1
Q
nominal rate of return
A
investment return / opening asset balance
2
Q
Change in Endowment Foundation Value
A
= income from gifts = Spending + net investment return
3
Q
real rate of return
A
real rate = [(1+ nominal rate) / (1 + inflation rate)] -1
4
Q
spending rate fixed percentage
A
market value of the asset (t-1) * spending rate
5
Q
spending rate - trailing market value rate
A
spending ={ [market value (t-3) + market value (t-2) + market value (t-1)]/3}* spending rate
6
Q
liquidity requirement ratio
A
future capital commitment / total assets
7
Q
Bear put spread
A
Long a put with a high exercise price + selling a put option with a low exercise price
8
Q
Pension Fund Funded status
A
fair value of the plan asset less estimated pensions obligations
Plan assets < PBO : underfunded
Plan assets> pbo = overfunded
plan assetsd = pbo : fully funded
9
Q
Change in Liabilities (%)
A
= - modified duration * Yield change
10
Q
Expected Economic life
A
-{[1/ln(1+R)] * ln[(payment - R*assets)/payment]
R= expected return after fees and inflation
Payment=the annual spending in the first year of retirement
Assets=the value of the portfolio
11
Q
present value of annuity
A
= [payment/r]*{1-[1/(1+r)^n]}
r= discount rate
n=number of periods
12
Q
present value of growing annuity
A
= [initial payment/(r-g)]*{1-[(1+g)/(1+r)]^n}
r= discount rate
n=number of periods
13
Q
Future Value (growing annuity)
A
PV(growing annuity)*(1+r)^n
or
= [initial payment/r]*{[(1+r)^n-(1+g)^n]/(r-g)]}
14
Q
The utility function
A
E(U(W))= mu - (Lambda/2 )*var
lambda = constant degree of risk aversion
14
Q
Balance of payment
A
balance of payment = the current account + capital account + reserve account
15
Q
Utility function with Skewness and Kurtosis
A
E(U(W))= mu - (Lambda/2 )var + lambda2S - lamda3*K
lambda= preferred level of lambda
S=skeweness
K=kurtosis
16
Q
the utility function (Value at risk)
A
Var = mu - (Lambda/2 )*VaR(alpha)
mu: expected rate of return of the investment
lambda: constant degree of risk aversion to the VaR
17
Q
Risk aversion Level
A
Similar to Sharpe ratio but divided by the var not the stdev
(E(Rp)-r)/varp
r: risk free rate
varp: portfolio variance
18
Q
optimal risky asset weights in MVO
A
weight = (1/lambda)*{(expected risky return-risk free return)/var)
19
Q
optimal portfolio risky asset weights growing liabilities MVO
A
weight = (1/lambda){(expected risky return-risk free return)/var) +Ldelta/var
L= the relative size of the liabilities of the asset
delta= the covariance between the growth rates in the assets and liabilities
20
Q
MVO Hurdle rate
A
E[Rnew] - Risk free rate > (E[rp] - risk free rate)*beta new
E[Rnew] expected return on the new asset
beta new: beta of the new asset relative to the optimal portfolio
21
Q
Liquidity Penalty function
A
Max = avegRportfolio - lambda/2 var - PhiLp
Phi: investors level preference for liquidity
Lp: liquidity of the portfolio = sum(Wi*Li)
22
Q
factors exposures MVO
A
Ri = ai +bi*F + epsylon
If I want exposure to a factor below a target beta the function would be bp <= b con lineetta sopra
23
Q
standard error
A
=stdev/rad(n)
24
variance of the portfolio
varp = wi^2*vari +wj^2*varj +2wi*wj*stdev1*stdevj*r
r: correlation coefficient
25
risk budgeting with factors
stdevp=ro(1)*stdev(1)*B(1) + ro(2)*stdev(2)*B(2) +ro(epsylon)*stdev(epsylon)
26
risk parity - asset i contribution
asset i contribution = [delta(total risk)/delta(weight of asset i)]*weight of asset i
27
overcommitment ratio
total commitment of the fund / resources available for commitment
28
valuation of secondary PE prices
Po = sum{cash flows(t)/[(1+IRR(buyer))^t]}
0->T
T maturity of the fund
29
constant proportion portfolio insurance CPPI - floor value
F(t) = A(t) * e^(-r*(T-t))
F(t) floor value today
A(t) minimum value allowed at the end of the investment horizon
30
constant proportion portfolio insurance CPPI - target investment
target investment = M*(V-F)
M proportion of cushion invested in the asset (the multiplier)
V value of the portfolio
F floor value
31
optimal position in the liquid asset (futures position)
F(t) approx [(alpha/r(qt))+ beta]*(Kt-Wt)
F(t): optimal future position
alpha: alpha of the illiquid position
r: excess return of the futures positions
beta: beta of the illiquid asset compared to the futures contract
k: will depend on the strategy being followed
w: weighting of the risky asset in the portfolio
32
Hedging Portfolio - Beta on the new portfolio
beta new = beta(port) + (F/P)* beta(futures)
F/P : ratio of futures contracts on the size of the portfolio
beta(futures): beta on the futures contracts
33
Put call parity
C + X/(1+r)^T = P + S
C cost of the call
* present value of the purchase in the underlying bond
p cost of the put
S underlying asset
34
options greeks - delta
Change in the price of the option/change int he price of the underlying
=C+ - C- / S+ - S-
N(d1) delta for calls
N(d1) - 1 delta for puts
35
geometric mean return on an investment (compounded) approx
per period aritmetic mean return - (1/2 )*var
36
Active Return
Actual REturn - Benchmark return
37
Ex post CAPM time series model
R(t) - Rf = alpha + beta*(R(mkt) - Rf) + epsylon(t)
38
Capitalization Rate
NOI/Valuation
39
Valuation (cap rate)
Valuation = NOI/Cap rate
40
Expected return
Rf + Risk premium
41
Margin to equity ratio
Initial Margin / NAV
42
Value at risk(X%)
Asset Value * [ expected return% - (Zvalue* Stdev)]
43
Value at risk (variance calculation)
var = 1/T * sum((mu-Rt)^2)
44
Var -EWMA - exponentially weighted moving average
var = (1-lambda)*var(t-1) + lambda*(Rt - mu(t-1))^2
lambda?
45
Omega ratio
average upper partial movement / average lower partial movement
upper partial movement = max( realized return - Target, 0)
lower partial movement = Max(Target - realized return, 0)
46
Price reported P(t, reported) alpha
alpha*P(t,true) + alpha*(1-alpha)*P(t-1,true) .....
alpha decay value
47
true price P(t,true)
P(t-1, reported) + (1/alpha) * (P(t,reported) - P(t-1,reported))
48
Raported price
| (autocorrelation)
approx = (1-ro)*R(t,true) +ro*R(t-1,reported)
ro first order autocorrelation
49
Unsmoothed return (autocorrelation)
R(t,true) = (R(t,reported) - ro*R(t-1,reported))/(1-ro)
50
Variance unsmoothed
= var(reported) *[(1+ro)/(1-ro)]
51
Unsmoothed Beta
beta(true) = beta(reported)/(1-ro)
52
Vasicek model
r(t+1) = r(t) + kappa*(mu - r(t)) + stdev*epsilon(t+1)
r(t+1) next period short term interest rate
kappa: speed of the mean reversion factor
mu: mean rate
stdev: volatility of the change in the interest rates (constant)
epsilon: small random walk movement
53
Vasicek model rules - will be given in the exam
| what happens to R in time
if the short term rate> long term rate : there will be negative drift adjustment
if the short term rate< long term rate : there will be positive drift adjustment
54
CIR Model - Cox Ingersoll Ross model
r(t+1) = r(t) + kappa*(mu - r(t)) + stdev*epsilon(t+1)* rad(r(t))
r(t+1) next period short term interest rate
kappa: speed of the mean reversion factor
mu: mean rate
stdev: volatility of the change in the interest rates (constant)
epsilon: small random walk movement
55
Ho Lee model
r(t+1) = r(t) + Phi(t) +
56
Black Derman Toy (BDT) model
Two-year average return = 0.50[(1 + r0)(1 + rU)-1] +0.50[(1 + r0)(1 + rD)-1]
57
Black Derman Toy (BDT) model - rate calculation
upper rate from the lower rate
lower rate from the upper rate
58
Loss Given Default
LGD = 1 - recovery rate
59
Expected Loss
EL = EAD * PD * LGD
60
Merton Model Equity Value
in the merton model Equity can be seen as call option on the difference between asset and debt
E(t)= max(0,A(t)-K)
K face value of the debt
61
Merton model Debt Value
D(t) = K - Max(0, K - A(t))
62
Black scholes pricing model - equity value
Given in the exam
E(t)=A(t)*N(d) - K*[e^(-r*T)]*N(d-stdev*rad(T))
d formula given
63
Credit spread (calculation 1) - Given in the exam
S(t) = -(1/T) * [(N(d-stdevA*rad(T)) + (A/K)*[e^(r*t)]*N(-d)]
64
Value of a bond
D(t) =K*e^[-(r+s(t))*T]
65
Credit spread - calculation 2
Credit spread = lambda * (1-RR)
66
The risk neutral default intensity
lambda = (credit spread)/ (1-RR)
67
KMV credit risk model
Kealhover mcquown vasicek
stdev(E) = [A(T)/E(T)]*Delta *stdev(A)
Delta: sensitivity of the firms' equity to changes in asset value
68
DD distance to default
DD= (A(t)-K)/(A(t)*stdev(A))
K is the default trigger
69
EDF expected default frequency
(number of firms that defaulted within the year - DD=n) / (total firms in the population with a DD=n)
70
Probability of Survival p(t)
=e^(-lambda*t)
lambda: the default intensity that determines the expected time to default as well as the probability of survival
71
Expected time to default
= 1/ lambda
72
Binomial tree - downward movement
d=1/u
u=e^stdev*rad(T)
73
Binomial tree - upward movement
u=e^stdev*rad(T)
74
Binomial Tree probabilities
P(up) = (1+r-d)/ (u-d)
r risk free rate
75
Fama French Model
Ri= Rf + (β(market)* Market factor) + (β(SMB)* SMB) + (β(HML)* HML) + εi
76
Carhart Model
REquired return = Rf + (β(market)X Market factor) + (β(SMB)X SMB) + (β(HML)x HML) + (β(momentum)*Momentum factor)
momentum factor (UMD)
77
Fama French Extended
Ri= Rf + (β(market)* Market factor) + (β(SMB)* SMB) + (β(HML)* HML) + (β(RMW)* RMW) +(β(CMA)* CMA) + εi
RMW risk premium attributable to robust factor
CMA risk premium attributable to conservative versus aggressive factor
78
Moving Average
t->n SMA = sum(P(t-1))/n
79
Signal to noise ratio SNR
SNRt(n) = [P(t)−P(t−n)]/∑[P(t−1)−P(t−i−1)]
80
Market divergent Index MDI
MDI(n) = (1/M) *sum(SNR(n)) n->M
M the number of markets
n the lookback period
81
Gordon Growth Model
V(0) = D(1) / (r-g)
82
Free Cash Flow to Firm FCFF
Ni + NCC + Interest expenses *(1-Tax rate) - FCInv - WC
NI net income attributable to common shareholders
NCC net non cash charges
FCInv fixed investment
WC working capital investment
83
Firm Value
[FCFF*(1+g)]/(WACC-g)
84
Equity value
Firm value - market value of debt
85
Return on Equity
net income / average total Equity
86
Return on equity 2 (developed)
(net income/ net sales)*(net sales/total average assets)*(total average assets/total average equity)
net profit margin * total asset turnover * financial leverage
87
ROA
Net income / average total assets
88
P/E Ratio
Market Price per share / earning per share (12 months)
89
Stock to flow measure
Total Supply / annualized issuance
90
Metcalfe law
n*(n-1) / 2
number of nodes and number of correlation coefficients for n assets
91
Partial Autocorrelation Coefficients
R(t) = a + autocorrelation coefficient(-1)*R(t-1) etc + E(t)
92
Dummy variable regression model
R(i,t) - Rf = alfa(i) + [beta(i,d) + D1 * beta(I DIFF)*[R,-Rf]] + e
93
cointegration equation
Ln(x) - a*ln(t) - u(t)
u is a stationary variable
non la chiederanno mai
94
Substitution spread statistic
ln(closing price of commodity A/closing price of commodity B)
95
100 day Substitution spread statistic
[sobstitution statistic - (100 day moving average of the substitution statistics) ] / 100 day standard dev of the substitution statistic
96
Covered interest rate parity (currency and return)
(1+r(domestic risk free security))*(Forward rate of exchange / spot rate of exchange) = (1 + r(foreign))
97
present value of the tax shield
sum((depreciation * Tax rate) /[(1+r)^t])
97
Depreciation Tax shield
depreciation * tax rate
98
after tax rate of return
rate of return * (1-T)
99
rate of return with tax deferral
[1+[(1+r)^T -1]*(1-tax rate)]^(1/T)-1
100
leveraged return on equity
ROA*L
101
volatility of the leveraged return on equity
stdev lev = stadevasset*L
102
leverage factor
asset / equity
103
the weights of the risky assets
| multifactor model
R(t, HF)–r = β1x (F(1,t) -r) + β2x (F(2,t) -r) +...+ βkx (Fkt -r) + ε
r short term risk free rate
beta are the risky assets weights
104
R^2 not sure
explained variation / total variation
105
Cash weighting estimate
Beta cash = 1- sum (beta(i))
106
Non linear replication model
Rt, HF-r= β1x (F1t -r) + β2x (F2t -r) +...+ βkx max(Fkt -r) + ε
107
Equal weighting diversification approach
W = 1/N
108
Equal risk weighting
(1/stdev annual (i)) / sum(1/stdev annual(i))
109
ETF daily leveraged returns
R(tL)=AxRt + (1−A)r
A degree of leverage
r daily cost of borrowing
110
long term ETF value
photo 7/3
111
Long-term ETF value –with volatility (variance)
photo 7/3
112
DPI
sum of distributions / sum of paid in capital
113
RVPI
NAV after distributions/ paid in capital
114
TVPI
RVPI + DPI
115
PME Ratio
FV distributions + NAV / FV contributions
116
Equially weighted IRR
IIRR = (1/N )*sum(IRR)
117
commitment weighted IRR
1/CC * sum(IRR*CC)
118
Accounting Equation
Assets = equity + liabilities
119
ROE with leverage
ROA*L -r*(L-1)
120
equity
L-1
121
Vega
change in option price /change in a unit of stdev (volatiliy)
122
BSM vega
vega = price of the underlying * N'(d) * rad(T)
N'(d) the noncumulative probability density function of the normal distribution at d
T option time to expiry
123
vega in basis points
vega = price of the underlying * N'(d) * rad(T) / 100
124
Impact of vega on option price
delta p = vega * delta st dev
125
gamma
second derivative of the option price / the second derivative of the stock price
126
gamma non divident paying stock with BSM
N'(d)/ S*stdev*rad(T)
127
Change in volatility Jump process
σt+∆^(−σt)=𝛾∆+𝛿∆𝑌+φΔJ
Where:
𝛾= The expected change in volatility
𝛿+φ= The volatility of the change
∆Y= Small random changes in volatility
∆J= Large, positive unexpected jumps in volatility
128
Long stradde
Long call option at the money + long put option at the money
129
Short stradde
Short at the mony put option + sshort at the money call option
130
Short strangle
Short an out-the-money put option + *Short an out-the-money call option
131
Long strangle
long an out-the-money put option + Long an out-the-money call option
132
Short Iron Butterfly
Short bull call spread + short a bear spread (the middle strike price needs to be identical
133
Short iron condor
short a bull spread OTM + short a bear spread OTM
134
Bull call spread
long call + short call different SP
135
The VIX 30 day contract price
photo 7/3
136
Correlation swap - payoff
notional principal ^ ( ro realized - ro fixed)
137
realized correlation
138
realized correlation - approximation
ro average = variance portfolio / variance i
139
Tax free wrap funds
return pretax equal return post tax
140
Fully taxed wrap funds
return after taxes = r*(1-T)
141
standard deviation of a global real estate investment correlation = 1
st dev in local currency terms = str dev of local asset return + standard deviation of the foreign exchange rate
142
standard deviation of a global real estate investment correlation = -1
st dev in local currency terms = abs(str dev of local asset return - standard deviation of the foreign exchange rate)
143
standard deviation of a global real estate investment correlation = 0
rad(var of local asset return + var of the foreign exchange rate)
144
Fully taxed wrap funds
r* = [w(i) x r x (1 –To)] + [w(capital gain)x r x (1 –Tcg)]
145
tax deferred wrap funds
r* = {1 + [(1 + r)^N–1](1-T)}^(1/N) -1
146
tax deferred and deduction wrap funds
147
put call parity
148
value of global real estate investment
investment * (1+return on investment)*(1+ return on forex exchange rate)
149
value of the global real estate investment (approximation)
initial investment * (1 + r + fx)
150
Variance of a global real estate investment
Caia Level 2
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