MATH419: Actuarial Science. Exam-FM Formulas - Texas A&M University

[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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