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 = 11+i discount d = 1 − v. constant force of interest δ = ln(1 + i).

· varying force of interest δ(t) = dA/dtA(t) . separate and integrate A(t) = A(0)e

∫ t0 δ(s)ds.

· interest earned from a to b = A(b)−A(a). X deposited at a accumulated till b is A(b) = Xe∫ ba δ(s)ds

Level Annuities: 5-button formula PV = PMTan + Fvn

· PV immediate an = 1−vni PV due an = (1 + i)an continuously paid an = an

(iδ

)· FV sn = (1 + i)nan = (1+i)n−1

i sn = (1+i)n−1d perpetuity a∞ = 1

i a∞ = 1d

· a(m)n 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 11+j = 1+e

1+i .

· arithmetic: init P , increase Q: PV = Pan + Qi (an − nvn)

Q is negative for decrease. P can be zero. ctsly payable - multiply by(iδ

)· ctsly compounding, ctsly payable f(t): PV =

∫ n0 f(t)vtdt

· varying force of interest δ(t): PV =∫ n0 f(t)e−

∫ t0 δ(r)drdt FV =

∫ n0 f(t)e

∫ nt δ(r)drdt

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 = PMT . It = iOBt−1.

· prospective: OBt = PMTan−t, present value of remaining payments. Pt = PMTvn−t+1,

· retrospective: OBt = L(1+ i)t−PMTst, FVloan - FVpayments made. Pt = (1+ i)t−1(PMT −Li)

Bonds: F = par = face, C = redemption amount, r = coupon rate, i = yield rate.

· bond price PV = Fran + Cvn, book value is outstanding balance

· write down is principal repaid: Pt = (Fr − 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/diP , 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 PVt∑PVt

· asset-liability matching: Asset income equals Liability due at all t.

· Redington immunization: PVA = PVL at i0 and PVA > PVL for i near i0.

· duration of assets = duration of liabilities dPVAdi = dPVL

di , and

· convexity of assets > convexity of liabilities d2PVAdi2

> d2PVLdi2

.

· 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 = FP − e−rtK

· prepaid forward price FP : 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 − δ, 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 dailylosses can be taken. It does earn interest.

- maintainance margin: minimum proportion of the initial margin that must be maintainedthroughout 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)+

X2

(1 + i2)2+

X3

(1 + i3)3+ · · · =

X

(1 + i1)+

X

(1 + i2)2+

X

(1 + i3)3+ . . .

· interest rate swap: fixed rate R equivalent to varying rates, where fi is the forward rate for thatperiod.

i1(1 + i1)

+f2

(1 + i2)2+

f3(1 + i3)3

+ · · · =R

(1 + i1)+

R

(1 + i2)2+

R

(1 + i3)3+ . . .

· interest rate swap payment: difference between actual interest payment due and the interest dueaccording to the swap.