Stella section Flashcards

1
Q

Define asset price

A

the amount paid for an asset, represents amount of value the market has been assigned to an asset

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2
Q

Define log returns

A

The log return n is defined as the logarithmic price changes on an asset, with appropriate adjustments for any dividend
payments. Let pt and rt denote the price and the log-return at time t respectively. If ignoring dividends, then rt = log(pt/pt−1).

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3
Q

Define gross return

A

Ratio of two prices

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4
Q

Define simple returns

A

simple return Rt of an asset over time period t can be measured
by the sum of the change in its market price plus any income received
over the holding period divided by its price at the beginning of the
holding period

Rt = (pt − p_(t−1) )/ p_(t−1)

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5
Q

Relate simple returns to log returns

A

rt = log(1 + Rt)

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6
Q

Give the 11 stylised facts

A

AHGAIVSCLVA

  • absence of autocorrelations
  • heavy tails
  • gain/loss symmetry
  • aggregational gaussanity
  • intermittency
  • volatility clustering
  • slow decay of autocorrelation in absolute terms
  • conditional heavy tails
  • leverage effect
  • volume/volatility correlation
  • asymmetry in time scales
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7
Q

State the three most important stylised facts

A
  • The first important stylized fact for returns is that their distribution is not normal.
  • The second important stylized fact is that the sample autocorrelations of
    returns are generally close to zero, regardless of the time lag.
  • The third major stylized fact is about the positive dependence between absolute returns or
    squared returns on nearby day
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8
Q

Give the RWH for modelling returns

A

rt = µ + et, t = 0, ±1, ±2, . . . ,
where {et} is weakly stationary with E[et] = 0 and Var(et) < ∞

note et = σtεt

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9
Q

What is the H0 for testing the RWH in modelling returns?

A

Cov(rt,rt+h) = 0 for h≠0, i.e., {et} ∼ WN(0, σ^2)

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10
Q

Give the Q-test for the RWH

A

Qτ = n(n + 2)*sum(k=1 to τ) ρˆ2_k/(n − k) ≈
sum(k=1 to τ) nρˆ2_k

Under the RWH, Qτ→ χ^2_τ

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11
Q

Give the variance-ratio test for modelling returns by the RWH

A

VR(N) = V(N) / NV(1) = 1 + 2/N * sum(τ=1 to N-1) (N − τ )ρτ

zN = (VR( ˆ N) − 1) / sqrt(v_N/n)∼ N(0, 1)

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12
Q

When does the variance ratio test perform better than the q-test?

A

If the null hypothesis is heteroskedastic

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13
Q

Give the sample skewness formula

A

β_2 = 1/n * sum(i=1 to n) (ri − rbar)^3 / S^3

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14
Q

Give the sample kurtosis formula

A

κ_2 = 1 / )n−1) sum(i=1 to n) (ri − rbar)^4 / S^4

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15
Q

Give the JB test statistic

A

JBn = n*(β^2/6 + (κ_2 − 3)^2 /24→ χ_2

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16
Q

Define volatility

A

Volatility is a measure of price variability over some period of time. It is defined as the standard deviation of the change in the logarithm of a price
during a stated period of time

17
Q

Define conditional volatility

A

the standard deviation of a future log return that is conditional on known information such as the history of previous returns

18
Q

Define unconditional volatility

A

the standard deviation of a log return without

conditioning on the history of previous returns

19
Q

Define a GARCH model

A

rt = µt + et, et = σtεt
where
- εt is a white noise,
- µt is a trend process (modelling the conditional mean).
- σt is a volatility process (modelling the conditional variance)

20
Q

Give the formula for sharpe ratio

A

SR = E(rt - rf) / sqrt( Var(rt)

21
Q

Define sharpe ratio

A

The ratio of the expected excess return of an investment to

its return volatility or standard deviation

22
Q

Give the single factor model for asset returns

A

rit = αi + βirmt + εit, (1)
where:

  • rit and rmt are the log returns of the ith stock and the common
    market index respectively
  • αi is the i-th stock specific effect, independent of market
    performance
  • εit represents a random error E[εit] = 0 and V[εit] = σ^2_ei,
  • βi is a factor loading that measures the expected change in rit for a unit change in rmt
  • εit’s uncorrelated with each other and uncorrelated with rmt
23
Q

Give expectation of a single factor model

A

E[rit] = αi + βiE[rmt]

24
Q

Give variance of a single factor model

A

V[rit] = βi^2 * σm^2 + σei^2

25
Q

Derive estimators of βi and αi

A

S = sum(i=1 to n) (rit − α − βi * rmt)^2

For β, dS/dβ = 0
For α, dS/dα = 0

βi =sum (t=1 to n) [(rit − rbar_i) * (rmt − rbar_m)] / sum(i=1 to n) (rmt − rbar_mt)^2

αi = rbar_i − βi*r¯bar_m

26
Q

Give the estimator for σei^2

A

1 /(n-2) * sum(t=1 to n) (rit − αˆi − βˆi*rmt)^2

27
Q

Give the estimator for σm^2

A

1 /(n−1) * sum (t=1 to n) (rmt − rbar_m)^2

28
Q

Give the estimator for σi^2

A

βiˆ2 * σm^2 + σε^2

29
Q

Give the estimator for σij

A

βiβjσm^2

30
Q

Define CAPM

A
E(Ra) = Rf + βa(E(Rm) − Rf)
where:
- Ra is the return on an asset
- Rf is the risk-free rate of return
- Rm is the return on the market
31
Q

Define CAPM in terms of excess returns

A

E(ra) = βaE(rm)

ra = Ra - Rf
rm = Rm - Rf
32
Q

in terms of CAPM give estimators for βa and αa

A
βa = Sam / Smm
αa = rbar_a - βa * rbar_m

where
Sam = sum(i=1 to n) (r_at - rbar_a) * (r_mt - rbar_m)

Smm = 1/n * sum(i=1 to n) (r_mt - rbar_m)^2

33
Q

Give an estimator for var(αa)

A

1/(n-2) * (Saaf - Sam^2/Smmf)

34
Q

Give the test statistic for testing CAPM

A

t = αa / sqrt ( Var(αa)hat)

we reject H0 if |t| > tn−2(η/2)

35
Q

Define value at risk

A

For a confidence level α, the Value-at-Risk, VaR(α), is chosen so that the probability of a loss (over a given time horizon) larger than the VaR(α) is equal to α

If L is the loss over the holding period, then VaR(α) is the α-th upper quantile of L

VaR(α) = inf{x : P(L > x) ≤ α}.