Aktualisiert: Aug 2
Content: Many do often have the opinion that one is either more talented in the mathematical area or in the area of language. This article should show why you should be interested in mathematics or rather the financial world, regardless of what type you are. Since we all have access to the stock exchange, this article explains what the risk and return calculation is all about. If you want to invest your money and thereby increase it in the long term, it makes perfect sense to be on the stock exchange. Therefore try to understand what you are doing on the stock exchange and ensure this way your long-term success.
Introduction: We don't work so hard for nothing, but with the aim of investing the hard-earned money properly so that our money works for us. It does not matter whether we invest the money in capital investments, whether we make investments in tangible and financial assets or invest in companies and real estate, the only thing that counts is the return and the risk taken. We understand risk to mean the possibility of losing all or part of the cash invested, i.e. the risk of losing capital. In other words, risk is the possibility of a positive or negative deviation of a consequence of action from your expected value. Still curious?
What are historical annual returns?
The real historical returns of the different asset classes are important, because this enables you to develop an understanding of how the earnings potential and risk of these classes develop over time in the long term and how they will fluctuate. Whether stocks (blue chips - standard stocks, small stocks, value stocks, groth stocks, emerging market stocks), bonds (government bonds, bond certificates, corporate bonds, convertible bonds, inflation-linked bonds), real estate (residential real estate, commercial real estate), raw materials (oil, natural gas, coal) , Precious metals, agricultural goods, cotton, wood) or gold, using the average inflation-adjusted annual returns of the past few years, the range of fluctuation, i.e. volatility, is analyzed in relation to the average annual return, with costs and taxes not yet deducted.
For equities, the simple period return is the basis of all average return considerations. Average returns can be calculated time-weighted (geometric mean) or money-weighted (capital value method / IRR). The discrete equity returns can be converted into constant returns by being logarithmized.
For bonds, the yields are determined using the present value model. With a given market price, the yield on expiry IRR can be calculated. The sensitivity of a bond to changes in interest rates can be quantified using the duration or the modified duration.
So that our money can work for us, we ideally create a broad portfolio in which we take transaction costs, asset classes and the weighting of these into account. In this way, we can tailor the risk to be taken and the return achieved on our well-being and minimize the overall risk of the portfolio by diversifying, i.e. spreading the risk. The less the correlation between the different classes, the better. So can we draw an exciting conclusion from the historical figures? Well we already have experienced a global economic crisis, the oil crisis, two world wars, the dotcom crisis and also the financial crisis. So for now the tip can be given that with the current low interest rate phase, stocks or stock EFTs bring a good return.
Mathematical characteristics of historical returns and their calculation:
Big differences in the spread of the two time series
Span: max (ri) - min (ri)
Standard deviation = volatility = risk
The likelihood of a negative return on stocks is significantly greater than that on bonds
Example No.1 for calculating the risk and the arithmetic mean:
An investment A has achieved the following steady returns in the following years:
ri: 1995 -> 10% ; 1996 -> -4% ; 1997 -> 14% ; 1998 -> 8% ; 1999 -> -7%
We take the information from ri (returns) and r dash (arithmetic mean also called mean) and insert it into our formula of standard deviation, marked with the σ "sigma" to calculate the volatility and thus our risk.
Example No. 2 for calculating the constant return and its risk:
The following information is available on the UBS share price:
1995 => Market value end 19,52
1996 => Market value end 15,95 with dividend per year from 0,6
1997 => Market value end 17,94 with dividend per year from 0,65
1998 => Market value end 12,235 with dividend per year from 0,7
1999 => Market value end 12,225 with dividend per year from 0,3521
We are looking for r here, so we create an equation that can be solved for r. Basically you would expect the sum symbol, but because the dividends are different every year, this cannot be done. At Dt we add the different dividends in a first step and add the fraction in which we return the final capital. The calculator will display 2 values as solutions, but since 1.88 ... is unrealistic, the value for r is only 0.0755.
Example No. 3 for calculating historical stock returns:
The share price on January 1, 2019 was CHF 12.235 and on January 1, 20 CHF 12.225, the dividend in 2020 was CHF 0.3521. Calculate the historical return on shares:
If the return is to be calculated over several years, we use the calculation method using IRR:
Example No. 4 Calculation of historical equity returns over several years IRR:
The following information is given, calculate the return using IRR:
The share price was CHF 19.52 in 2015, CHF 15.95 in 2016, CHF 17.94 in 2017, CHF 12.235 in 2018 and CHF 12.225 in 2019, with dividends of CHF 0.6 in 2016, CHF 0.65 in 2017, CHF 0.7 in 2018 and CHF 0.3521 in 2019
With this solution variant, the dividend of 0.3521 and Kt was combined in the fourth year, since it makes no difference whether two separate fractions with the same denominator are added or whether they are combined into one fraction.
Example No. 5 Calculation of the multi-period return on shares - the geometric mean, through the Time Weighted Return TWR:
Here, all annual returns are first calculated and then geometrically averaged, as in the compound interest calculation.
We calculate annual returns using the formula:
We need the compounding factor for the geometric mean so that we can omit the -1 and only:
In a next step, the geometric mean is determined:
The return is -7.49%
Example No. 6 Calculation of the average return on shares with steady returns:
The constant return Rt can be calculated using the logarithm transformation from the simple return rt. We use the following formula:
the mean historical return r dash is the arithmetic mean of the steady returns:
To calculate the return using IRR:
Finally, the arithmetic mean is determined:
Mathematical characteristics of bonds - modified duration:
It becomes interesting if one can find out in practice more about the relative change in the bond price depending on a change in the market interest rate level. The modified duration "MD" indicates the percentage by which the bond price changes when the market interest rate level rm changes by one percentage point i.e. it measures the price effect triggered by a marginal change in interest rates and thus represents a kind of elasticity of the bond price from the market interest rate. The change in the market interest rate of
Delta rm results in a change in the present value of the Delta PV / PV obligation of approximately -MD * Delta RM.
Example No. 7 Calculation of modified duration for bonds
The purchase and buyback value is CHF 1,000 and the annual coupon is CHF 70.00, the modified duration is to be calculated. The market interest rate is at 2% over the next 5 years.
The sum is formed of all final values that have been multiplied by the year and this sum is multiplied by the purchase value so that we get the duration:
The modified duration can be calculated from this value, because you only have to:
The conclusion is: If the market interest rate increases by 0.5%, the purchase price will be reduced by 2.655%.
Mathematical characteristics of bonds - present value model:
If the nominal value corresponds to the initial value and also the repurchase value, then the coupon interest rate (Co / Nv) corresponds to the market interest rate. If the purchase value is smaller than the buyback value, which corresponds to the nominal value, the coupon interest rate is lower than the market interest rate. If it is the opposite, it is larger.
Example No. 8 Calculation of the initial value
The initial value is to be calculated if the redemption value is CHF 1,100 and the annual coupon CHF 70 is paid out over the next 5 years. The market interest rate is 2%.
Because the coupon is stable, we can use the sum sign to calculate the solution:
Conclusion: The purchase price must be less than 1,326.25 so that it pays off, since the yield is higher than the market interest rate. Tip: When calculating, don't forget, only the front part is calculated with the sum sign, the repayment value divided by 1.02 ^ 5 is added without the sum sign.
Example No. 9 Calculating the present value of all future payments
Present value = fair price
C0 = Coupon equal postpayments
RV = redemption value, does not have to be identical to NV, i.e. nominal value
rm = market interest rate / discount rate constant over the entire term
Calculate present value if the following is known: RV = 102, C0 = 7 , rm = 7,5% , n = 3
The present value model is a dynamic model, it also includes time. We use it to calculate what the highest purchase price may be, so that at least the market interest rate is given.
Mathematical characteristics of bonds - payment during the year:
This results in the new formula for calculating the coupon payment during the year:
M is now added, which represents the number of coupon payments per year. The annual coupon Co is divided by the number of periods m. The nominal market interest rm is divided by the number of periods m. And the number of years n is also counted with the number of interest-bearing periods n * m.
Example No. 10 Calculation of the initial value for coupon payments during the year
The initial value is to be calculated if the surrender value is CHF 1,100 and the annual coupon CHF 70 is paid out over the next 5 years. The market interest rate is 2%. It is billed quarterly.
Mathematical characteristics of bonds - yield on maturity:
The bond is bought today and is held until repayment, which means that all cash flows are known and we are interested in the IRR, ie the yield on maturity "Yield to Maturity". The return can be determined based on the present value formula by replacing K0 with the purchase price or stock exchange price and resolving it to r:
Example No. 11 Calculation of the yield on maturity
Enter everything in the brackets in the calculator and press enter with the solver function in the calculator. This gives the yield on the expiry, see above in the formula behind the equal, of 0.075776.
Mathematical characteristics of bonds - duration - average duration of capital commitment:
Unforeseen changes in interest rates have two opposing effects on the final value of an interest-bearing security, for example a bond. An interest rate hike leads to a lower present value, due to the reinvestment premise, future coupon payments receive higher interest, and this leads to a higher final value. The duration in which the market value of the bond returned to its original value after an increase in interest rates due to the reinvested coupons is called duration. The duration D represents a measure of the sensitivity of the bond value PV to changes in interest rates: the larger D, the more strongly PV reacts to changes in interest rates. D is calculated as the sum of the time-weighted discounted cash flows PVt based on the present value PV.
Mathematical characteristics of bonds - static return:
Example No. 12 - static return - same purchase and redemption value
The static return is to be calculated if the purchase and redemption value is CHF 1,000 and the annual coupon is CHF 70.00.
Example No. 13 - static return - different purchase and redemption values
The static return is to be calculated if the purchase price is CHF 1,000, the repayment value is CHF 1,100 and the annual coupon is CHF 70 over the next 5 years.
C0 = coupon with the same amount of subsequent payments
K0 = purchase price
RV = redemption value
n = (remaining) term
Mathematical characteristics of stocks - future yield calculation:
We take into account the probability of occurrence of future (single) period returns when calculating the expected returns E (r):
pi = Probability that ri will arrive
ri = Return of the scenario i
n = Number of scenarios
Example No. 14 - future return calculation for stocks
The expected value for share A should be calculated:
So if the estimated probability for the negative economic system is sought, we know that in total the estimated probability must always be 100%. The expected value is then calculated from the estimated probability and the estimated return by multiplying both values together.
That means we expect a negative return of 0.5%. For all those who do not understand how to get 0.5% if the result is 0.0051: percent means one hundred, and the 0.0051 is written in decimal numbers. That means that you take the 100, this has two zeros, so you have to move 2 zeros to the right with the decimal at the decimal number, so that 0.0051 -> 0.51, and that's when you move to 1 position the comma writes 0.5%.
Mathematical characteristics of bonds - static yield calculation:
If the purchase price is identical to the redemption value, these forms can be used:
Otherwise, if the purchase price and redemption value differ, this formula should be used:
rsimple = simple return
C0 = same high, subsequent payments -> coupon
K0 = purchase price
RV = redemption value
n = (remaining) term of the bond in years
Mathematical features of the calculation of the standard deviation of future returns:
The standard deviation is often used as a measure for the quantification of the risk. Here, too, the probability of future "ex ante" period returns is taken into account in the calculation. The formula is:
pi = Probability that ri will arrive
ri = Return of the scenario i
n = Number of scenarios
Example No. 15 Calculating the standard deviation of future returns:
Example No. 16 Risk calculation:
The return and the risk for the investment below must be calculated:
So the future risk is 0.014387 and our return is 0.51%.
Mathematical characteristics of the normal distribution of returns:
The distribution of steady returns can be described in a good approximation by a normal distribution. The known properties of the normal distribution also apply to returns, namely:
2/3 of the returns are within the range
approx. 95% of the returns are within the range
approx. 99% of the returns are within the range
The mean and standard deviation completely determine the distribution:
If x is normally distributed with mu and sigma then z = ((x-mu) / sigma) is standard normally distributed with mu = 0 and sigma = 1. We determine the quantiles with the Excel function = norminv (...) and = normvert (.. .). If we look for x1 and x2, we take x1 = norminv (alpha; mean; standard deviation) and for x2 we take = standnorminv (alpha) * standard deviation + mean. If x is given and we are looking for the alpha, we make for a1 = normvert (x; mean; standard deviation; true) and for a2 = standard normvert ((x-mean) / standard deviation). We differentiate between the frequency distribution in the ex-post analysis, the probability calculation with the ex-ante analysis and the normal distribution that enables the uncertainty to be quantified.
Mathematical characteristics of the shortfall risk:
The "shortfall risk" denotes the probability that a return is below a given "threshold return", ie the target return or, statistically speaking, the quantile of the normal distribution. In other words, it is the probability that a certain return, e.g. Falls below 10%. With the shortfall risk, we want to determine the probability, i.e. the area marked gray with the normal distribution, that the stock will fall below a certain rate of return. The marginal return from which the probability is to be calculated is referred to as the threshold return.
Example No. 17 Calculate normal distribution with mu, sigma and threshold return:
The mu is 8%, the sigma is 21% and the threshold return is -10%. We calculate for 5 years.
First we add the 5 years to the values:
Using the standard normal distribution using the Z transformation, we calculate by entering the results in the formula:
The interpretation is:
So we take the received value of -1,916 and calculate the cdf of the standard normal distribution from this value. The Excel function = standard norm (-1.916) can be used for this. This gives us the value of 0.028, which indicates that there is a 2.8% probability that the threshold return will fall below -10%. Alternatively, the calculation of the probability can also be solved with the calculator by calculating the probability:
The window can be opened in the menu, Statistics area, Distribution area, Normal Cdf. For the upper bound I entered -9E999 for the calculations instead of the infinite character, otherwise an error was displayed. The upper limit was -50, the must be 40 and the sigma was 46.96.
Instead of a certain expected return r *, the nominal capital preservation r * = 0, the inflation rate (real capital preservation) or the risk-free interest rate are used as the threshold.
Mathematical features of Value at Risk:
While the shortfall risk describes the probability with which a certain loss (return) is achieved, the value at risk describes how high the loss, amount of money, is with a certain probability in a certain time horizon.
Example No. 18 the Value at Risk is to be calculated:
Continuous calculation method of the value at risk:
A portfolio with a value of 100,000 is invested in the Pictet-Rätzer share index for 1 year with mu = 7.25% and sigma = 19.3%. How big is the value at risk at a one-sided confidence level of 95%?
We first calculate the 5% quantile of the standard normal distribution by doing the following:
or using the calculator:
Calculator: Menu statistics, distributions, inverse normal distribution => the area is 0.05, the mean is 0.0725 and the standard deviation is 0.193, we get the value of -0.244957 as a solution, so the marginal return must not fall below this value. Then we calculate the amount of money:
I.e. we made a maximum loss of CHF 21,726.20 at a 95% level, so with a probability of 95% a maximum loss of 21,726.20 was made.
Linear calculation method of the value at risk (linearized approximation to calculate the value at risk):
We gave the risk position, i.e. our portfolio, with 100,000. The yield is 7.25% and volatility is 19.3%, both on an annual basis. The liquidation period is 1 year. The 5% quantile of the standard normal distribution, i.e. the number of standard deviations, is -1.64. We now also know that the critical annual return is -0.244.
After 1 year and in 5% of all cases, the value of the portfolio is:
1 - 0.244 = 0.7560 and thus Value at Risk 24'400, because we have 100'000 * 0.7560 = 75'600 thus results in 100'000 - 75'600 = 24'400.
Example No. 19 Calculate the probability that the level of the index will decrease
How likely is the Pictet-Rätzer index to be below 18,000 monetary units in one year? The current level is 20'322 with a return of 7.59% and a risk of 18.7%.
In a first step, we calculate the threshold return using the formula of constant interest:
Next, the probability is calculated:
It is exciting to know that with confidence a higher confidence level leads to a higher value at risk. In practice, 99% or 95% is often used as a confidence level. A longer time horizon leads to a greater value at risk. In practice, 1 or 10 days are used in practice, during which 1 year is usually taken in asset management. With value at risk, no statement is made about the expected loss, only the statement about the threshold.
Note: The risk, i.e. the standard distribution, must be coordinated with the time horizon. Important to know:
The expected value grows in proportion to the time
The variance grows in proportion to time
The standard deviation grows in proportion to the root of time
It follows that the risk, i.e. our volatility, grows more slowly than the expected value.
Example No. 20 Calculating the daily risk
We gave: return of 7.25% and a risk of 19.3%
How the time horizons are calculated:
So if we have to calculate for a day, we take the value of one year, the 7.25% -> 0.0725, and divide this by 365 days, which is one year, and thus get 0.000199. The calculation for the daily risk can either be done via the variance or as the root of the days:
We square to get the variance and then calculate it down to one day. Then we take the root of the result to get the risk.