Actuarial Science Reference Sheet Author: Daniel Nolan Email: [email protected]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
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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
• 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
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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
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– 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)
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– 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(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
– 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).
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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
– 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)
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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.
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• 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
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.
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.)