FRM P1
Edgy Rhinx
27-09-2021
1
Enron
Disasters
corp. governance, Arthur
Andersen lost license to SEC,
Northern Rock
repo financing, funding
JPM and City were
liquidity
SG
counterparties, Agency risk,
Jerome Kerviel, fake offsetting
SOX 2002
transactions
UBS
incorrect modelling of
long-dated options
Drysdale
collateral
Kidder
Jett, artificial, sell forward
Barings
1995 Leeson, long-long Nikkey
Allied Irish
Rusnak, FX, fictioneous trades
MGMR
1993 stack and roll oil, f.
World.com
corp. governance
Global Crossing
corp. governance
SWIFT
2016 Bangladesh bank , 81m
Volkswagen
2015 emission scandal
Nabisco
lev. buyout, spread 100 to
350bp
MF Global
CEO on the Board invested in
EU during Greek crisis
liquidity, backward. →
contango
LTCM
stressed correlations, f.
liquidity, shorted govt. US and
2
Germany vs flight to quality
Bankers Trust
P&G, Gibson, complex
Orange County
1994 R. Citron, complex
CML: E(Rp ) = r +
Cont. Illinois
MBS and real estate f. by 1d
Rm −r
σm
i
σp
T P Im = Rm − r
E(RI )
repo
βi = b =
rapid growth during 1970
Jensen α = E(Rp ) − [r + βp (Rm − r)]
p
E(Rp )−E(Rb )
IR =
ε = V ar(Rp − Rb )
ε
q P
n
ε = n1 t=1 (Rp,t − Rb,t )2 =
p
E[(Rp − Rb )2 ] − (E[Rp − Rb ])2
boom, in 1984 Penn Square
failed → loss of confidence
Niederhoffer
1997 lost on naked puts
JPM
2012 Bruno Iksil, London
Cov(i,m)
2
σm
E(R )−R
p
min
Sortino =
q P σ1/2
n
σ1/2 = n1 t=1 [(Rp,t − Rmin )− ]2
whale, corp. governance,
Sachsen
h
E(Rp )−r
Sharp =
σp
E(Rp )−r
T reynor =
βp
E(Rp )−r
βp = Rm −r , Rm ≈
inverse swaps
Lehman
CAPM
complex cred. derivatives
Fama and French
2007 Landesbank, leveraged
E[Rp ] = α + β1 (Rm − r) + β2 SM B + β3 HM L +
AAA, CEO Ken Law also
β4 RM W + β5 CM A,
chaired
1
size
small – big
exhaustive
capitalization
high – low
P (A|C) × P (B|C) = P (A ∩ B|C) – conditionally
operational PL
robust – weak
independent, taller students do better given age 8
investments
conservative – agressive
TPI > Market > Sharp: positive α, lacks
40% of (blue and red) are convertibles gives no
diversification.
Conditional E[x] is E[x] weighted by conditional
additional info about blue and red events.
probabilities.
3
Funds
Open-end funds – not on exchange, but sold at
5
NAV, transactions at 16:00, no stop-orders
σp2 = w12 σ12 + w22 σ22 + 2w1 w2 p12 σ1 σ2
(unknown price).
Covs (X, Y ) = βs2X =
Closed-end – on exchange, can be shorted, below
Covariance
n
n−1 Cov(X, Y
) = RsX sY
n
Covs (X, Y ) = n−1
[E(X, Y ) − E(X)E(Y )] =
hP
i
(XY )
n
− xy
n−1
n
NAV.
ETF – on exchange, shorted, at NAV, disclosed
Spearman’s rank correlation
twice a day.
Kendal’s τ =
Undesirable
Late trading, after 16:00
Market timing, stale stocks
nc −nd
n(n−1)/2
=
nc −nd
nc +nd +nt
nc
concordant
Xi > Xj → Yi > Yj
nd
discordant
Xi > Xj → Yi < Yj
nt ties
Xi = Xj → Yi = Yj
Pn
Sum of mean deviations is 0: i=1 (xi − µ̂) = 0
Pn
0] = 12 (nσ 2 ) =
Cov
V ar(µ̂) = n12 [ i=1 V ar(Xi ) + n
Front running
Direct brokerage
Defences
σ2
n
Hurdle rate
High watermark clause
Minimum Variance Portfolio
Clawback clause
2
σ12
= w2 σ12 + (1 − w)2 σ22 + 2w(1 − w)σ12 ,
Backfill bias – hedgefund reports results when
w=
σ22 −σ12
σ12 −2σ12 +σ22
6
Hypothesis
profitable.
Incentive Fees
H0 : x = x0 , x ≤ x0 , x ≥ x0
√
√
[µ − tα σ/ n, µ + tα σ/ n]
2 + 20%
Prior
Post
0.8(R − 0.02), R > 0.02
P (T ype I, rejected true H0 ) = α = P (Crit. value)
0.8R − 0.02(1 + R), R > 0
P (T ype II, f ailure to reject f alse H0 ) = β
With probabilities
Significance level = α (5%)
Overall = RP (R) − LP (L)
95% = 1 − α, degree of confidence
To fund = 0.02 + 0.2P (R)(R − 0.02)
Power of test = 1 − β
To investor = 0.8P (R)(R − 0.02) + P (L)(−L − 0.02)
4
Equality of two means (vs Normal)
T =
Probability
µz
σ
√z
n
=
µx −µy
r
x−µ
√ ,
s/ n
2 +σ 2 −2Cov(x,y)
σx
y
n
P (A ∪ B) = P (A) + P (B) − P (A ∩ B)
tstat =
P (A ∩ B) = P (A|B)P (B) −−−→ P (A)P (B)
(size < 30 and Normal)
P (B|U ) =
P (U |B)P (B)
P (U )
=
p-value = P (T eststat )
indp.
P (U ∩B)
P (U )
c
−→
iid
µ −µy
r x
2
σx
nx
σ2
+ nyy
when (V unknown and size ≥ 30) or
zstat , when (V known and Normal) or (V unknown
c
P (U ) = P (U |A)P (A) + P (U |A )P (A ) =
and n is large)
P (U ∩ A) + P (U ∩ Ac ) – mutually exclusive and
χ2 =
2
(x−1)s2
σ02
F =
s2big
2
ssmall
Survivorship
N (−1)
0.159
Simultaneity
N (−1.28)
0.1
Omitted variable
N (−1.645)
0.05
Attenuation or fading, measured with ε,
N (−1.96)
0.025
N (−2.33)
0.01
N (−2.58)
√
εµ = σ/ n
leads to underestimation of regression coefficients
Omitted variable: βˆ1 = β1 + β2 ρ12 σ1
σ2
Cook’s Distance > 1 → outlier
Pn (Yˆj )2
Dj = i=1 ksi 2 > 1,
0.005
√
εσ = σ/ 2n
f
k – n of variables
s2f – squared residuals of the full model
7
β=
Regression
Cov(X,Y )
V ar(X)
Estimator properties
α = Y − βX
2
R2 ∈ (−∞, 1] = rXY
= rY2 Ŷ for multiple
P
ESS = (Ŷ − Y )2
P
2
T SS = (Y − Y ) =
+
P
RSS = (Y − Ŷ )2
1=
ESS+RSS
T SS
RSS
T SS
2
Adjusted R = 1 −
= 1 − R2
RSS
T SS
×
Unbiased
expected val. = parameter, E(µ̂) − µ = 0
Efficient
best possible, least V ar(ε) estimation
Consistent
sample size increases, bias goes to zero
Linear Transformation Properties
n−1
n−k−1
Y = a + bX
Dtstat = β−0
ε
ESS/k
Fall = RSS/(n−k−1) , n − k − 1 degrees of freedom
(RSS −RSS )/q
Fpartial = RSSfp/(n−kff−1) , q – number of removed
E(Y ) = a + bE(X)
2
V (Y ) = b2 σX
Cov[a + bX1 , c + dX2 ] = bd Cov[X1 , X2 ]
variables (restrictions)
p
SEE = RSS/(n − k − 1)
SY = sign(b)SX
KY = KX
Ft > Fcrit full model contributes meaningfully
High F → superior
fit → rejects H0 : βi = 0
F rejects
multicollinearity, P (T ype 2 error) ↑
T does not ≈ 0
No individual variables effects, but common source
8
Cov. stationary if the first 2 moments
{µ, Cov(Xt , Xt−1 )} are stationary
explains.
F does not
Forecasting, Stationarity
PACF – partial autocorr. function of Yt and Yt−h
Yule-Walker equations (YW) estimate autocorr.
variable has small effect
T rejects
Gauss-Markov theorem: OLS is BLUE, ε ∼ N (iid),
function
yt = Φ|t| y0 ,
no outliers, E(ε) = 0 → relax normality assumption
√
2
σ
x = σy = 1 → b = R
σ
xy
β = σ2
σ 2 σxy
x
y
σxy σx2
α = Y − βX
Violations
Heteroskedasticity
σε2 , OLS → W LS
Multicollinearity
Corr(β1 , β2 ) > 0.9, V IF > 10
t = 0, 1, 2, .. 0.651 , 0.652 , ..
y1 = ϕ1 y0 + ϕ2 y−1
ρ1 = y1 /y0 =
ϕ1
1−ϕ2
E(Yt ) = d + ϕYt−1 + ϵt ,
P
stationary
|ϕ| < 1, absolute values of slope
coefficients.
long-run µ =
d
1−ϕ ,
d – intercept
Serial correlation
of ε
1
Variance inflation V IF = 1−R
2 , omit the variable
V (Yt ) =
to cure multicollinearity
MA is covar. stationary
Biases
Function cuts off at order of the process
i
3
σε2
1−ϕ2
ACF
→ it’s MA process
Unit root
PACF
→ it’s AR (seasonality)
Yt = Yt−1 + ε
both
no good fit
Box-Pierce and Ljung–Box tests
Ph
Qbp = n i=1 ρ2i , h – number of lags, ρ –
autocorrelation
Ph Ph
2
Qlb = n i=1 n+2
i=1
n−i ρi = n(n + 2)
Qc ∼
χ21−α,h ,
Yt = Y0 + ε1 + ε2 + .. + εt
(1 − L)(1 − 0.65L)Yt = εt
9
ρ2i
n−i
Sequences
MA(1) Yt = µ + θεt−1 + εt
h degrees of freedom
AR(1) Yt = α + ϕYt−1 + εt
H0 : The data is independently distributed (i.e.
E(Yt ) = µ =
residual correlations = 0, any observed corr. from
α
1−β
V (Yt ) = γ0 =
σ2
1−β 2 , β
=ϕ
ARMA(1) Yt = α + ϕYt−1 + θεt−1 + εt
Ps
Seasonality Yt = β(t) + i=1 γi (Di,t )) + εt
randomness of the sampling process)
Ha : The data is not independently distributed,
EWMA
exhibit serial correlation
2
σt2 = 0 + (1 − λ)rt2 + λσt−1
Model parameters selection to Adj.R2
Covt = 0 + (1 − λ)rt,1 rt,2 + λCovt−1
Akaike’s information criteria (AIC) = 2k − 2 ln(L̂)
2
AIC = ln RSS
n−k + n k
GARCH
Bayesian/Schwarz (BIC/SIC) = k ln(n) − 2 ln(L̂),
2
σt2 = ywvol + αrt2 + βσt−1
L̂ max value of the model’s likelihood function
Covt = ywcov + αrt,1 rt,2 + βCovt−1
k – number of parameters
y – pull towards long-run mean, reversion
n – sample size
BIC = ln RSS
n−k +
Long-run variance VL =
k
n
w
1−α−β
=
w
y
Stable if α + β < 1
ln(n)
SICk = nd/n = 50025/500 , d – degrees of freedom
Augmented Dickey-Fuller (ADF) unit root test
2
R in trending timeseries → 1
H0 : unit root exists, in yYt−1 y = 0
Box–Jenkins model selection
y = 0 means no predictive value from the past, e.g.
1. Stationarity via ACF PACF plots, ADF for unit
random walk.
roots, Jarque-Bera for N (0, 1) of returns
Lagged level, deterministic, lagged differences;
2. Parameters estimation (AIC, BIC): OLS for AR,
random walk with drift.
max likelihood for AR, MA, ARMA
∆Yt = α + βt + yYt−1 + σ∆Yt−1 + .. + εt
HA : y < 0
3. Residuals diagnostics (Ljung–Box for white noise)
Conditional heterosk. is worse than unconditional.
10
Distribution
Heterosk.: coefficients are unbiased and consistent,
f (x) = P DF = CDF ′
st. ε are unreliable.
CF D(x) = P (X < x), probability of outcome
AR process is stationary when its lagged
strictly less than x
polynomial is invertible.
PMF dice: f (x) = 61 ,
Treating non-stationarity
Q(x) = N −1 (x) = F −1 (x), inverse CDF, quantile
F (x) =
x
6
Q(50%) – median
R
µ = xf (x)
R
P
σ 2 = (x − µ)2 f (x)dx =
(x − µ)2 P (x)
Trend → estimating or removing
Seasonality → dummies
Unit root (spurious relationship, no mean
reversion) → differencing
Random Variables
U = µ1 + σ1 Z1
V = µ2 + σ2 (ρZ1 +
p
1 − ρ2 Z2 )
LLN: limn→∞ Ê[f (x)] = E[f (x)], µ̂ → µ
4
CLT: V (Ê[f (x)]) = σf2 /n, Ê[f (x)] ∼ N (µ, σf2 /n)
m = ln
µ and σ 2 are finite.
s2 = ln(1 + w)
[Se(µ−σ
Mixture
Y = wX1 + (1 − w)X2
2
2
E[Y ] =
ρX12
+ (1 − ρ)X22
Y =
Y =
1
x
1 −β
,
βe
V ar[x] = np(1 − p)
P (x) =
px (1 − p)n−x
n
n!
x = x!(n−x)!
Pv
i=1
Zi2
V = 2v
Y = √ Z2
χ /v
E=0
V =
Portfolio default rate
v
v−2
p
1 loan loss σ1 = (1 − RR) p(1 − p)
p
σp = σ1 n(1 + (n − 1)p) =
p
(1 − RR) np(1 − p)(1 + (n − 1)p)
√
σ
n(1+(n−1)p)
σ
αp = np = 1
n
Power Law
Beta
Moments
1
α−1
(1
Beta(α,β) p
K – scale
α – fatness
µ = E[X]
− p)β−1 , mass ∈ [0, 1]
V = E[(X − µ)2 ]
E[(X−µ)3 ]
σ3
E[(X−µ)4 ]
=
σ4
Loss frequency
K
(from exponential)
K(X 2 Y 2 ) – parabola on corr/kurtosis plot
λx e−λ
x!
K(XY 3 ) and K(X 3 Y ) – lines
H0 : S = 0, K = 3
Uniform
2
JB = (T − 1)( Sk6 +
(b−a)2
12
V =
P (l < x < u) =
(K−3)2
)
24
vs χ2
Positive skew: mode, median, mean – mean is
U2 = a + (b − a)U1
affected by outliers.
min(u,b)−max(l,a)
b−a
11
Normal
E=µ
v−2
v−4
Jarque-Bera N(0,1) test
S=
E=
K=
Fat tails P (v > x) = Kx−α ,
Poisson (discrete)
a+b
2
V = 1/λ2
E = 1/λ
Student
P (X = x) =
] – confidence interval
1/β = λ
V = β2
E=v
Binomial (discrete)
E = V = λ = np =
σ
µ
Chi
P (x) = px (1 − p)1−x
t
β
2
P D12 = SP1 − SP2 = e−h1 t1 − e−h2 t2
Bernoulli
f (p) =
,w =
Survivalto 6 = e−6/5 , β = 5
fx1 ,x2 (X1 ,X2 )
fx2 (X2 )
n
x
√
/2)t±zα σ t
E=β
Conditional
E[x] = np
µ
(1+w)
Exponential
2
V (Y ) = E[Y ] − E[Y ]
fx1 |x2 =
2
√
VAR
r = ln P1 − ln P0 ,
V = σ2
R=
P1
P0
− 1,
r = ln(1 + R)
P1
P1 = P0 (1 + R) = P0 eln(1+R) = P0 eln(1+ P0 −1)
P
Q
rperiod =
r Rperiod = (1 + R)
Lognormal
Loss severity
Coherent:
Monotonicity
E = µ + σ 2 /2
R1 ≥ R2 → p(R1 ) ≤ p(R2 )
p(R1 + R2 ) ≤ p(R1 ) + p(R2 )
Sample from Poisson → get n losses
Subadditivity
Sample n from N (0, 1)
Pn
2
Loss = i=1 eN (m,s )
Positive homogenity
Translation invariance
5
p(βR) = βp(R), β > 0
p(R + c) = p(R) − c
VAR coherent when N(0, 1); ES always
−z 2 /2
Scenarios
−1.652 /2
e
e
√
√
ES = µ + σ (1−x)
= µ + σ (1−0.95)
2π
2π
DFAST
1/2y, no cap. plan, [10, 50b]
Spectral: ↑loss ↑weight
CCAR
1y, 9Q horizon, (50b, ∞)
Xa = F −1 (1 − U )
Uantithetic = 1 − U
Guassian Copula
CBB – circular block bootstrap with replacement,
√
block size = n
√
Simulations: εa = ε 1 + ρ, ε = √σn
√
V ARb = P V0 σy M Dzα t
U1 = a1 F +
p
1 − a21 Z1
Economy ↑ F ↑ Ui ↑
F0.1% = N −1 (0.001)
W CDR(F ) =
Conditional distribution has (µ, σ 2 ) conditioned on
√
N [N −1 (P D)− ρN −1 (0.001)]
√
1−ρ
EL = P D × LGD
p
2
U L = σEL = P DσLGD
+ LGD2 σP2 D
economic situation, it can be N(0, 1), but result in
fat-tailed unconditional distribution.
Control variable is effective when
q 2
σ
corr(x, y) > 0.5 σy2
Operational risk
∗
PA∗ = (PA − PBS ) + PBS
SMA: 7x + 7y + 5z,
Basic indicator (BI): 15% GI
x
y < 10m
u = eσ
Trees
t
e
pu =
of Gross Income (GI)
−d
u−d
12%
retail, asset management
rt
15%
commercial banking, agency
−d
Sert = Su eu−d
+ Sd u−e
u−d
Futures pu =
18% corp. finance, trading, payment and settlement
Approaches
1−d
u−d
fu −fd
Su −Sd
u −∆d
Γ= ∆
S u −S d
ud
S u = S2u +S
2
∆=
13
Basic indicator, firmwide, % of GI
Sd =
Standardized, business line ×β
S2d +Sud
2
Advanced measurement, operational VAR
AMA: Loss frequency =
Sensitivities
Loss severity
µ
µ̂ = ln √1+w
DV 01 = D × 0.0001 × P
∆P = D∆S + 12 Γ(∆S)2 + V ∆σ + T ∆t
mD =
B− −B+
2B0 ∆y
z < 100m
%
(r−q)t
rt
mC =
e−λ λn
n!
σˆ2 = ln(1 + w),
Estimated lossy = Observed lossx
B− +B+ −2B0
B0 ∆y 2
w = (σ/µ)2
0.23
Revenuey
Revenuex
Hedging
D and C
−V D0 − P1 D1 − P2 D2 = 0
15
V C0 + P1 C1 + P2 C2 = 0
Forward
Trailing hedge: ρ σσFS
F = S − I − Ke−rt = Se−qt − Ke−rt
t
1+R
,
F = (S + U − I) (1+c)(1+q)(1+l)
14
1 − a22 Z2
Ui ≤ N −1 (P D) → def ault
generating process in the underlying asset.
√
p
ρ = a1 a2
Contangion effect (σ ↑, ρ ↑) causes a different return
12
U2 = a2 F +
S
F
Capital
Financial Markets
U
- discounted storage costs
Capital8% = CET 14.5 + AT 11.5 + T 22
h −1
i
√
N (P D)+ ρN −1 (0.999)
√
W CDR = N
1−ρ
I
- discounted coupons
c
- convenience yield
Capitalirb = (W CDR − P D)LGD × EAD
p
2
U L = P DσLGD
+ LGD2 P D(1 − P D)
p
U Lp = U L21 + U L22 + 2p12 U L1 U L2
q
- dividend yield
l
- lease rate
HR = β =
a β
Est. Lossa = Observedb ( Revenue
Revenueb ) , β = 0.23
Cov(S,F )
2
σF
Hedge effectiveness
6
σ2
R2 = β 2 σF2
IR Collar f loor − cap
CT D : CP − CF × F
low
Risk reversal, Range forward chigh
otm − pitm
S
GP = CF × F + AI
CF =
GP −AI
100 , F
Cliquet – portfolio of ATM forward starting options
= 100
Asian (S − K)+
Rf wd = Rf ut − 12 σ 2 T1 T2 ,
σ – 1y volatility of future rate
Lookback
T2 = T1 + 90d
Fixed: c = (Smax − K)+
Rf ut = 100 − Zquoted
Floating: c = (Smax − ST )+
p = (K − Smin )+
Eurodollar Future
Gap has trigger K0 , payoff K1 , possible neg.
= 10000[100 − 0.25(100 − P %)] = 1m(0.75 + 0.25P )
premium
Compound – option on option
FRA
Shout c = max[(S − K)+ on tshout , (S − K)+ on T ]
FRA 1×4 – 3m forward rate in 1m
Rainbow – option on diff. assets
F −RK )τ −rt
F RA = L (R1+R
e
Fτ
Volatility swap L(σ − σK )
Option
2
Variance swap L(σ 2 − σK
)
E(St ) = S0 eµt
Warrant price delusion
E(lnSt ) = lnS0 + (µ −
Stock value decline S −
σ2
2 )t
n
n+m c
m
n+m c
V (lnSt ) = σ 2 t
Longevity L(Kmortality − R)
S − K ≤ C − P ≤ S − Ke−rt
c p C P
Chooser max(c, p) = c + max(0, c − p) =
c + (Ke−rt − S)+ = p + (S − Ke−rt )+
S
+
-
+
-
∆call − ∆put = 1
X
-
+
-
+
∆call = e−qt N (d1 )
T
?
?
+
+
∆f wd = e−qt
σ
+
+
+
+
vanna
∆ to σ
r
+
-
+
-
charm
∆ to time
Div
-
+
-
+
vomma
vega to σ
√i
2
lnSt ∼ N lnS0 + (µ − σ2 )t, σ t
√
)t−Zσ t
σ2
2
< St < elnS0 +(µ−
√
µ̂ = µ − σ 2 /2 σ̂ = σ/ t
∆f ut = e(r−q)t
′
σ2
2
′
(d2 )
√
gamma = e−qt NSσ(d√1t) = Ke−rt N
S2 σ t
√
√
vega = Se−qt tN ′ (d1 ) = Ke−rt tN ′ (d2 )
h
elnS0 +(µ−
∆put = e−qt [N (d1 ) − 1]
√
)t+Zσ t
rhoc = Kte−rt N (d2 )
rhop = −Kte−rt N (−d2 )
c = S −qt N (d1 ) − Ke−rt N (d2 )
C is exercised early when Dt > K(1 − e−r(T −t))
p = Ke−rt [1 − N (d2 )] − S −qt [1 − N (d1 )]
P is exercised early when Dt < K(1 − e−r(T −t))
d1 =
−qt
ln( S K
2
)+(r+ σ2
√
Asset or nothing put pays 1 unit of asset, when S <
)t
σ t
√
d2 = d1 − σ t
K (Z graph).
Fiduciary call c + Ke−rt = p + S Protective put
On stocks – American, on index – European.
Covered call S − c
Dividend in stock ≈ stock split → exchange adjusts
Box spread r
strike.
Straddle c + p
Rho is highest for ITM.
Strip 2p + c
Theta is highest for expiring ATM.
Strap p + 2c
Butterfly, calendar spread citm − 2catm + cotm
Diagonal = calendar with diff. K
CBOE Margin
high
high
low
Collar p + S − c ≈ clow
itm − cotm = potm − pitm Bull
−c : max(c + 0.2S + (S − K)−
otm , c + 0.1S)
call & put spreads (no S)
−p : max(p + 0.2S + (K − S)−
otm , p + 0.1K)
7
Forward KR010−1Y =
Bond
c2
2 + ..
(1+ R1 +0.0001
)
2
KR01 by 1bp, Dk r by 100bp
1+
MM and T-Bill
Act/360
T-Bond
Act/Act
Corporate
r
e = (1 + y/m)m
c1
R1 +0.0001
2
Yield curve:
Bull flattener
30/360
Bull steepener
n
n
= 1 − 0.068 360
P T Bill = 1 − Rq 360
Rq = (1 − P ) 360
n
Terms:
c
Par yield: 1 = A m
+d=
2(1−DT )
V =
AT
1+Rnom
Rreal = 1+Rinf
PT =
1+
P
c
DFi m
+d
c−PT
2
1+
+
short↓
long end↓↓
short↓↓
long end↓
MM
1Y
Short
1-5
Medium
5-12
Long
12+
E(Rbond ) = RT reasury + SDP − P D(1 − RR)
AT
Defaulted Zcpn claim receives issuing price +AI.
Mortgage
T-Bill rates go lower than OIS on FED fund rate
CP R = 1 − (1 − SM M )12
due to capital requirements.
1
SM M = 1 − (1 − CP R) 12
Pn
D = P1 i=1 ti ci e−yti M D =
Pn
C = P1 i=1 t2i ci e−yti M C =
D
1+y/m
PSA
← calculator
0.2% CPR to 30m increasing by 0.2% every year;
C
(1+y/m)2
6% flat afterwards.
1
a+be−cI
P repayment R =
PAC – planned amortization mortgage tranche
I = (W AC − R)ALS × A − K,
IO lower in value than PO, IO can receive less than
WAC – weighted avg. coupon
paid due to prepayment uncertainty.
A – annuity factor
Cash-out refinancing – extracting home equity.
K – cost to refinance
h
i
1
A= m
1
−
mT
y
(1+y/m)
h
i
1
c m
V alueA = m
y 1 − (1+y/m)mT
discounted using T-rate + OASestimate, until PV
= MTM.
c
y
V alueP erpetuity =
Japan bonds y =
OAS is computed via MC simulation where CFs are
c
p
+
2 MBS with equal credit quality, buy one with
100−p
pT
bigger OAS.
Cash & Carry = N ct − P0 rt
Rrealized =
OAS = Zero volatility spread – option cost
P1 −P0 +N ct−P0 rt
P0
P1 +Cash & Carry −P0 = Carry-roll-down +Rate
Change +Spread Change
Dollar roll
-Aug TBA +Sept TBA
Dollar roll value P0 − P1 − N ct + P0 rt
Prepayments↑: PO↑, IO↓
Corp Bond 105-07 = 105 78
7
T-Bond 105-07 = 105 32
18 6
Futures 105-187 = 105 32
256 = 105.5859375
1
1
256
V =
P
2
3
5
6
2
256
3
256
4
256
5
256
σi2
Factor
7
8
6
7
256
256
2
σ
importance= Vi
KR01 = P V1 − P V0
Dkr =
P V1 −P V0
P V0 ∆ykr
Dkr =
10000×KR01
P0
=
KR01
P V0 ∆ykr
D=
P
Dkr
8
16
SONIA, ESTR, EONIA are interbank rates
Timeline
Dutch auction sells at the first full offer P
1929
Wall Street crash
Yankee bonds – bonds by international
1933-
Glass–Steagall (GS) segregated inv.
organizations
banking
Make-whole call provision has no cost to
1970
inflation↑, FED short term rate↑
bondholders
1988
B1, capital 8%
Unsystematic risk is eliminated by 30 stocks and
1990
Brazil on loc. cur. debt
thousands of bonds
1993
Oil↓ $15/barrel
Pass-through mortgages carry prepayment risk
1997
Asian crisis, S&P -7%
TBA – to be announced, forward mortgage market
1998
Russia on loc. cur. debt, flight to safety,
Through the cycle PD: growth – overstated, crisis –
Oil↓
understated
-1999
GS repeal
RCSA – risk control and self assessment
2001
9/11, Enron
Knock-on effect – respose exacerbating adverse
2002
World Com, Global Crossing
condition
2003
Parmalat, Sarbanes–Oxley (SOX)
Asymptotically normal means becomes normal as
2007
US subprime, housing
n→∞
2008
Lehman counterparties, Bear Sterns,
Girsanov theorem σrn = σreal
Merrill Lynch, Libor-OIS spread 3.5%,
Spurious regression – false relationship
Northern Rock, Oil↓
Contraction risk – prepayment risk
2009
G20 limits bankers’ bonuses
BCBS operational risk types
2010
EU sovereign debt, Dodd-Frank, Volcher
Internal, ext. fraud, employee practises, and
rule: no prop. trading or fund ownership
workplace safety;
×2 Greece defaults on for. cur. debt, EU
Clients, products, and business practises;
bans uncovered CDS, LIBOR
Damage to physical assets;
manipulation scandal
Business disruption and system failures;
2012
2014
Oil↓, SEC rules that originators must
retain 5% of securitized product
Execution, delivery, and process management.
KMV model is PIT in contrast to agencies rating
2016
UK leaves EU, FRTB
models.
2020
Oil↓
Euler’s theorem to compute individual loan
contribution:
17
∆F
∆Xi /Xi ,
Terminology
∆F =
p
(X1 + ∆X1 )2 + X22 + ..
Orders
World’s theorem: every time series can be written
market
as a sum of deterministic and stochastic time series.
limit
Yt = εt + b1 εt−1 + b2 εt−2 + .. + ηt ,
stop-loss, sells below MV
ηt – deterministic sin process
∼ market if touched, sells above MV
Jensen’s inequality: convex → E[f ] > f (E)
stop-limit
PDF gives probability density of X, can be above 1
good till cancelled (open)
Control variables – var. with known relation to Y
fill or kill
VIX – 30d implied volatility of SP500.
Extraneous variables – superfluous var. with β = 0
SOFR repo-based secured ON financing rate
9
Positive definite – every linear combination of Xi
PCAOB – Public Companies Accounting Oversight
must have a non-negative variance.
Board (from SOX).
Matrix corrections
NINJA loan – no income, no job, no assets.
Equicorrel. – all correlations are equal
Liar loan – no evidence of employment.
TAF – term auction facility
2008 PDCF – primary dealer credit facility
Troubled assets relief program
Correlations are to the same factor ρij = Yi Yj
Uncovered interest rates parity – rates are the
same, but one currency depreciates.
IR↑
Bond↓
Forward↓
Futures↓↓, as loss has
to be financed at higher rate.
GARP
Knightian uncertainty – known unknowns
Conduct and integrity
RMP
Principles Conflict of interest
1. Identify
Confidentiality
2. Measure, manage, map
3. Operationalize risk app., distinguish EL and UL
Responsibility
Professional standards
Best practices
4. Address the relationships
5. Implement plan or strategy
6. Monitor and adjust
18
Insurance
Senior sets, BU implements, Fin. and operations
P D12 = SP1 − SP2 = (P D12 |SP1 )SP1
mitigate and transfer, RM supervises.
(P D23 |SP1 ) =
RM is more concerned with UL.
(P D12 |SP1 ))(P D23 |SP2 ) = 1 −
LGD↑ bankruptcy (liquidation) risk
Find Premium
Actual risks < Appetite < Capacity
1) P D23 |S12 = P D23 (1 − P D12 )
Concentration limits do not counter correlation risk
2) P V = N P D12 (1 +
R −1
2)
Financial position risk – balance sheet risk
3) n = 1 + 1 × S12 (1 +
R −2
2)
Dodd-Frank
4) P remium1Y = P V /n
1. FED oversights SIFIs (>50b)
Ratios
2. Ends to big to fail
Loss = Paid / Premium
3. Living will
Expenses = Expenses / Premium
4. Derivatives markets
Combined = (Paid + Expenses) / Premium
5. The Volker rule
Combined after dividends = (Paid + Expenses +
6. Consumer Financial Bureau
Dividends) / Premium
7. DFAST (10), CCAR (50)
Operating = (Paid + Expenses + Dividends -
CDO of mortgages is CMO.
Investment Income) / Premium
CLO of bank loans default less than mortgages due
Against moral hazard
SP2 −SP3
SP1
=
P D23
SP
1
= (1 −
SP1 −SP2
SP1
P D23
SP2
+ P D23 |S12 (1 +
R −3
2)
Deductibles
to better credit process.
Coinsurance provision
Reassignment – transfer risk to 3rd party in the
Policy limits
Mortality risk is bad for life insurance, good for
event of downgrade.
SIV – structured investment vehicle to profit from
annuity business.
spreads.
Mortality risk is mitigated by shorting longevity
Contango – no benefit of holding the asset.
derivatives (fixed - actual mortality) and survival
RDARR – risk data aggregation and risk reporting.
bonds (coupon is linked to the number of survivors).
10
Minimum CR = 0.25 to 0.45 × Solvency capital
19
requirement
Derivatives & Integrals
Below SCR → plan to increase
(x)′ = 1
R
Below MCR → business operations restricted
(af )′ = af ′
R
1 = x [+C]
R
af = a f
Plans
(ax)′ = a
R
a = ax
Defined benefit plan (employee benefit fixed)
(f ± g)′ = f ′ ± g ′
R
f ±g =
Defined contribution plan (employee benefit
(f g)′ = f ′ g + f g ′
unknown)
( fg )′ =
f ′ g−f g ′
g2
a
Rb
(f (g))′ = f ′ (g)g ′
Life insurance
Whole life
on death, fixed premium
(f n )′ = nf n−1 f ′
Term
death in period, fixed premium
(xn )′ = nxn−1
Endowment
at the end or on death
(ln f )′ =
f′
f
opposite, stops on death
(ln x)′ =
1
x
= x−1
(loga x) =
x ′
( ln
ln a )
Annuity
Rb
f±
R
c = c(b − a)
xn =
xn+1
n+1
R
1
ax+b
=
R
ln x = x ln x − x
R
loga x =
(ef )′ = ef f ′
R
ef =
(eax )′ = aeax
R
eax = a1 eax
(bax )′ = a ln(b)bax
R
bax =
=
1
x ln a
g
f = F (b) − F (a)
R
′
11
a
R
1
a
ln(ax + b)
x ln x−x
ln a
1 f
f′ e
1
ax
a ln b b