MATH419: Actuarial Science. Exam-FM Formulas
[Pages:3]MATH419: Actuarial Science.
Exam-FM Formulas
Interest: sum of geometric series Sn = a(1 - rn)/(1 - r) ? Compound: A(t) = A(0)(1 + i)t = A(0)(1 - d)-t Simple: A(t) = A(0)(1 + it)
?
v=
1 1+i
discount d = 1 - v.
constant force of interest = ln(1 + i).
?
varying
force
of
interest
(t)
=
dA/dt A(t)
.
separate
and
integrate
A(t)
=
A(0)e
t 0
(s)ds.
? interest earned from a to b = A(b)-A(a).
X
deposited
at
a
accumulated
till
b
is
A(b)
=
Xe
b a
(s)ds
Level Annuities: 5-button formula P V = P M T an + F vn
?
PV
immediate
an
=
1-vn i
PV due a?n = (1 + i)an
continuously paid an = an
i
?
FV
sn
= (1 + i)nan
=
(1+i)n-1 i
s?n
=
(1+i)n-1 d
perpetuity
a
=
1 i
a?
=
1 d
? a(nm) means m payments per year for n years.
i(12)
nominal
means
i(12) 12
interest
per
month
Varying Annuities: CF button, to enter PMTs and frequency.
?
geometric:
increase
e%
per
payment,
calculate
new
interest
rate
1 1+j
=
1+e 1+i
.
?
arithmetic:
init
P,
increase
Q:
PV
= P an
+
Q i
(an
-
nvn)
Q is negative for decrease.
P can be zero.
ctsly payable - multiply by
i
? ctsly compounding, ctsly payable f (t):
PV =
n 0
f
(t)vtdt
? varying force of interest (t):
PV =
n 0
f
(t)e-
t 0
(r)dr
dt
FV =
n 0
f (t)e
n t
(r)dr
dt
Loans: AMORT button after entering info into 5-buttons
? L is principle, OBt is outstanding balance just after payment at t,
? It is interest in tth payment, Pt is principle repaid tth payment. Pt + It = P M T . It = iOBt-1. ? prospective: OBt = P M T an-t, present value of remaining payments. Pt = P M T vn-t+1, ? retrospective: OBt = L(1 + i)t - P M T st, FVloan - FVpayments made. Pt = (1 + i)t-1(P M T - Li) Bonds: F = par = face, C = redemption amount, r = coupon rate, i = yield rate. ? bond price P V = F ran + Cvn, book value is outstanding balance ? write down is principal repaid: Pt = (F r - Ci)vn-t+1, amortization of bond.
? premuim=price-redemption.
discount=redemption-price.
NPV & IRR: CF, NPV, IRR (finds solution closest to zero only).
? IRR is rate at which PV of flows equals 0, interest rate = cost of capital
? dollar-weighted: simple interest rate that must have been in effect. solve for i.
? time-weighted: (b/a)(c/b)(d/c) = 1 + i where a grew to b, b grew to c etc. solve for i.
? investment year: interest rate depends on when deposited (row).
? portfolio method: interest rate depends on current year (column).
? new money rate: investment year rate for money deposited this year.
Varying Rates: (1 + st-1)t-1(1 + ft) = (1 + st)t. spot rate: st rate for term t starting at 0.
? forward rate: fa,b rate for term starting at a and ending at b. ft = ft-1,t.
?
modified
duration
DM
=
-dP P
/di
,
equals
t/(1 + i)
for
constant
i
and
term
t.
? duration (Macaulay) D = (1 + i)DM , equals t for constant i and term t. D =
t P Vt P Vt
? asset-liability matching: Asset income equals Liability due at all t.
? Redington immunization: P VA = P VL at i0 and P VA > P VL for i near i0.
?
duration
of
assets
=
duration
of
liabilities
dP VA di
=
dP VL di
,
and
?
convexity
of
assets
>
convexity
of
liabilities
d2P VA di2
>
d2P VL di2
.
? full immunization: Asset income greater than or equal to Liability due for any i.
Ch1: Derivatives: value determined by price of something else. long: buyer. short: seller.
? insurance is risk-sharing. Insurance firms use reinsurance to share risk of extreme events.
? diversifiable risk is unrelated to other risks and can be shared. non-diversifiable risk does not vanish when shared (it already affects everyone).
? bid: price can sell at, ask: price can buy for. You always pay more than you get so ask > bid.
Ch2: Forwards and Options: call: right to buy, put: right to sell, forward: obligation.
? European: exercise at end. American: exercise anytime. Bermudan: exercise specified times between.
? Option profit = payoff - FV(option price). Options are insurance, strike = value-deductable.
Ch3: Insurance and Collars: Put-Call Parity C - P = F P - e-rtK
? prepaid forward price F P : current price less the PV of dividends.
? forward price F : FV of prepaid forward price.
Ch4: Hedging ? reasons to hedge: risk-aversion, distress costs, costly external financing, increase debt capacity, tax. ? reasons NOT to hedge: transaction costs, bid/ask spread, needs more expertise, regulating, reporting.
Ch5: Forwards and futures ? cost-of-carry: r - , cost of holding long position. ? futures: - mark-to-market: settled daily so no money is owed. When asset looses value, buyer pays out. - margin: deposit from both buyer/seller left with broker when buying future, from which daily losses can be taken. It does earn interest. - maintainance margin: minimum proportion of the initial margin that must be maintained throughout the contract period - on S& P 500: only sold in bundles of 250
Ch8: Swaps settles throughout the term. like a set of forwards. ? prepaid commodity swap: single payment at time 0 equivalent to varying payments ? commodity swap: swap price is the level payment X equivalent to varying payments Xi
X1 (1 + i1)
+
(1
X2 + i2)2
+
(1
X3 + i3)3
+
???
=
(1
X + i1)
+
(1
X + i2)2
+
(1
X + i3)3
+
...
? interest rate swap: fixed rate R equivalent to varying rates, where fi is the forward rate for that period.
(1
i1 + i1)
+
(1
f2 + i2)2
+
(1
f3 + i3)3
+
???
=
(1
R + i1)
+
(1
R + i2)2
+
(1
R + i3)3
+
...
? interest rate swap payment: difference between actual interest payment due and the interest due according to the swap.
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