Actuarial Science Reference Sheet

Author: Daniel NolanEmail: daniel_nolan@msn.com

The purpose of this document is to provide entry-level actuarial students with a sneak preview of the mathematics used in the syllabi for SOA exams P and FM. It can also server as a refresher for students who have already covered the material in some depth.

1

1 Probability

1.1 Preliminaries

• Indicator Function IA(ω) = I(ω ∈ A), where I(p) =

1 if p

0 if ¬p• Delta “Function”

1. δ(x) =

∞ if x = 0

0 otherwise

2.∫∞−∞ δ(x)f(x)dx = f(0) for any function f , and in particular

∫∞−∞ δ(x)dx = 1

3. δ(x) = du/dx, where u(x) = I(x ≥ 0)

• Gamma and Beta Functions

– Γ(x) =∫∞

0tx−1e−tdt, in particular Γ(1) = 1 and Γ(1/2) =

√π

– Γ(x+ 1) = xΓ(x) and therefore Γ(n) = (n− 1)! for any positive integer n

– Γ′(x) =∫∞

0tx−1e−t log tdt, in particular Γ′(1) = −γ, where γ = limn→∞ γn and γn =

∑ni=1 1/i− log n

– Incomplete Gamma Function

∗ Ix(y) = 1Γ(x)

∫ y0tx−1e−tdt

∗ Ix+1(y) = Ix(y)− yxe−y

Γ(x+1)

– B(x, y) = Γ(x)Γ(y)/Γ(x+ y) = B(y, x)

– Incomplete Beta Function

∗ Ix(r, s) = 1B(r,s)

∫∞0tr−1(1− t)s−1dt, 0 ≤ x ≤ 1

∗ Ix(r, 1) = xr and Ix(1, s) = 1− (1− x)s

∗ Ix(r, s) = Γ(r+s)xr(1−x)s−1

Γ(r+1)Γ(s) + Ix(r + 1, s− 1)

• Monotonic Sequences of Sets

– A1 ⊂ A2 ⊂ · · · =⇒ An → A =⋃∞i=1Ai

– A1 ⊃ A2 ⊃ · · · =⇒ An → A =⋂∞i=1Ai

• DeMorgan’s Laws:(⋃

i∈I Ai)c

=⋂i∈I A

ci and

(⋂i∈I Ai

)c=⋃i∈I A

ci

1.2 Probability Spaces

• Probability Space (Ω,A,P)

– sample space Ω = set of all possible outcomes ω

– events A ∈ A ⊂ P(Ω), where A is a σ-algebra, i.e.

1. ∅ ∈ A2. A ∈ A =⇒ Ac ∈ A3. Ai ∈ A, i = 1, 2, . . . =⇒ ⋃∞

i=1Ai ∈ A– probability measure P : A → R such that

1. P(A) ≥ 0 for all A ∈ A2. P(Ω) = 1

3. A1, A2, . . . pairwise disjoint =⇒ P (⋃∞i=1Ai) =

∑∞i=1 P (Ai)

• Misc. Properties

1

– P (∅) = 0

– A ⊂ B =⇒ P(A) ≤ P(B)

– 0 ≤ P(A) ≤ 1

– P(Ac) = 1− P(A)

– A ∩B = ∅ =⇒ P(A ∪B) = P(A) + P(B)

– P(A ∪B) = P(A) + P(B)− P(A ∩B), and in general

P

(n⋃

k=1

Ak

)=

n∑

k=1

(−1)k−1∑

I⊂1,...,n|I|=k

P

(⋂

i∈IAi

)

• Continuity of Probabilities An → A =⇒ P(An)→ P(A)

• Independence of Events

– A and B are independent if P(A ∩B) = P(A)P(B), and in this case we write AqB

– a collection Ai : i ∈ I is independent if P(⋂

j∈J Aj

)=∏j∈J P(Aj) for every finite subset J of I

– if Ai : i ∈ I is independent, then P(⋃

i∈I Ai)

= 1−∏i∈I [1− P(Ai)] by DeMorgan

– disjoint events with positive probabilities are not independent

• Conditional Probability

– if P(B) > 0, then probability of A given that B has occured P(A|B) = P(A ∩B)/P(B)

– P(·|B) satisfies the axioms of probability for a fixed event B

– AqB iff P(A|B) = P(A)

– P(A ∩B) = P(A|B)P(B) = P(B|A)P(A)

• Law of Total Probability Let A1, . . . , An be a partition of Ω. Then for any event B, P(B) =∑ni=1 P(B|Ai)P(Ai).

• Bayes’ Theorem Let A1, . . . , An be a partition of Ω such that P(Ai) > 0 for each 1 ≤ i ≤ n. If P(B) > 0, then

posterior︷ ︸︸ ︷P(Ai|B) =

P(B|Ai)prior︷ ︸︸ ︷P(Ai)∑n

j=1 P(B|Aj)P(Aj)

1.3 Random Variables (RVs)

• Random Variable X : Ω→ R such that ω : X(ω) ≤ x ∈ A for all x

• Cumulative Distribution Function (CDF) FX : R→ [0, 1] defined by FX(x) = P(X ≤ x)

• Theorem Let X ∼ F and Y ∼ G. If F (x) = G(x) for all x, then P(X ∈ A) = P(Y ∈ A) for every measurable event A.

• Theorem F : R→ [0, 1] is the CDF for some probability measure iff F satisfies the following:

1. x < y =⇒ F (x) ≤ F (y) (increasing)

2. F (x)→ 0 as x→ −∞ and F (x)→ 1 as x→∞ (normalized)

3. F (x) = F (x+) for all x, where F (x+) denotes limy→xy>x

F (y) (right-continuous)

• X discrete if X : Ω→ x1, x2, . . ., and in this case we define the probability mass function fX(x) = P(X = x)

• X continuous if there exists a function fX such that fX(x) ≥ 0 for all x,∫∞−∞ fX(x)dx = 1, and for every a ≤ b,

P(a ≤ X ≤ b) =

∫ b

a

fX(x)dx

2

and in this case fX is called the probability density function (PDF). Note that

FX(x) =

∫ x

−∞fX(s)ds

and fX(x) = F ′X(x) wherever F ′X exists.

• Lemma Let X ∼ F . Then

1. P(X = x) = F (x)− F (x−), where F (x−) denotes limy→xy<x

F (y)

2. P(x ≤ X ≤ y) = F (y)− F (x)

3. S(x) = P(X > x) = 1− F (x)

4. X continuous =⇒ F (b)− F (a) = P(a < X < b) = P(a < X ≤ b) = P(a ≤ X < b) = P(a ≤ X ≤ b)• X mixed if X neither discrete nor continuous

– let FX have jump discontinuities at a1, a2, . . .

– define k =∑∞i=1 ki, where ki = FX(ai)− FX(a−i )

– F (x) = [FX(x)−∑∞i=1 kiu(x− ai)] /(1− k) is a continuous CDF, hence F = FC for some continuous RV C

– FX(x) = (1−k)FC(x)+∑∞i=1 kiu(x−ai) = (1−k)FC(x)+kFD(x), where D is a discrete RV with P(D = ai) = ki/k

– FX is a weighted average of continuous and discrete CDFs

– fX(x) = (1− k)fC(x) +∑∞i=1 kiδ(x− ai)

• Quantile Function F−1 : [0, 1]→ R defined by F−1(q) = infx : F (x) > q– F−1(1/4) = 1st quartile

– F−1(1/2) = median

– F−1(3/4) = 3rd quartile

• Survival and Hazard Functions

– SX(x) = P(X > x) = 1− FX(x) has the following properties:

1. x < y =⇒ SX(x) ≥ SX(y) (decreasing)

2. SX(x)→ 1 as x→ −∞ and SX(x)→ 0 as x→∞ (normalized)

3. SX(x) = SX(x+) (right-continuous)

– λX(x) = limh→0 P(x < X < x+ h|X > x)/h = −S′X(x)/SX(x) =⇒ SX(x) = exp[−∫ x−∞ λX(s)ds

]

• Bivariate Distributions

– F (x, y) = P(X ≤ x, Y ≤ y)

– f(x, y) is such that

1. f(x, y) ≥ 0 for all (x, y) ∈ R2

2.∫∫

R2 f(x, y)dxdy = 1

3. P((X,Y ) ∈ A) =∫∫Af(x, y)dxdy

• Marginal Distributions

– fX(x) =∫∞−∞ f(x, y)dy

– fY (y) =∫∞−∞ f(x, y)dx

• Independent RVs

– X q Y if, for every pair of events A and B, P(X ∈ A, Y ∈ B) = P(X ∈ A)P(Y ∈ B)

– Theorem Let X and Y have joint PDF fX,Y . Then X q Y iff fX,Y (x, y) = fX(x)fY (y) for all (x, y) ∈ R2.

– Theorem Suppose (X(x), Y (y)) : (x, y) ∈ R2 is a rectangle. Then f(x, y) = g(x)h(y) =⇒ X q Y .

3

• Conditional Distributions

– fX|Y (x|y) = fX,Y (x, y)/fY (y), assuming fY (y) > 0

– P(X ∈ A|Y = y) =∫AfX|Y (x|y)dx

• Multivariate Distributions and IID Samples

– random vector X = (X1, . . . , Xn) with PDF f(x1, . . . , xn), where X1, . . . , Xn are RVs

– X1, . . . , Xn independent ⇐⇒ P(X1 ∈ A1, . . . , Xn ∈ An) =∏ni=1 P(Xi ∈ Ai) ⇐⇒ f(x1, . . . , xn) =

∏ni=1 fXi

(xi)

– X1, . . . , XnIID∼ F signifies that X1, . . . , Xn are independent and identically distributed, each with CDF F

• Transformations of RVs

– univariate: Y = r(X), e.g. Y = X2 or Y = eX

1. Ay = x : r(x) ≤ y2. FY (y) = P(Y ≤ y) = P(X ∈ Ay) =

∫AyfX(x)dx

3. fY (y) = F ′Y (y)

– bivariate: Y = r(X1, X2), e.g. Y = X1 +X2 or Y = minX1, X21. Ay = (x1, x2) : r(x1, x2) ≤ y2. FY (y) = P(Y ≤ y) = P((X1, X2) ∈ Ay) =

∫∫AyfX1,X2

(x1, x2)dx1dx2

3. fY (y) = F ′Y (y)

1.4 Expectation

• Expected Value, or Mean µ = EX =∫xf(x)dx =

∫S(x)dx, assuming

∫|x|f(x)dx <∞

• Y = r(X) =⇒ EY =∫r(x)f(x)dx

• E|X|k <∞ and j < k =⇒ E|X|j <∞• E (

∑ni=1 aiXi) =

∑ni=1 aiEXi

• X1, . . . , Xn independent =⇒ E (∏ni=1Xi) =

∏ni=1 EXi

• Geometric Expectation

– for any positive RV X, Eg[X] = expE[logX]

– X discrete =⇒ Eg[X] =∏ni=1 x

pii , where pi = P(X = xi)

– (Arithmetic-Geometric Means Inequality) Eg[X] ≤ EX

– logEg[X] = E(logX) and expEX = Eg[expX]

• kth Moment = EXk and kth central moment = E(X − µ)k

• Variance σ2 = VX = E(X − µ)2

– σ2 = E(X2)− µ2

– V(aX + b) = a2VX

– X1, . . . , Xn independent =⇒ V (∑ni=1 aiXi) =

∑ni=1 a

2VXi

• Standard Deviation sd(X) =√VX

• Sample Mean Xn = 1n

∑ni=1Xi

• Sample Variance s2n = 1

n−1

∑ni=1(Xi −Xn)2

• Theorem Let X1, . . . , Xn be IID. Then EXn = µ, VXn = σ2/n, and Es2n = σ2.

• Covariance Cov(X,Y ) = E[(X − µX)(Y − µY )] = E(XY )− E(X)E(Y )

• Correlation ρ(X,Y ) = Cov(X,Y )/σXσY

4

• Theorem The correlation satisfies −1 ≤ ρ ≤ 1. If Y = aX + b, then ρ = sgn(a). If X q Y , then Cov(X,Y ) = 0 andtherefore ρ = 0 as well.

• V (∑i aiXi) =

∑i a

2iVXi +

∑i

∑j<i aiajCov(Xi, Xj)

• Skewness

– γ = EZ3, where Z = (X − µ)/σ

– γ = 0 if X is symmetric, i.e. if f(µ+ x) = f(µ− x) for all x

– Y = aX + b =⇒ γY = sgn(a)γX

– Y = X1 +X2 =⇒ γY = (γ1σ31 + γ2σ

32)/σ3

Y if X1 qX2

• Multivariate Expectation

– random vector X = (X1, . . . , Xn)′

– µX = (µ1, . . . , µn)′, where µi = EXi

– variance-covariance matrix Σ defined by Σij = Cov(Xi, Xj), in particular Σii = VXi

– Lemma If a is a vector and X is a random vector with mean µ and variance Σ, then E(a′X) = a′µ andV(a′X) = a′Σa. If A is a matrix, then E(AX) = Aµ and V(AX) = AΣA′.

• Conditional Expectation

– E(X|y) =∫xfX|Y (x|y)dx and E[r(X,Y )|y] =

∫r(X, y)fX|Y (x, y)dx

– whereas EX is a number, E(X|y) is a function of y

– Rule of Iterated Expectations For RVs X and Y , assuming the expectations exist, we have that E[E(X|Y )] =EX. More generally, for any function r(X,Y ),

EE[(r(X,Y )|X] = E[r(X,Y )]

– V(X|y) =∫

[x− µ(y)]2fX|Y (x, y)dx, where µ(y) = E(X|y)

– Theorem For any RVs X and Y , we have

VX = EV(X|Y ) + VE(X|Y )

• Moment Generating Function

– MX(t) = E[etX]

= E[1 +X + (Xt)2

2! + · · ·]

– M(n)X (0) = EXn, n = 0, 1, 2, . . .

– Y = aX + b =⇒ MY (t) = ebtMX(at)

– Y = X1 +X2 =⇒ MY (t) = M1(t)M2(t) if X1 qX2

• Cumulant Generating Function

– ψX(t) = logMX(t)

– ψ(n)X (0) =

0 n = 0

µ n = 1

σ2 n = 2

σ3γ n = 3

– Y = aX + b =⇒ ψY (t) = btψX(at)

– Y = X1 +X2 =⇒ ψY (t) = ψ1(t) + ψ2(t) if X1 qX2

5

1.5 Important Distributions

1.5.1 Discrete Distributions

• Point Mass Distribution X ∼ δa– PMF: f(x) = I(x = a)

– CDF: F (x) = I(x ≥ a)

– EX = a and VX = 0

• Uniform Distribution X ∼ Uniformx1, . . . , xn

– PMF: f(x) =

1/n x = x1, . . . , xn

0 otherwise

– CDF: F (x) = 1n

∑ni=1 I(x ≥ xi)

– EX = 1n

∑ni=1 xi and VX = 1

n

∑ni=1 x

2i −

(1n

∑ni=1 xi

)2

• Bernoulli Distribution X ∼ Bernoulli(p)

– X represents outcome of single trial, where P(success) = p

– PMF: f(x) =

px(1− p)1−x if x = 0 or x = 1

0 otherwise

– CDF: F (x) = (1− p)I(x ≥ 0) + pI(x ≥ 1)

– MGF: MX(t) = pet + (1− p)– EX = p and VX = p(1− p)

• Binomial Distribution X ∼ Binomial(n, p)

– X represents number of successes in n independent Bernoulli trials, each with P(success) = p

– PMF: f(x) =(nx

)px(1− p)n−x, x = 0, 1, . . . , n

– MGF: MX(t) = [pet + (1− p)]n

– EX = np and VX = np(1− p)– X ∼ Binomial(m, p) and Y ∼ Binomial(n, p) and X q Y =⇒ X + Y ∼ Binomial(m+ n, p)

• Poisson Distribution X ∼ Poisson(λ)

– X represents the number of occurences of a rare event during some fixed time period in which the expected numberof occurences is λ and individual occurences are independent of each other

– Poisson RVs are used in the insurance industry to represent the number of claims in a large group of policies forwhich the expected number of claims is known and claims occur independently and infrequently

– PMF: f(x) = e−λλx/x!, x = 0, 1, 2, . . .

– MGF: MX(t) = exp[λ(et − 1)]

– EX = VX = λ and γ = 1/√λ

– X ∼ Poisson(λ) and Y ∼ Poisson(µ) and X q Y =⇒ X + Y ∼ Poisson(λ+ µ)

• Negative Binomial Distribution X ∼ NB(r, p)

– X represents the number of failures that occur in a sequence of independent Bernoulli trials before the rth success,where P(success) = p in each of the trials

– PMF: f(x) = Γ(r+x)Γ(r)Γ(x+1)p

r(1− p)x, x = 0, 1, 2, . . .

– MGF: MX(t) =(

p1−(1−p)et

)r, (1− p)et < 1

– EX = r(1− p)/p and VX = r(1− p)/p2

6

– X ∼ NB(r, p) and Y ∼ NB(s, p) and X q Y =⇒ X + Y ∼ NB(r + s, p)

• Geometric Distribution X ∼ Geometric(p)

– X represents the number of failurs that occur in a sequence of independent Bernoulli trials before the first success,where P(success) = p in each of the trials

– PMF: f(x) = p(1− p)x, x = 0, 1, 2, . . .

– MGF: MX(t) = p/[1− (1− p)et], (1− p)et < 1

– EX = (1− p)/p and VX = (1− p)/p2

– P(X > s+ t|X > t) = P(X > s) for all positive integers s and t

– X1, . . . , XrIID∼ Geometric(p) =⇒ ∑r

i=1Xi ∼ NB(r, p)

1.5.2 Continuous Distributions

• Exponential Distribution X ∼ Exponential(λ)

– X represents one of the following:

∗ time until first arrival when arrivals are such that the number of arrivals in [0, t] is Poisson(λt)

∗ lifetime of an item that does not age

– PDF: f(x) =

λe−λx x ≥ 0

0 x < 0

– MGF: MX(t) = λ/(λ− t), t < λ

– EX = 1/λ, VX = 1/λ2, and γ = 2

– X ∼ Exponential(λ) =⇒ aX ∼ Exponential(λ/a), where a > 0

– X,YIID∼ Exponential(λ) =⇒ X + Y ∼ Gamma(2, λ)

– X ∼ Exponential(λ) and Y ∼ Exponential(µ) and X q Y =⇒ min(X,Y ) ∼ Exponential(λ+ µ)

• Gamma Distribution X ∼ Gamma(r, λ)

– X has the following interpretations when r is a positive integer:

∗ time until rth arrival when arrivals are such that the number of arrivals in [0, t] is Poisson(λt)

∗ ∑ri=1Xi, where X1, . . . , Xr

IID∼ Exponential(λ)

– PDF: f(x) = λr

Γ(r)xr−1e−λx, x > 0

– MGF: MX(t) = [λ/(λ− t)]r, t < λ

– EX = r/λ, VX = r/λ2, and γ = 2/√r

– X ∼ Gamma(r, λ) =⇒ aX ∼ Gamma(r, λ/a)

– X ∼ Gamma(r, λ) and Y ∼ Gamma(s, λ) and X q Y =⇒ X + Y ∼ Gamma(r + s, λ)

• Beta Distribution X ∼ Beta(α, β)

– typically used as prior distributions in Bayesian statistics

– PDF: f(x) = Γ(α+β)Γ(α)Γ(β)x

α−1(1− x)β−1, 0 < x < 1

– MGF: MX(t) = 1 +∑∞n=1

(∏n−1k=0

α+kα+β+k

)tn

n!

– EX = α/(α+ β), VX = αβ/(α+ β)2(α+ β + 1)

• Pareto Distribution X ∼ Pareto(α, β)

– similar to exponential, except with heaver tail

– PDF: f(x) = αβ (1 + α/β)−(α+1)

7

– EXk = βkk!(α−1)···(α−k) , k < α, in particular EX = βα/(β − 1), β > 1 and VX = αβ/(β − 1)2(β − 2), β > 2

– X ∼ Pareto(α, β) =⇒ bX ∼ Pareto(α, bβ)

• Weibull Distribution X ∼Weibull(α, β)

– X has the following interpretations:

∗ lifetime of an item whose instantaneous risk of failure is given by a power function, i.e. λX(t) = ktβ−1

∗ positive power of an exponential RV, in particular αY 1/β , where Y ∼ Exponential(1)

– PDF: f(x) = βα (x/α)β−1 exp[−(x/α)β ], x ≥ 0

– EXk = αkΓ(1 + k/β)

– X ∼Weibull(α, β) =⇒ aX ∼Weibull(aα, β) and Xr ∼Weibull(αr, β/r)

• DeMoivre Distribution X ∼ DeMoivre(ω)

– X represents continuous quantities whose values we consider to be “equally likely” in the sense that all intervalsof the same length have equal probability, or lifetimes for which failures are uniformly distributed

– PDF: f(x) =

1/ω 0 < x < ω

0 otherwise

– MGF: MX(t) = (etω − 1)/tω

– EX = ω/2 and VX = ω2/12

– X ∼ DeMoivre(ω) =⇒ aX ∼ DeMoivre(aω)

• Normal Distribution X ∼ Normal(µ, σ2)

– X has the following interpretations:

∗ continuous analog of binomial RV with p = 1/2

∗ measurements of a continuous quantity in a scientific experiment

∗ limiting distribution for sum of any collections of IID RVs

– PDF: ϕ(x) = (2πσ2)−1/2 exp[(x− µ)2/2σ2]

– CDF: Φ(x) =∫ x−∞ ϕ(t)dt

– MGF: MX(t) = exp(µt+ σ2t2/2)

– EX = µ and VX = σ2

– X ∼ Normal(µ, σ2) =⇒ aX + b ∼ Normal(aµ+ b, a2σ2), in particular (X − µ)/σ ∼ Normal(0, 1)

– X ∼ Normal(µ1, σ21) and Y ∼ Normal(µ2, σ

22) and X q Y =⇒ X + Y ∼ Normal(µ1 + µ2, σ

21 + σ2

2)

• Student’s t-distribution X ∼ tp– similar to Normal distribution, except with heavier tails (Normal corresponds to tp with p =∞)

– PDF: f(x) =Γ( p+1

2 )Γ(p/2) (1 + x2/p)−(p+1)/2

– EX = 0, p > 1 and VX = p/(p− 2), p > 2

• Log-normal Distribution X ∼ Log-normal(µ, σ2)

– X represents the limiting distribution of the product of any collection of positive IID RVs

– PDF: f(x) = 1xσ√

2πexp[−(log x− µ)/2σ2], x ≥ 0

– EXk = exp(µk + σ2k2/2), in particular EX = exp(µ+ σ2/2) and VX = (expσ2 − 1) exp(2µ+ σ2)

– X ∼ Log-normal(µ, σ) =⇒ aXb ∼ Log-normal(log a+ bµ, b2σ2)

– X ∼ Log-normal(µ1, σ21) and Y ∼ Log-normal(µ2, σ

22) and X q Y =⇒ XY ∼ Log-normal(µ1 + µ2, σ

21 + σ2

2)

• Chi-square X ∼ χ2p

8

– if Z1, . . . , ZpIID∼ Normal(0, 1), then

∑pi=1 Z

2i ∼ χ2

p

– PDF: f(x) = 1Γ(p/2)2p/2x

p/2−1e−x/2, x > 0

– MGF: MX(t) = (1− 2t)−p/2, t < 1/2

– EX = p and VX = 2p

– X ∼ χ2p and Y ∼ χ2

q and X q Y =⇒ X + Y ∼ χ2p+q

1.5.3 Multivariate Distributions

• Multinomial X ∼ Multinomial(n, p), where X = (X1, . . . , Xk) and p = (p1, . . . , pk)

– X summarizes the results of a sequence of n identical random experiments with k outcomes each; specifically, ifY1, . . . , Yn are IID RVs with P(Yi = j) = pj , then Xj =

∑ni=1 I(Yi = j) ∼ Binomial(n, pj)

–∑ki=1Xi = n and

∑ki=1 pi = 1 and Cov(Xi, Xj) = −npipj

– PMF: f(x) =(

nx1...xk

)∏ki=1 p

xii , where

(n

x1...xk

)= n!

x1!···xk!

– MGF: MX(t1, . . . , tk) =(∑k

i=1 pieti)n

• Multivariate Normal X ∼ Normal(µ,Σ)

– PDF: f(x) = 1√(2π)k|Σ|

exp[− 1

2 (x− µ)′Σ−1(x− µ)]

– Theorem If Z ∼ Normal(0, I) and X = µ + Σ1/2Z, then X ∼ Normal(µ,Σ). Conversely, if X ∼ Normal(µ,Σ),then Σ−1/2(X − µ) ∼ Normal(0, I).

– Theorem Let X ∼ Normal(µ,Σ). Suppose we partition X = (Xa, Xb). We can partition µ = (µa, µb) as well as

Σ =

(Σaa Σab

Σba Σbb

)

We have the following:

1. Xa ∼ Normal(µa,Σaa)

2. Xb|Xa = xa ∼ Normal(µb + ΣbaΣ−1aa (xa − µa),Σbb − ΣbaΣ−1

aa Σab)

3. a′X ∼ Normal(a′µ, a′Σa)

4. (X − µ)′Σ−1(X − µ) ∼ χ2k

1.5.4 Overview

Table 1: Overview of Important Distributions

Distribution f(x) EX VX MX(t)

Bernoulli(p) px(1− p)1−x x = 0, 1 p p(1− p) pet + (1− p)Binomial(n, p)

(nx

)px(1− p)n−x x = 0, 1, . . . , n np np(1− p) [pet + (1− p)]n

Poisson(λ) e−λλx/x! x = 0, 1, . . . λ λ exp[λ(et − 1)]

Geometric(p) p(1− p)x x = 0, 1, . . . (1− p)/p (1− p)/p2 p/[1− (1− p)et] t < log 11−p

Exponential(λ) λe−λx x ≥ 0 1/λ 1/λ2 λ/(λ− t) t < λ

Gamma(r, λ) λr

Γ(r)xr−1e−λx x > 0 r/λ r/λ2 [λ/(λ− t)]r t < λ

Normal(µ, σ2) (2πσ2)−1/2 exp[

(x−µ)2

2σ2

]µ σ2 exp

(µt+ σ2t2

2

)

tpΓ( p+1

2 )Γ(p/2) (1 + x2/p)−(p+1)/2 0 p/(p− 2)

χ2p

1Γ(p/2)2p/2x

p/2−1e−x/2 x > 0 p 2p (1− 2t)−p/2 t < 1/2

9

0 10 20 30 40

0.00

0.05

0.10

0.15

0.20

0.25

Binomial

x

PM

F

n = 40, p = 0.3n = 30, p = 0.6n = 25, p = 0.9

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

Geometric

x

PM

F

p = 0.2p = 0.5p = 0.8

0 5 10 15 20

0.0

0.1

0.2

0.3

Poisson

x

PM

F

λ = 1λ = 4λ = 10

0 1 2 3 4 5

0.0

0.5

1.0

1.5

2.0

Exponential

x

PD

F

β = 2β = 1β = 0.4

0 5 10 15 20

0.0

0.1

0.2

0.3

0.4

0.5

Gamma

x

PD

F

α = 1, β = 2α = 2, β = 2α = 3, β = 2α = 5, β = 1α = 9, β = 0.5

−4 −2 0 2 4

0.0

0.2

0.4

0.6

0.8

Normal

x

φ(x)

µ = 0, σ2 = 0.2µ = 0, σ2 = 1µ = 0, σ2 = 5µ = −2, σ2 = 0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

Log−normal

x

PD

F

µ = 0, σ2 = 3µ = 2, σ2 = 2µ = 0, σ2 = 1µ = 0.5, σ2 = 1µ = 0.25, σ2 = 1µ = 0.125, σ2 = 1

0 2 4 6 8

0.0

0.1

0.2

0.3

0.4

0.5

χ2

x

PD

F

k = 1k = 2k = 3k = 4k = 5

1.6 Inequalities

• Markov’s Inequality Let X be a non-negative RV such that EX exists. Then for any t > 0, P(X > t) ≤ EX/t.

• Chebyshev’s Inequality P(|X − µ| ≥ t) ≤ σ2/t2, in particular P(|Z| ≥ t) ≤ 1/t2

• Hoeffding’s Inequality

– Let X1, . . . , XnIID∼ Bernoulli(p). Then for any ε > 0, P(|Xn − p| > ε) ≤ 2 exp(−2nε2).

– εn =√

log(2/α)/2n =⇒ P(|Xn − p| > εn) ≤ α

• Mill’s Inequality P(|Z| > t) ≤√

2/π exp(−t2/2)/t

• Cauchy-Schwartz Inequality If X and Y both have finite variance, then E|XY | ≤√EX2EY 2.

• Jensen’s Inequality Let Y = g(X). If g is convex, then Eg(X) ≥ g(EX). If g is concave, then Eg(X) ≤ g(EX).

10

2 Financial Mathematics

• Simple interest is more beneficial to the lender for fractions of a conversion period, i.e. for 0 < t < 1.Proof. Let i and t both belong to (0, 1). Then

(1 + i)t =

∞∑

k=0

ak = 1 + it+

∞∑

k=1

bk

where

ak =t(t− 1) · · · (t− k + 1)

k!ik

andbk = a2k + a2k+1

Note that ti2k/(2k)! is positive, while(t− 1) · · · (t− 2k + 1)︸ ︷︷ ︸

odd no. of neg. terms

is negative, so a2k < 0. Note also that ti2k+1/(2k + 1)! is positive, and so is

(t− 1) · · · (t− 2k)︸ ︷︷ ︸even no. of neg. terms

hence a2k+1 > 0, and therefore

a2k+1 = |a2k| ·∣∣∣∣t− 2k

2k + 1i

∣∣∣∣

= |a2k| ·∣∣∣∣2k − t2k + 1

∣∣∣∣ · |i|

< |a2k|

Finally,bk = −|a2k|+ a2k+1 < 0

so we have

(1 + i)t = 1 + it+

∞∑

k=1

bk < 1 + it

which completes the proof.

• Force of Interest δt = ddt logA(t) =⇒ A(t) = A(0) exp

∫ t0δsds

• Determination of Time (See Table 2)

– Exact Interest “actual/actual”

– Ordinary Interest “30/360” (∆t = 360∆y + 30∆m+ ∆d)

– Banker’s Rule “actual/360” (always more beneficial to lender)

• Rule of 72: the amount of time it takes an investment to double at a given rate of interest is approximately 0.72/i.(Most accurate for rates between 0.04 and 0.1.)

• an i =∑nt=1 ν

t = 1−(1+i)−n

i

• sn i = an i(1 + i)n = (1+i)n−1i

• an =∑n−1t=0 ν

t = (1 + i)an

• sn = (1 + i)sn

• 1an

= 1sn

+ i (relevant to sinking funds)

12

Table 2: Table Numbering Days of the Year

Day of

Month Jan Feb* Mar Apr May Jun Jul Aug Sep Oct Nov Dec

1 1 32 60 91 121 152 182 213 244 274 305 335

2 2 33 61 92 122 153 183 214 245 275 306 336

3 3 34 62 93 123 154 184 215 246 276 307 337

4 4 35 63 94 124 155 185 216 247 277 308 338

5 5 36 64 95 125 156 186 217 248 278 309 339

6 6 37 65 96 126 157 187 218 249 279 310 340

7 7 38 66 97 127 158 188 219 250 280 311 341

8 8 39 67 98 128 159 189 220 251 281 312 342

9 9 40 68 99 129 160 190 221 252 282 313 343

10 10 41 69 100 130 161 191 222 253 283 314 344

11 11 42 70 101 131 162 192 223 254 284 315 345

12 12 43 71 102 132 163 193 224 255 285 316 346

13 13 44 72 103 133 164 194 225 256 286 317 347

14 14 45 73 104 134 165 195 226 257 287 318 348

15 15 46 74 105 135 166 196 227 258 288 319 349

16 16 47 75 106 136 167 197 228 259 289 320 350

17 17 48 76 107 137 168 198 229 260 290 321 351

18 18 49 77 108 138 169 199 230 261 291 322 352

19 19 50 78 109 139 170 200 231 262 292 323 353

20 20 51 79 110 140 171 201 232 263 293 324 354

21 21 52 80 111 141 172 202 233 264 294 325 355

22 22 53 81 112 142 173 203 234 265 295 326 356

23 23 54 82 113 143 174 204 235 266 296 327 357

24 24 55 83 114 144 175 205 236 267 297 328 358

25 25 56 84 115 145 176 206 237 268 298 329 359

26 26 57 85 116 146 177 207 238 269 299 330 360

27 27 58 86 117 147 178 208 239 270 300 331 361

28 28 59 87 118 148 179 209 240 271 301 332 362

29 29 88 119 149 180 210 241 272 302 333 363

30 30 89 120 150 181 211 242 273 303 334 364

31 31 90 151 212 243 304 365

*For leap years, add 1 to the number of each day after Feb 28.

13

• Let an i = g. Then

i ≈ 2(n− g)

g(n+ 1)

If instead sn i = g, then

i ≈ 2(g − n)

g(n− 1)

• Increasing Annuities

1. arithmetically increasing annuities

0 1 2 3 n− 1 n

P P +Q P + 2Q P + (n− 2)Q P + (n− 1)Q

Psn| +Qsn|−n

i Pan| +Qan|−nνn

i

(a) (Ia)n =an −nν

n

i (Is)n =sn −ni

(b) (Da)n =n−ani (Ds)n =

n(1+i)n−sni

2. geometrically increasing annuities

0 1 2 3 n− 1 n

P P (1 + r) P (1 + r)2 P (1 + r)n−2 P (1 + r)n−1

P[1−( 1+r

1+i )n

i−r

]P

[(1+i)n−(1+r)n

i−r

]

• See Table 3

Table 3: Amortization Schedule for a Loan of an Repaid Over n Periods

Period Pmt amount Interest paid Principal repaid Balance

0 an

1 1 1− νn νn an−1

2 1 1− νn−1 νn−1 an−2

· · · · ·· · · · ·· · · · ·t 1 1− νn−t+1 νn−t+1 an−t

· · · · ·· · · · ·· · · · ·

n− 1 1 1− ν2 v2 a1

n 1 1− ν ν 0

Total n n− an an

• See Table 4

14

Table 4: Bond Terminology

P price of bond

F par value

C redemption value

r coupon rate

Fr amount of coupon

g modified coupon rate, i.e. rate such that Fr = Cg

i yield rate, often called yield to maturity (YTM)

n no. of coupon payment periods remaining

K present value, computed at yield rate, of redemption value

G base amount of bond, i.e. amount such that Gi = Fr

• It is customary to use semiannual compounding so that an 8% bond has r = 0.08/2 = 0.04.

• P = Fran + Cνn︸ ︷︷ ︸basic

= C + (Fr − Ci)an︸ ︷︷ ︸premium/discount

= G+ (C −G)νn︸ ︷︷ ︸base amount

= K +g

i(C −K)

︸ ︷︷ ︸Makeham

• Premium = P − C = C(g − i)an• See Table 5

Table 5: Bond Amortization Schedule for a $1 n-period Bond with Coupons at g Bought to Yield i.

Period Coupon Interest Earned Principal Adjustment Book Value

0 1 + (g − i)an1 g i[1 + (g − i)an ] (g − i)νn 1 + (g − i)an−1

2 g i[1 + (g − i)an−1 ] (g − i)νn−1 1 + (g − i)an−2

· · · · ·· · · · ·· · · · ·t g i[1 + (g − i)an−t+1 ] (g − i)νn−t+1 1 + (g − i)an−1

· · · · ·· · · · ·· · · · ·

n− 1 g i[1 + (g − i)a2 ] (g − i)ν2 1 + (g − i)a1

n g i[1 + (g − i)a1 ] (g − i)ν 1

Total ng ng − p p = (g − i)an

• See Table 6

Table 6: Valuation Between Coupon Payment Dates. Bft+k = Bmt+k + Frk

Method Flat Price Bft+k Accrued Coupon Frk Market Price Bmt+k

Theoretical Bt(1 + i)k Fr[

(1+i)k−1i

]Bt(1 + i)k − Fr

[(1+i)k−1

i

]

Practical Bt(1 + ik) kFr Bt(1 + ik)− kFrSemi-theoretical Bt(1 + i)k kFr Bt(1 + i)k − kFr

15

• Approximate YTM: i ≈ g− kn

1+ n+12n k≈ g − k/n

1 + k/2︸ ︷︷ ︸Bond

Salesman’sFormula

, where k = P−CC

• Callable Bonds

1. If i < g, i.e. if the bond sells at a premium, then assume the redemption date will be the earliest possible date.

2. If i > g, i.e. if the bond sells at a discount, then assume the redemption date will be the latest possible date.

• If 1. i > −1 exists such that NPV = 0, and 2. for such i, Bt > 0 for t = 0, 1, . . . , n − 1, then i is unique. (Bt denotesthe outstanding investment balance at time t.)

• See Figure 1

0 1 2 n− 1 n

i i i i

1

1 + isn|j

(a) An investment of 1 for n periods at rate i. The interestis reinvested at rate j.

0 1 2 3 n− 1 n

i 2i (n− 2)i (n− 1)i

1 1 1 1 1

n+ (Is)n−1|j

(b) An investment of 1 at the end of each period for n periods, atrate i. The interest is reinvested at rate j.

Figure 1: Examples Involving Reinvestment Rates

• See Table 7

Table 7: Interest Measurement Terminology

A beginning balance

B ending balance

I amount of interest earned during the period

Ct net amount of principal contributed at time t (0 ≤ t ≤ 1)

C total amount of principal contributed during the period, i.e. C =∑t Ct

aib amount of interest earned by 1 invested at time b over the following period oflength a, where a+ b ≤ 1

• B = A+ C + I

• iDW ≈ IA+

∑t Ct(1−t) ≈

2IA+B−I , assuming that on average net principal contributions occur at time t = 1/2

• iTW =∏mk=1(1 + jk)− 1, where 1 + jk =

B′kB′k−1+C′k−1

• See Table 8

• 1 + i′ = 1+i1+r =⇒ i′ = i−r

1+r ≈ i− r, where r denotes inflation, and i′ is called the real rate of interest

• PV of ordinary annuity for which payments are indexed to reflect inflation: R(1 + r)1−( 1+r

1+i )n

i−r = Ran i′

• Normal Yield Curve (Increasing)

– Expectations Theory

– Liquidity Preference Theory

– Inflation Premium Theory

• Inverted Yield Curve (Decreasing): Fed may set high short-term rates in order to fight inflation or to remove excessliquidity from the economy. Long-term rates may be lower due to expectations of inflation or the possibility of arecession.

• (1 + sn)n = (1 + sn−1)n−1(1 + fn−1) =∏n−1k=0(1 + fk)

16

Table 8: Illustration of Investment Year Method.

z 8.00 8.10 8.10 8.25 8.30 8.10 z +5

z +1 8.25 8.25 8.40 8.50 8.50 8.35 z +6

z +2 8.50 8.70 8.75 8.90 9.00 8.60 z +7

z +3 9.00 9.00 9.10 9.10 9.20 8.85 z +8

Calendar yearof portfoliorate y +5

Calendar yearof original

investment y

Investment year rates Portfolioratesi y +5i y (1) i y (2) i y (3) i y (4) i y (5)

z +4 9.00 9.10 9.20 9.30 9.40 9.10 z +9

z +5 9.25 9.35 9.50 9.55 9.60 9.35 z +10

z +6 9.50 9.50 9.60 9.70 9.70

z +7 10.00 10.00 9.90 9.80

z +8 10.00 9.80 9.70

z +9 9.50 9.50

z +10 9 00z +10 9.00

• Method of Equated Time t =∑nt=1 tRt/

∑nt=1Rt

• Macaulay Duration d =∑nt=1 tν

tRt/∑nt=1 ν

tRt

– i = 0 =⇒ d = t

– ∂d/∂i < 0

– If there is only one future cash flow, then d is the time at which it occurs.

– See Figure 2

– Duration of Level Annuity: R(Ia)n

– Duration of a Coupon Bond: Fr(Ia)n + nCνn

Figure 2: Duration exhibits discontinuities on payment dates.

• Volatility (Modified Duration) v = −P ′(i)/P (i) = d/(1 + i), where P (i) =∑nt=1(1 + i)−tRt.

• continuous compounding =⇒ v = d

• Convexity c = P ′′(i)/P (i)

• P (i+ h) ≈ P (i)(

1− hv + h2

2 c)

• dvdi = v2 − c

• Interest Sensitive Cash Flows

– Assume the following quantities are known:

17

P (i) = current price at yield rate i

P (i+ h) = price if yield rate increases by h

P (i− h) = price if yield rate decreases by h

– Effective Duration de ≈ P (i−h)−P (i+h)2hP (i)

– Effective Convexity ce ≈ P (i−h)−2P (i)+P (i+h)h2P (i)

– P (i± h) ≈ P (i)(

1∓ hde + h2

2 ce

)

• For a portfolio consisting of m securities:

– P =∑mk=1 Pk

– v =∑mk=1

Pk

P vk

– c =∑mk=1

Pk

P ck

• Redington Immunization

– yield curve assumed to be flat

– Rt = At − Lt for t = 1, 2, . . . , n

– P (i) = 0

– P (i+ h) = P (i) + hP ′(i) + h2

2 P′′(ξ), where 0 < |ξ| < |h|

– Choose asset portfolio such that P ′(i) = 0 and P ′′(i) > 0, i.e. such that

PV of assets equals PV of liabilities

vA = vL

cA > cL

– The value of the resulting portfolio increases under small changes in the interest rate.

• Full Immunization

– use force of interest δ equivalent to i

– liability Lk at time k

– hold two assets providing cash inflows of A at time k − a and B at time k + b

– solve the following equations simultaneously:

P (δ) = Aeaδ +Be−bδ − Lk = 0

P ′(δ) = Aaeaδ −Bbe−bδ = 0

– If the two known quantities are: (1) a, b; (2) B, b; (3) A, a; or (4) A, b; then a unique solution exists. However,for the cases: (5) a, B; and (6) A, B; a unique solution fails to exist. (In cases 5 and 6, solutions may be severalor nonexistant.)

– repeat the process for each liability Lk

18