ln(ap):x] sigma = {[(aq):x]^s / (aq):x} * [-ln(ap):x] where (aq):x = [(aq):x]^d + [(aq):x]^s

Question 34

Independence of decrements

Answer 34

(ap):x =m(p1):x * (p2):x * ... * (pm):x

Question 35

Decrements dependent probability integral

Answer 35

(aqj):x = int(0,1)[t(ap)x * muj:x+t dt]

Question 36

Multiple decrement definitions

Answer 36

(al):x = expected number of active lives at exact age x (adj):x = expected number of decrements due to cause j over the year of age x to x + 1, given a radix of (al):alpha lives active at age alpha.

Question 37

Multiple decrement probabilities

Answer 37

(aqj):x = (adj):x / (al):x n(ap):x = (al):(x+n) / (al):x n(aqj):x = {sum(r=0,n-1)[(adj):(x+r)]} / (al):x n|(aqj):x = (adj):(x+n) / (al):x

Question 38

Multiple decrement tables construction

Answer 38

(adj):x = (al):x * (aqj):x (al):(x+1) = (al):x - sum(j=1,m)[(adj):x]

Question 39

Accumulation with-profits (AWP)

Answer 39

F:t = (F:(t-1) + p) * (1 + g) * (1 + b:t) where g is the guaranteed annual interest and b:t is the bonus annual interest for year t

Question 40

Projecting cashflows - conventional products

Answer 40

(+) premiums (-) expenses (+) investment income (-) benefit payouts (death, maturity, surrender) (-) increase in reserves = profit vector where increase in reserves = {probability of staying in force over the year} * {reserve per policy at end of year} - {reserve at start of year} - {interest on start year reserve}

Question 41

Projecting cashflows - unit-linked

Answer 41

(+) Premium * allocation rate * (1 - bid-offer spread) (+) unit fund from end of previous year (+) expected unit fund growth (-) management charge = fund at end of year after charges

Question 42

Projecting cashflows - unitised with-profits

Answer 42

(+) premium less cost of allocation (-) expenses (+) interest on non-unit fund (-) non-unit benefit costs (+) non-unit surrender profit (+) management charge transferred from unit fund (+) other charges from unit fund, if applicable = non-unit cashflow vector (= profit vector in absence of any non-unit reserves)

Question 43

Profit margin

Answer 43

NPV / EPV premiums

Question 44

Risk discount rate

Answer 44

cost of capital + risk premium

Question 45

NPV

Answer 45

sum(all events)[{present value of profit|event occurs} * P(event occurs)]