Lecture 1: Asset returns and volatility portfolios
Lecture 1.1: Recap Finance 1
𝐶𝑡
When recapping finance 1; we learned about the time value of money: 𝑃𝑉 = 𝑡
(1+𝑟)
So, the present value (PV) is calculated by dividing a certain cashflow by 1 + r. Here r represents a
certain discount rate.
Furthermore, we learned about growing perpetuity. Perpetuities are securities or cash flows that
payout for an infinite amount of time. A growing perpetuity is a cash flow that grows at the same rate
𝐶
of growth forever. The present value of growing perpetuity is: 𝑃𝑉 = 𝑟 −𝑔
Here g is the growth rate of
the perpetuity.
Next we discussed annuity. It is an insurance contract between you and a provider company in which
you purchase a stream of payments to yourself over time to protect against outliving your income.
Annuity can be useful when you have a certain project (investment) that stops after a certain amount
of time. The present value of annuity is calculated with: 𝑃𝑉 = 𝐶 ×
1 ⎡1 − 1 ⎤
𝑟 ⎢ 𝑡 ⎥
⎣ (1+𝑟) ⎦
The dividend discount model is a more applied measure of the growing perpetuity. The formula for
𝐷𝑖𝑣1
this is: 𝑃0 = 𝑟𝑒−𝑔
So the price (at t=0) of a share is calculated by dividing the dividends at t=1
by the discount rate minus the growth rate. 𝑟𝑒 is the discount rate that belongs to the share. g
represents the growth factor in dividends in this case.
Finance cares about market values, not book values. So, when talking about debt, equity, etc. we are
always talking about the market values and not the book values. Furthermore, finance cares about
cash and not about profits. We can specify when certain cash flows are paid out and discount these
cash flows appropriately to get an insight about the future payouts.
We also talked about some investment decision rules in Finance 1. So, most importantly; take a
positive NPV project, because this will generate a positive cash flow in the future. Another investment
decision rule that has been talked about is the internal rate of return.
Lecture 1.2: Expected returns and volatility
What we are going to discuss here is how risk and return interact and how you can measure this risk
and return if you know the distribution of your returns.
If we would go back in time, how would we invest 100$? The options are:
- S&P500: industry leaders and among the largest firms traded on U.S. markets.
- Small stocks: securities traded on the NYSE with market capitalization in the bottom 20%
- World portfolio: international stocks from all the world’s major stock markets in North
America, Europe and Asia.
- Corporate bonds: rather than investing in equity (previous 3 options) we can also invest in
bonds. Here we have the option to invest in either corporate bonds (so lending money to
companies) or in the form of treasury bills (lending money to governments).
What we see is that Small stocks would have generated the highest return. The riskier the investment
the higher the return. This is in a nutshell the relation between risk and return.
Small stocks had the highest long-term returns, while t-bills (treasury bills) had the lowest long-term
returns. Small stocks had the largest fluctuations in price, while T-bills had the lowest.
We now want to know how to measure these returns and risks.
, - Returns: when investing in a stock returns come from 2 places: capital gain and dividends.
𝐷𝑖𝑣𝑡+1 𝑃𝑡+1−𝑃𝑡
To calculate the return we use: 𝑅𝑡+1 = 𝑃𝑡
+ 𝑃𝑡
𝐷𝑖𝑣𝑡+1 𝑃𝑡+1−𝑃𝑡
The first part of the formula ( 𝑃𝑡
) is called the dividend yield. The second part ( 𝑃𝑡
) is
called the capital gains.
When we assume that returns are a random variable then they typically follow a probability
distribution. When we know the probability distribution we can calculate what the expected return is
and the standard deviation. The standard deviation can tell us something about how spread out the
returns are and we can use this to get an indication about the interval in which most of the returns are.
To calculate the expected return we use: 𝐸(𝑅) = ∑ 𝑝𝑖 × 𝑅𝑖
Here 𝑝𝑖 is the probability of a certain return and 𝑅𝑖 is the return that belongs to that probability.
So how much do the actual returns (R ) deviate from the expected return (E(R)). The standard
deviation is the square root of this. The standard deviation represents the volatility in finance.
Volatility is mostly measured in % per annum. It is a measure of uncertainty about asset returns. It can
be scaled with different horizons: σ𝑇 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 = σ1 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝑇
- From daily to annually: σ𝑎𝑛𝑛𝑢𝑎𝑙 = σ𝑑𝑎𝑖𝑙𝑦 252 (there are 252 trading days a year)
- From monthly to annually: σ𝑎𝑛𝑛𝑢𝑎𝑙 = σ𝑚𝑜𝑛𝑡ℎ𝑙𝑦 12
- From weekly to annually: σ𝑎𝑛𝑛𝑢𝑎𝑙 = σ𝑤𝑒𝑒𝑘𝑙𝑦 52
Lecture 1.3: Returns and volatility from historical data
In the last part we discussed how to determine the volatility (standard deviation) when the distribution
was known. Now we are going to extend this to a situation where the distribution is not known. In this
case we use historical data to get an approximation of the distribution.
First we are interested in the expected return (E(R)), like in the last part. Therefore we will look at the
average. So, we take the average from a large number of historical observations and this average will
be the expected return we will use. In this case the following formula is used:
𝑅=
1
𝑇 (𝑅1 + 𝑅2 +... + 𝑅𝑇)
We are also interested in determining the variance, so we can determine the standard deviation and
therefore the volatility. We estimate the variance by also using the historical data, especially the
realised returns in the past. The formula we use when estimating the variance is:
𝑇 2
𝑉𝑎𝑟(𝑅) =
1
𝑇−1 (
∑ 𝑅𝑡 − 𝑅
𝑡=1
)
The T-1 represents the degrees of freedom.
Furthermore, the standard error is often mentioned. It does not tell us as much about the distribution
as the standard deviation, but it gives a good indication about how good the estimation is. When the
standard error is low, the estimation is quite good and therefore close to the actual outcome. The
standard error is calculated with:
𝑆𝐷(𝑅)
𝑆𝐸 = here SD(R) is the standard deviation of R
𝑇
With the standard error we can make confidence intervals. A confidence interval of 95% is
calculated with:
𝐻𝑖𝑠𝑡𝑜𝑟𝑖𝑐𝑎𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 ± 1. 96 × 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟
, Lecture 1.4: Portfolios
A portfolio is a collection of assets that you hold as an investor. The portfolio weight in total is 1 or
𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑖
100%. To calculate the weight of an asset we use: 𝑥𝑖 = 𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜
Then the return on the portfolio, Rp, is the weighted average of the returns on the investments in the
portfolio, where the weights correspond to portfolio weights. This is calculated with:
(
𝑅𝑃 = 𝑥1𝑅1 + 𝑥2𝑅2 +... + 𝑥𝑖𝑅1 = ∑ 𝑥𝑖 · 𝑅𝑖 ) [ ]
The expected return of a portfolio is the weighted average of the expected returns of the investments
within it. Calculated with:
[ ]
𝐸 𝑅𝑃 = ∑ 𝑥𝑖 · 𝐸 𝑅𝑖 [ ]
Lecture 1.5: Diversification
Diversification of a portfolio means that the portfolio consists of more assets. When a portfolio is
highly diversified it means that it consists of many assets (stocks). Diversification leads to a decrease
in volatility while return stays the same.
Diversification lowers risk in both directions; downward and upward. So, diversification also prevents a
big return. So the key principle of diversification is eliminating the individual risk in a portfolio.
Lecture 2: portfolios & CAPM
Lecture 2.1: Covariance correlation
To find the risk of a portfolio we must know the degree to which the assets’ returns move together. For
[ ][
this the covariance is used. It is calculated with: 𝐶𝑜𝑣(𝑅𝑖, 𝑅𝑗) = 𝐸 𝑅𝑖 − 𝐸(𝑅𝑖) 𝑅𝑗 − 𝐸(𝑅𝑗) ]
Historical covariance (estimated) between returns 𝑅𝑖, 𝑅𝑗 is calculated with:
𝑇
1
𝐶𝑜𝑣(𝑅𝑖, 𝑅𝑗) = 𝑇−1
∑ ⎡⎢𝑅𝑖,𝑡 − 𝑅𝑖⎤⎥⎡⎢𝑅𝑗,𝑡 − 𝑅𝑗⎤⎥
𝑡=1⎣ ⎦⎣ ⎦
If the covariance is positive, the two returns tend to move together. If the covariance is negative the
two returns tend to move in opposite directions.
Correlation is a measure of the common risk shared by stocks that does not depend on their
volatility. It is calculated with:
𝐶𝑜𝑣(𝑅𝑖,𝑅𝑗)
𝐶𝑜𝑟𝑟(𝑅𝑖, 𝑅𝑗) = ρ𝑖,𝑗 = 𝑆𝐷(𝑅𝑖) · 𝑆𝐷(𝑅𝑗)
The correlation between stocks will always be between -1 and +1. It measures a linear relationship
between 𝑅𝑖 and 𝑅𝑗. If 𝑅𝑖 changes by p%, we expect 𝑅𝑗 to changes by ρ · 𝑝%
𝑖,𝑗
A variance-covariance matrix shows in the diagonals (0.00055) the covariance
between A and B and the other two values are the variances of A (top left →
0.0013) and B (bottom right → 0.0007). If you want to calculate the correlation
from a variance-covariance matrix, note that the square root should be taken from
variances in the calculation.
To calculate the volatility (standard deviation) of a two stock portfolio we use the formula:
2 2
𝑉𝑎𝑟(𝑅𝑃) = 𝑥1 𝑉𝑎𝑟(𝑅1) + 𝑥2 𝑉𝑎𝑟(𝑅2) + 2𝑥1𝑥2 𝐶𝑜𝑣(𝑅1, 𝑅2)
Here 𝑅𝑃 is calculated as discussed in part 1.4 → 𝑅𝑃 = 𝑥 𝑅1 + 𝑥2𝑅2
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