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:
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Investment environment, Grafiken und Mindmaps von Energietechnik / Energiesysteme
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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
- 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