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Edge Designations – CAIA LI
Formula sheet
Topic 2 – Introduction to Alternative Investments
Reading 2.1 – WHAT IS AN ALTERNATIVE INVESTMENT?
None
Reading 2.2 – THE ENVIRONMENT OF ALTERNATIVE INVESTMENTS
1. Short squeeze ratio
Short squeeze ratio =
short squeeze).
Number of shares being sold short
Total free float
(The higher this ratio becomes, the greater the likelihood of a
Reading 2.3 – ACCESSING ALTERNATIVE INVESTMENTS
2. Leverage
Leverage =
Assets
Net borrowings
Reading 2.4 – QUANTITATIVE FOUNDATIONS
3. Effective annual rate (EAR)
EAR = (1 + periodic rate)m − 1
Where:
Periodic rate = stated annual rate / number of periods
m = number of compounding periods per annum
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4. Future value of a single sum
FVN = PV(1 + r)N
Where:
FVN
PV
R
N
= The future value of the amount invested ($…)
= The amount invested today ($…)
= The annual interest rate on the investment (e.g., 12%)
= The period of the investment (e.g., 10 years)
5. Future value – multiple periods
r
FVN = PV(1 + )m X N
m
Where:
FVN
PV
R
N
m
= The future value of the amount invested ($…)
= The amount invested today ($…)
= The annual interest rate on the investment (e.g., 12%)
= The period of the investment (e.g., 10 years)
= The number of compounding periods
6. Continuously compounded rates of return
R = eR − 1
7. Geometric return calculation
T
∏(1 + R t )1/T − 1
t=1
8. Fully collateralized return
Rfullycoll = ln(1 + R) + Rf
9. Partially collateralized return
Rpartiallycoll = [L x ln(1 + R)] + Rf
Where:
L = leverage factor =
P = collateral %
1
p
10. Internal rate of return
NPV = CF0 +
CF1
(1+ IRR)1
+
CF2
(1+ IRR)2
+ ⋯+
CFn
(1+ IRR)n
Thus:
NPV = ∑nt=0
CFt
(1+IRR)t
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11. Money-weighted rate of return
Money-weighted rate of return = IRR
12. Time-weighted rate of return
Step 1: Calculate the holding period return (HPR) for each holding period
HPR =
Inflows over holding period
Outflows over holding period
Step 2: Take the geometric mean of the annual returns to find the annualized time-weighted rate of return over the
measurement period.
(1 + time-weighted return)
Number of holding periods
= HPR1 x HPR2
13. Distribution to Paid-In (DPI) Ratio
DPI =
Cumulative distributions
Paid in capital
14. Residual value to paid-in capital (RVPI)
RVPI =
NAV after distributions
Paid in capital
15. Total value to paid-in capital (TVPI)
TVPI = DPI + RVPI
16. The Public Market Equivalent (PME)
PME ratio =
FV of distributions + NAV
FV of contributions
Where:
I
FV of distributions = ∑Tt=0 Dt x T
It
I
FV of contributions = ∑Tt=0 Ct x T
It
Dt
Ct
IT
It
= The distributions at time y
= The contributions at time y
= The value of the publicly traded index at time period T
= The value of the publicly traded index at time period T
A PME ratio > 1 = the fund outperformed the public market index.
A PME ratio < 1 = the fund underperformed the public market index.
17. Hurdle rates
Soft hurdle rate – The GP can share in all profits if the fund’s performance is above the hurdle rate.
Hard hurdle rate – The GP can only share in profits that are in excess of the hurdle rate.
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Reading 2.5 – STATISTICAL FOUNDATIONS
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18. Ex-post versus Ex-ante
Ex-post returns are historical returns (post-fact).
Ex-ante returns are expected / forecasted returns.
19. Lognormal distribution – mean and variance
2
The mean of a lognormal distribution = eμ+0.5σ
2
2
The variance of a lognormal distribution = e2μ+σ + eσ −1
20. Discretely compounded rate of return
R = [S1/So] – 1
21. Continuously compounded rate of return
ln(1+R) = ln(S1/So)
22. Population mean
N
 Xi
μ= 1
N
Where x1, x2, x3, x4, x5, ………xN are the actual data values of the population.
23. Sample mean
n
åX
x= 1
i
n
24. Expectations
The expected value on random variable X is defined as:
E(X) = P(xi)xi = P(x1)x1 + P(xn)xn
Where:
E(X)
x1
= the expected value on asset X
= the observed value for random variable X
25. Variance
σ2 =
 (x − μ) 
2
i
N
where:
μ = is the population mean
N is the population size
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26. Standard deviation
σ= √
∑(Xi – 𝜇)2
N
27. Sample variance
S2 =
∑(Xi – x̅)2
n−1
28. Sample standard deviation
σ= √
∑(Xi – 𝜇)2
n−1
29. Skewness
=
E[(X− μ)3 ]
σ3
30. Kurtosis
=
E[(X− μ)4 ]
σ4
31. Covariance
Cov(Ri,Rj) = E{[Ri - E(Ri)] × [Rj – E(Rj)]}
32. Correlation coefficient
R=
Covariance of X and Y
(Standard deviation of X)(Standard deviatian of Y)
33. Spearman rank correlation
𝜌S = 1 −
6 ∑ d12
n(n2 − 1)
34. Beta
Beta =
Covi,m
𝜎 2m
Beta =
𝜎market 𝜎stock pmarket,stock
𝜎 2 market
Beta = (
𝜎stock
) (pmarket,stock )
𝜎market
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35. Approximate Durbin Watson (DW)
DW ≈2(1-r)
Where:
r = the correlation coefficient of the residuals from current period to previous period
In the event that DW is less than 2 = positive serial correlation.
In the event that DW is greater than 2 = negative serial correlation.
36. Second-order partial autocorrelation coefficient
𝜌 − 𝜌12
= 2
1− 𝜌12
37. Variance of a portfolio
σ2 port = w12 σ12 + w22 σ22 + 2w1 w2 r1,2 σ1 σ2
Where:
σ2 port
σ12
W1
σ22
W2
r1,2
𝜎1
𝜎2
= Variance on the portfolio
= Variance on Asset 1
= Weighting of Asset 1 in the portfolio
= Variance on Asset 2
= Weighting of Asset 2 in the portfolio
= Correlation coefficient – Asset 1 and 2
= Standard deviation on Asset 1
= Standard deviation on Asset 2
38. Standard deviation of a portfolio
σport = √w12 σ2i + w22 σ22 + 2w1 w2 r1,2 σ1 σ2
Or
1
σport = [w12 σ2i + w22 σ22 + 2w1 w2 r1,2 σ1 σ2 ]2
39. Annual standard deviation of a portfolio
Daily standard deviation x √T
This rule is called the square root of time rule.
40. Jarque-Bera (JB) test
n
K2
6
4
JB = (S 2 +
)
Where:
S = the skewness for the sample data
K = the excess kurtosis for the sample data
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Reading 2.6 – FINANCIAL ECONOMICS FOUNDATIONS
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41. Present value - discrete
PV =
Expected cash flow in periodt
(1+i)t
42. Present value - continuous
PV = FV x e−rT
43. Fisher effect
n=r+ i
Where:
n=
r=
i=
nominal interest rate in a given country
real interest rate
anticipated inflation rate
44. Modified Fisher effect
Modified Fisher equation =
r+i
1−T
Where:
T=
income tax rate
45. Forward rates
R f = R 2 + ((R 2 − R1 ) x
T1
)
T2 − T1
46. Binomial model – up and down moves - discreet
u=
d=
S+
S
S−
S
47. Binomial model – up and down moves - continuous
Up move = e𝜎√t
1
Down move = e−𝜎√t = 𝜎√t =
e
1
Up move
48. Risk-neutral probability - discreet
The risk neutral probability of an upwards move is calculated as follows:
πU =
1+r−d
u−d
The risk neutral probability of an downwards move is calculated as follows:
πD = 1 − πU
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49. Risk-neutral probability - continuous
The risk neutral probability of an upwards move is calculated continuously as follows:
πU =
ert −d
u−d
The risk neutral probability of a downwards move is calculated as follows:
πD = 1 − πU
50. Price of a call option
c=
πc+ + (1 – π)c−
1+r
Where:
π=
1+r−d
u−d
51. Risk neutral probability
λ≈
s
1 − RR
52. Credit spread
s ≈ λ(1 − RR)
53. Expected loss
Expected loss = PD x EAD x LGD
Where:
PD = The probability of default (PD) can be defined as the probability that the issuer will not make timely
payment of interest and principal.
LGD = The loss given default (LGD) is the amount that the investor stands to lose in the event of default by the
issuer.
This can be calculated as follows:
Loss given default = 1 – recovery rate.
EAD = Exposure at default (EAD). The EAD is the value of the outstanding debt at the time of the default.
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54. Macaulay duration
(
1 + r 1 + r + [N X (c − r)}
t
−
)−
r
c X {(1 + r))N − 1
T
Where:
c = coupon rate per period
r = yield to maturity
t = time period in days since last payment
T = time period in days for each coupon period
N = number of periods until maturity
55. Modified duration
Modified duration =
Macaulay duration
(1 + yield to maturity)
56. Approximate modified duration
Modified duration =
Where:
V- =
V+ =
V0 =
Δy =
V− − V+
2(V0 )(∆y )
price if yields decline by Δy
price if yields increase by Δy
initial price
change in the yield
57. Approximate price change
Approximate % price change = - duration X Δy* X 100
Where:
Δy* = the yield change for which the estimated % price change is sought
58. Capital Asset Pricing Model (CAPM)
E(r) = Rf + ß(Rm - Rf)
Where:
E(r)
Rf
ß
Rm
= The return required by equity holders
= The risk free rate (This is often displayed as the rate of return provided by a government bond, both of
which are considered to be almost risk free.)
= The beta of the share (The beta of a share is an indicator of risk. Thus a share with a beta of 2 compared
to a beta of 1 will on average be expected to generate twice the return on the market compared to the risk
free rate. Thus a beta of 2 shows more risk than a beta of 1)
= Return on the market portfolio
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59. Ex-post Capital Asset Pricing Model (CAPM)
Ri,t - Rf = Rf + ß(Rm - Rf) + εi,t
Where:
Ri,t
Rf
Rm
εi,t
= Realized return
= Risk-free rate
= Market return
= Error term / unexplained return due to idiosyncratic risk
60. The test statistic
Test statistic =
Sample statistic − popuplation value of the statistic under H0
Standard error of the sample statistic
61. Hypothesis – error types
Type I error – Rejecting the null hypothesis when it is in fact true. The probability of this error is denoted .
Type II error – Not rejecting the null hypothesis when it is false. The probability of this error is denoted .
Reading 2.7 – DERIVATIVES AND RISK NEUTRAL VALUATION
62. Forward price – generic
F0 = P0erT
Where:
F0
T
p0
r
= the forward contract price - the notation means the price of the forward contract initiated at T0 (now)
and expiring in time T (expiration date).
= expiration date
= the spot price at T0 (today)
= the risk-free rate
63. Forward price – with cash flows on the underlying
F0 = (P0 – I)erT
Where:
F0
T
P0
r
I
= the forward contract price - the notation means the price of the forward contract initiated at T0 (now)
and expiring in time T (expiration date).
= expiration date
= the spot price at T0 (today)
= the risk-free rate
= the present value of the interim cash flows
64. Forward price – commodity contract
F0,T = Poer T
65. Forward price – commodity contract + storage
F0,T = Poe(r + c)T
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66. Forward price – commodity contract + storage + convenience yield
F0,T = Poe(r + c - y)T
Where:
F0,T
Po
r
c
y
T
= the forward price
= the spot price today
= the continuously compounded interest rate
= the continuously compounded cost of storage
= the continuously compounded convenience yield
= expiry date of the contract
67. Value of a long position in a forward contract
V0,T = Pte(r + c - y)T-t – F0,T
Where:
V0,T
Pt
r
c
y
T
F0,T
= the forward value
= the spot price at time period t
= the continuously compounded interest rate
= the continuously compounded cost of storage
= the continuously compounded convenience yield
= expiry date of the contract
= the forward price
68. Options - Payoff values – call options
An options value at expiration is called its payoff.
At expiry the call option is worth the greater of:
i) Zero, Or
ii) The difference between the underlying price and the exercise price
CT = Max(0, ST – X)
69. Options - Payoff values – put options
At expiry the put option is worth the greater of:
i) Zero, Or
ii) The difference between the exercise price and the underlying price
PT = Max(0, X - ST)
70. Options – profit calculation – call option
Call option
Profit = Call option payoff – Call option premium
Profit = CT – C0
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71. Options – profit calculation – put option
Put option
Profit = Put option payoff – Put option premium
Profit = PT – P0
72. Covered call
Selling an out-of-the-money call + purchase of the current stock.
73. Protective puts
Long stock + Long put option.
74. Bull call spread
The trade is constructed as follows:
Long a call option, called CL, with a low exercise price = XL.
Short a call option, called CH, with a high exercise price = XH.
75. Bear call spread
The trade is constructed as follows:
Selling a call option with a low exercise price.
Long a call option with a high exercise price.
76. Bear put spread
A bear put trade is constructed as follows:
Long a put option with a high exercise price.
Selling a put option with a low exercise price.
77. Long straddle
A straddle is a simple volatility trade.
A long straddle strategy =
•
•
Long an at-the-money put option +
Long an at-the-money call option
78. Short straddle
A short straddle strategy =
•
•
Short an at-the-money put option +
Short an at-the-money call option
79. Long strangle
A long strangle strategy =
•
•
Long an out-the-money put option +
Long an out-the-money call option
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80. Collars
Buy an out-of-the-money (OTM) put option (protective put)
Sell an out-of-the-money call option (covered call).
81. Put-call parity
c0 +
X
(1+r)T
= p0 + S0
Where:
c0
= cost of the call
= present value of the purchase of the underlying bond
X
(1+r)T
p0
S0
= cost of the put
= underlying asset
82. Black-Scholes-Merton (BSM) formula
Call option =
S0N(d1) – Xe-rcT N(d2)
Put option =
Xe-rcT [1 - N(d2)] – S0[1 - N(d1)]
Where:
c=
p=
the value of the call option
the value of the put option
S
d1 =
d2 =
σ=
X=
rc =
T=
σ2
⬚
ln( X0 )+ [rC + ( 2 )] T
σ√T
d1 - σ√T
the annualized standard deviation of the continuously compounded return on the stock
the exercise price
the continuously compounded risk-free rate
time to expiry
N(d1) and N(d2) represent normal probabilities based on the values of (d1) and (d2).
83. Delta
Delta =
Change in the options price
Change in the underlying price
84. Interest rate caps and floors
The cap and floor payments are equal to:
Cap payment = max[(reference rate – strike rate), 0] x notional value / m
Floor payment = max[(strike rate - reference rate), 0] x notional value / m
85. Interest rate swap – fixed payment
1.0 – B0 (hn )
FS(0, n , m) = ∑n
j=1 B0 (hj )
Where:
FS(0, n , m)
B0(hn)
= Fixed payment amount that corresponds to $1 of principal
= Present value factor for the final $1 of principal
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Reading 2.8 – MEASURES OF RISK AND PERFORMANCE
86. Semivariance
Semivariance =
∑Rt <E(R)[Rt −E(R)]2
Sample semivariance =
T
∑Rt <E(R)[Rt −E(R)]2
T−1
Where:
Rt
E(R)
T
= return on the asset in time period t
= expected return
= number of observations
87. Semistandard deviation
Semistandard deviation = √
∑Rt <E(R)[Rt −E(R)]2
T
Sample semistandard deviation = √
∑Rt <E(R)[Rt −E(R)]2
T−1
88. Semivolatility
Semistandard deviation = √
∑Rt <E(R)[Rt −E(R)]2
T∗ −0.5
Where:
Rt
E(R)
T*
= return on the asset in time period t
= expected return
= number of observations less than the mean / threshold
89. Shortfall risk
Shortfall risk measures the probability of the investment returns falling below the target return.
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90. Tracking error
∑T (Rt −RB −M)2
T−1
TE = √ t=1
Where:
TE
M
Rt
RB
t=
= Tracking error
= Mean difference between Rt and RB (in most cases it is assumed to be zero)
= Portfolio return
= Benchmark return
= Number of observations
OR
TE = s(Rp – RB)
Where:
TE
s
Rp
RB
= Tracking error
= Sample standard deviation
= Portfolio return
= Benchmark return
91. Dollar VaR (method 1) – Excluding expected returns
VaR(X%) = Asset value X (Zvalue x σ)
Where:
VaR(X%) = Probability % value at risk
Zvalue = Critical z-value (obtained from the z-tables)
σ = Standard deviation of returns
92. Dollar VaR (method 1) – Including expected returns
VaR(X%) = Asset value X [Expected return% - (Zvalue x σ)]
Where:
VaR(X%) = Probability % value at risk
Zvalue = Critical z-value (obtained from the z-tables)
σ = Standard deviation of returns
93. Aggregating the VaR measure
Perfectly positive correlation.
VaRportfolio = VaRA + VaRB
Perfectly negative correlation.
VaRportfolio = VaRA - VaRB
No correlation.
VaRportfolio = √VaR2A + VaR2B
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94. Active return
The active return = Actual return - Benchmark return
95. Jensen’s Alpha (also known as ex post alpha or simply alpha)
αP = R Pt − R P
Where:
= Ex post alpha on the portfolio
= Actual return on the portfolio in period t
= R F + βP (R M − R F ) = The account return predicted by the SML and CAPM; where R F is the risk-free rate
of return, βP is the portfolio’s beta (systematic risk) and R M is the expected market return.
αP
R Pt
RP
96. The Treynor Measure
TP =
E(Rp) − Rf
βA
Where:
E(R p ) = Portfolio return
Rf = Risk-free return
βA = Portfolio beta
97. The Sharpe Ratio (SR)
SP =
E(Rp) − Rf
σP
Where:
E(R p )
Rf
σP
= Portfolio return
= Risk-free return
= Portfolio standard deviation
We can also calculate a monthly SR as follows:
SR monthly =
E(Rp) − Rf⁄
12
σP
⁄
√12
98. The Information ratio
IR =
Portfolio mean return−Benchmark mean return
Tracking error
99. Return on VaR (RoVaR)
RoVaR =
E(Rp )
VaR
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100.
The Sortino ratio
Sortino ratio =
Rp −MAR
Standard deviation of returns below the MAR
Where:
MAR
= Minimum acceptable or target return
(RPt −Rmin )2
Standard deviation of returns below the MAR = √
101.
N
The M-squared measure
M 2 = Rf + (Rp − Rf)
σm
σp
Where:
σm = Standard deviation of the market returns
σp = Standard deviation of the portfolio returns
102.
Stale (reported) pricing
The model for stale pricing is as follows:
r*I,t = αri,t + (1 – α)ri,t-1
Where:
r*I,t
α
rI,t
= Reported price at time period t (this is the stale price)
= Parameter that relates to the relationship between current and past true prices
= True price at time period t
103.
Mean of a stale return series
The following equation will be used to calculate the mean of a stale return series:
1
u* = u + [(1 – α)(ri,o – ri,T)]
𝑇
104.
Stale prices and volatility
We make use of the following equation:
σ(r*i,t) =
σ𝑟𝑖,𝑡
√N
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Reading 2.9 – ALPHA, BETA, AND HYPOTHESIS TESTING
105.
18
Ex-post alpha
Ri,t - Rf = Rf + ß(Rm - Rf) + εi,t
Where:
Ri,t
Rf
Rm
εi,t
= Realized return
= Risk-free rate
= Market return
= Error term / unexplained return due to idiosyncratic risk
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