Investment environment, Grafiken und Mindmaps von Energietechnik / Energiesysteme

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2025/2026

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Investments
Günter Strobl
University of Vienna
Fall 2025
Risk and Return
2Risk and Return
Holding-Period Return
Gross Return:
Pt+1: ending price
Pt: beginning price
Dt+1: cash dividend (paid at the end of the period)
Net Return:
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff

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Investments

Günter Strobl

University of Vienna

Fall 2025

Risk and Return

Holding-Period Return

 Gross Return:

  • Pt+1: ending price
  • Pt: beginning price
  • Dt+1: cash dividend (paid at the end of the period)

 Net Return:

Risk and Return 3

Multi-period Returns and Compounding

Assuming that we reinvest dividends, the return over

the two periods is:

P 0 P 1 P 2 D 1 D 2

Multi-period Returns and Compounding

 Example:

  • Suppose you earn 10% in year 1 and 20% in year 2
  • Then, your 2-year return is (1 + 0.1)(1 + 0.2) – 1 = 0.32 = 32%
  • What is the average annual return?
    • Because of compounding, it is not rA = (10% + 20%) / 2 = 15% (arithmetic average)
    • Also, it is not 32% / 2 = 16%
    • What we want is a rate rG such that (1 + rG)^2 = 1.32 → rG = 14.9%
    • rG is the geometric average

Historical Record

Equity returns around the world (1900–2000):

Risk and Return 7

Stocks for the Long Run

Risk and Return 9

Risk

 Stock returns are more volatile

  • While stocks do better on average, investors know that in any one year stocks may do much worse than bonds

Risk Measures

 Variance of return distribution (σ^2 )

  • Expected value of squared deviations from the mean
  • Variance can be estimated from a sample of size T as: where the sample mean is:
  • Standard deviation is the square root of the variance (σ)
  • These estimates are unbiased and consistent
  • Excel functions: VAR.S, STDEV.S

Risk and Return 13

Compensation for Risk

 Finance theory says that average returns over long

periods of time are determined by risk

  • Stocks offer higher expected returns than bonds because people are risk averse

 Two fundamental questions of finance:

  • What is risk and how should we measure it?
  • How much extra return do we need to compensate for the additional risk? - Is an equity premium of 6% enough? - Is it too much?

Equity Risk Premium

 Equity premium = E[rstocks – rT-bills]

 Equity premium has been high in the 20th^ century

  • Over the period 1926-2000, it was 6.2% in the US
  • Real bond returns have fallen sharply in the 20th^ century
  • By contrast, real stock returns have been relatively stable

 Low in the early 2000s, increased again recently

  • From 2000 to 2010, the equity premium in the US market was essentially zero
  • However, after the financial crises (i.e., from 2010 to
    1. it increased to about 15% for the US market

Equity Premium from Surveys of CFOs

Risk and Return 15

Equity Premium and Volatility

Risk and Return 19

Skewness

 Measures the asymmetry of a distribution

  • Excel function: SKEW
  • Skewness can be positive or negative
  • When the skewness is negative, the standard deviation underestimates risk mean median mean median Skew < 0 (left skewed) Skew > 0 (right skewed)

Kurtosis

 Measures the “peakedness” of a distribution

  • Positive kurtosis indicates “fat tails” (more probability mass in the tails than predicted by the normal dist.)
  • Excel function: KURT
  • When the kurtosis is positive, the standard deviation underestimates the likelihood of extreme events mean Kurtosis > 0 (“fat tails”)

Risk and Return 21

Skewness & Kurtosis of Daily Returns

Probability of ’87 Crash under Normality

 On 10/19/87, the price of IBM shares fell by 26%

 How likely is such a price movement under the

normal distribution?

  • Let’s standardize the return using the sample mean of 0.06% and the sample standard deviation of 1.54%: (-0.26 – 0.0006)/0.0154 = -16.
  • What is the probability of a 17-sigma event under the standard normal distribution? Pr (X < -16.84) = 6.2 × 10 -64!!!
  • Excel function: NORM.S.DIST (-16.84)

Risk and Return 25

t-Statistic and t-Test

 Whether the observed sample mean is far from E[r]

under the null or not depends on its variance:

Var(r) = σ^2 /T

  • T = number of observations (“sample size”)
  • Why? Think about it (and have a look at the appendix!)

 Standardize the sample mean using the population

mean under the null and the variance from above:

  • Since σ is unknown, we have to use the estimate
  • Note: is a random variable, is its standard error

t-Statistic and t-Test (cont.)

 This test statistic is called a t-statistic and follows a

t distribution with T-1 degrees of freedom

  • The t distribution is symmetric and resembles a normal distribution with heavier tails
  • It converges to a normal distribution as T → ∞

 Decision rule

  • Fix a significance level α
    • α = probability of mistakenly rejecting the null hypothesis
  • Find the critical value
    • Depends on α and degrees of freedom (df)
    • Excel function: T.INV.2T (α, df)
  • Reject the null hypothesis if |t| > critical value

Risk and Return 27

t-Statistic and t-Test (cont.)

 Example: monthly IBM return

  • T = 1,006, = 1.38%, = 7.13%, H 0 : E[r] = 0
  • t-statistic: t = 6.
  • Significance level α = 5%
    • That is, we reject the null hypothesis only when the data suggest that the likelihood with which the null is true is 5% or less
  • Critical value for t that corresponds to α = 5% and df = 1,005 is 1. - Rejection region: t < -1.96 or t > 1.
  • Since t = 6.1, we reject the null hypothesis that E[r] = 0
    • It is not likely to observe a sample average of 1.38% from 1, observations just because of sampling error

p-Value

 A p-value is the probability of observing a test

statistic as extreme as or more extreme than the

observed value under the null hypothesis

 In other words, a p-value is a measure of how much

evidence we have against the null hypothesis

  • E.g., a p-value of .02 indicates that we have only a 2% chance of drawing the sample if the null is true

 p-values are compared to a selected significance

level α, and H 0 is rejected if p < α

  • Excel function (t-test): T.DIST.2T