T distribution table
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This table provides critical values (tα/ν) of the Student's t-distribution for various confidence levels (α) with degrees of freedom (ν) ranging from 1 to infinity. The t-distribution is used to test hypotheses about the mean of a population when the population standard deviation is unknown. The table allows researchers to determine if a calculated t-statistic is greater than the critical value and thus determine if the null hypothesis can be rejected for a given confidence level and degrees of freedom.
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USEFUL FORMULAS
Measures of risk
Expected returns: ∑ ×= ssi RobPr)r(E
Variance of returns: ∑ −×=σ
2
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2
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Covariance between returns:
( )( ))r(Er)r(ErobPr)r,r(Cov jjsiissji −−∑ ×=
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)r(Var
)r,r(Cov
M
Mi
i =β
Portfolio Theory
Expected rate of return on a portfolio with weights w in securities i and
j:
)r(Ew)r(Ew)R(E jjiip +=
Variance of portfolio consisting of securities i and j:
)r,r(Covww2ww jiji
2
j
2
j
2
i
2
i
2
p ×××+σ+σ=σ
Covariance/Correlation coefficient:
j,ijiji Corr)r,r(Cov ×σ×σ=
Minimum variance portfolio:
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Present value of $1
Discrete period compounding:
T)r1(
1
PV
+
=
Continuous compounding: rTePV −=
Forward rate of interest for period T:
1T
1T
T
T
T )y1(
)y1(
f
−
−+
+
=
Real interest rate: 1
i1
r1
R −
+
+
=
where r is the nominal interest rate; and
i is the inflation rate
…/continued overleaf
)RR( COV 2 - )RVAR( + )RVAR(
)RR( COV - )RVAR( = W
BABA
BAB
2
Duration of a security:
Equity Analysis
Constant growth dividend discount model:
gk
D
gk
)g1(D
P 100 −
=
−
+
=
Growth rate of dividends: bROEg ×=
Price-earnings multiple:
bROEk
b1
EP
×−
−
=
Present value of growth opportunities: PVGO
k
E
P 10 +=
Derivative Assets
Put-call parity: SEeCP rT −+= −
Black-Scholes formula:
)d(NEe)d(SNC 2
rT
1
−−=
T
T)2r()ESln(
d
2
1
σ
σ++
=
Tdd 12 σ−=
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Sharpe’s measure:
p
fp
p
rr
S
σ
−
=
Treynor’s measure:
p
fp
p
rr
T
β
−
=
Jensen’s measure, or alpha:
)]rr(r[r fMfpp −β+−=α
…/continued overleaf
∑
=
×=
n
t
t t
iceMarket
CFPV
D
1 Pr
)(
3
CONTINUED OVERLEAF...
TABLE 1:
Period 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%
1 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.909 0.901 0.893 0.885 0.877 0.870 0.862 0.855 0.847 0.840 0.833
2 0.980 0.961 0.943 0.925 0.907 0.890 0.873 0.857 0.842 0.826 0.812 0.797 0.783 0.769 0.756 0.743 0.731 0.718 0.706 0.694
3 0.971 0.942 0.915 0.889 0.864 0.840 0.816 0.794 0.772 0.751 0.731 0.712 0.693 0.675 0.658 0.641 0.624 0.609 0.593 0.579
4 0.961 0.924 0.888 0.855 0.823 0.792 0.763 0.735 0.708 0.683 0.659 0.636 0.613 0.592 0.572 0.552 0.534 0.516 0.499 0.482
5 0.951 0.906 0.863 0.822 0.784 0.747 0.713 0.681 0.650 0.621 0.593 0.567 0.543 0.519 0.497 0.476 0.456 0.437 0.419 0.402
6 0.942 0.888 0.837 0.790 0.746 0.705 0.666 0.630 0.596 0.564 0.535 0.507 0.480 0.456 0.432 0.410 0.390 0.370 0.352 0.335
7 0.933 0.871 0.813 0.760 0.711 0.665 0.623 0.583 0.547 0.513 0.482 0.452 0.425 0.400 0.376 0.354 0.333 0.314 0.296 0.279
8 0.923 0.853 0.789 0.731 0.677 0.627 0.582 0.540 0.502 0.467 0.434 0.404 0.376 0.351 0.327 0.305 0.285 0.266 0.249 0.233
9 0.914 0.837 0.766 0.703 0.645 0.592 0.544 0.500 0.460 0.424 0.391 0.361 0.333 0.308 0.284 0.263 0.243 0.225 0. ...
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