Financial Management
Financial Management is a specific area of finance dealing with the financial decisions and the tools as well as analysis used to make these decisions. It has the goal to enhance corporate value by ensuring that return on capital exceeds cost of capital without taking excessive financial risks (2006).
Financial risks is defined as the unexpected variability or volatility returns, and thus includes both potential worse than expected as well as better than expected returns (2006).
Risk of a Single Asset
There are some ways to measure the risk of a single asset. This includes the probability of distribution, expected return, standard deviation, normal distribution, and coefficient of variation.
Probability Distribution
Assessing the risk of an asset requires having some sense for the range of possible outcomes. Predicting a range of outcomes and assigning to them different probabilities can give further insight into risk. Using a probability distribution, we can model different outcomes.
For example, the probability distribution of the results of throwing two dice is shown in the table below.
Table 1. Probability Distribution of Throwing Two Dices
Result
Probability
2
1/36 = 2.8%
3
2/36 = 5.6%
4
3/36 = 8.3%
5
4/36 = 11.1%
6
5/36 = 13.9%
7
6/36 = 16.7%
8
5/36 = 13.9%
9
4/36 = 11.1%
10
3/36 = 8.3%
11
2/36 = 5.6%
12
1/36 = 2.8%
Total
36/36 = 100%
The most likely throw is a 7 – but it only happens one-sixth of the time. Possible outcomes range from 2 to 12, although the likelihood of throwing a 2 or 12 is significantly lower than throwing a 7. Even though 7 would be the most frequent throw, you might throw less. In fact, there is a 41.6% chance that you will throw less than a 7.
Expected Return
Estimation of the value of an investment, including the change in price and any payments or dividends, calculated from a probability distribution curve of all possible rates of return. In general, if an asset is risky, the expected return will be the risk-free rate of return plus a certain risk premium.
Suppose the number you throw represents the return, and suppose you throw the dice thousands of times. We can compute the expected return for each possible result and then sum these results.
Table 2. Expected Return for Each Possible Result
Return
Probability
Weighted expected return
1/36 = 2.8%
.056
2/36 = 5.6%
.168
3/36 = 8.3%
.252
4/36 = 11.1%
.555
5/36 = 13.9%
.833
6/36 = 16.7%
.167
5/36 = 13.9%
.111
4/36 = 11.1%
.000
3/36 = 8.3%
.833
2/36 = 5.6%
.611
1/36 = 2.8%
.333
Total
36/36 = 100%
.00
The expected return is .00.
The average of a probability distribution of possible returns, calculated by using the following formula:
How do you calculate the average of a probability distribution? As denoted by the above formula, simply take the probability of each possible return outcome and multiply it by the return outcome itself.
From the above table, .00 is calculated using the above formula shown below.
= (0.028) (2) + (0.056) (3) + (0.083) (4) + (0.111) (5) + (0.139) (6) + (0.167) (7) + (0.139) (8) + (0.111) (9) + (0.083) (10) + (0.056) (11) + (0.028) (12)
= 7.00
Standard Deviation
Risk can be quantified. The most common measure of risk is the standard deviation — a numerical measure of the dispersion around the expected value. Standard deviation (designated by s ) is calculated for our two dice probability distributions as: compute first the expected return; then square the variance of each return from the expected return; multiply this by the probability; sum these weighted values; and finally calculate the square root of this sum (the standard deviation). This gives a numerical indication of how far the returns are dispersed from the average. There are various measures of risk: beta, standard deviation, R-squared.
Return
Square of variance
Probability
Weighted square
(7 – 2)2 = 25
1/36 = .028
0.694
(7 – 3)2 = 16
2/36 = .056
0.889
(7 – 4)2 = 9
3/36 = .083
0.750
(7 – 5)2 = 4
4/36 = .111
0.444
(7 – 6)2 = 1
5/36 = .139
0.139
(7 – 7)2 = 0
6/36 = .167
0.000
(7 – 8)2 = 1
5/36 = .139
0.139
(7 – 9)2 = 4
4/36 = .111
0.444
(7 – 10)2 = 9
3/36 = .083
0.750
(7 – 11)2 = 16
2/36 = .056
0.889
(7 – 12)2 = 25
1/36 = .028
0.694
Statistics:
Sum of weighted squares
5.833
Standard deviation
(square root of sum)
2.415
Normal Distribution
The standard normal distribution is the normal distribution with a mean of zero and a standard deviation of one. It is often called the bell curve because the graph of its probability density resembles a bell.
If the dice-throwing distribution were normally distributed (a classic “bell curve”), a standard deviation of 2.415 would indicate the following
- 38.3% of all returns are within one-half standard deviation of the expected return — in our example, from .793 to .207 (within .207 of .00)
- 68.3% of all returns are within one standard deviation of the average return — in our example, from .585 to .415 (within .415 of .00)
- 95.4% of the returns are within two standard deviations of the average return — in our example, from range from .170 to .830 (within .830 of .00)
The larger the standard deviation, the greater the dispersion and the greater the risk. In our example of the weird dice with only 3s and 4s, the standard deviation is 0.707.
Coefficient of Variation
The coefficient of variation is an attribute of a distribution: its standard deviation divided by its mean.
A statistical measure of the dispersion of data points in a data series around the mean. It is calculated as follows:
The coefficient of variation represents the ratio of the standard deviation to the mean, and it is a useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from each other.
For example below, suppose you are presented with two investment strategies.
Plan A
Plan B
Expected return
15%
20%
Standard deviation
5%
6%
Coefficient of variation
.333
.300
Plan A offers a lower expected return, but with less variability, than Plan B. But Plan B has less relative risk. The coefficient of variation — that is, the ratio of variability to return — is higher for Plan A (5/15 = .33) compared to Plan B (6/20 = .30). There is greater relative risk that returns will deviate from the expected return under Plan A. Therefore, the lower the coefficient of variation, the better is the risk-return tradeoff.
Risk Measures of a Portfolio
Risk is the chance that an investment’s actual return will be different than expected. This includes the possibility of losing some or all of the original investment. There are different types of risk. These include credit/default risk, country risk, foreign exchange risk, interest rate risk, political risk and market risk.
In Markowitz Portfolio Theory, it demonstrated that the variance of the rate of return is a meaningful measure of portfolio risk under reasonable assumptions. This theory derives the expected rate of return for a portfolio of assets and an expected risk measure. The portfolio variance formula shows how effectively diversify a portfolio ( 1952).
Standard Deviation
In finance, standard deviation is applied to the annual rate of return of an investment to measure the investment’s volatility (risk).
The portfolio standard deviation is shown below:
StdDev(r) = [1/n * (ri - rave)2]½
where the terms ri are actual values of the investment’s annual rate of return, taken over several years, n is the number of values of ri used, and rave is the average value of the ri.
For instance, the S&P 500, which has an 11.68% annualized return over the past 10 years with standard deviation of returns of 16.11%. If returns are “normally” distributed, this means the average return was 11.68%.
For one standard deviation, the range of possible returns is as low as -4.43% and as high as 27.79% (See Figure 1). One standard deviation describes approximately 68% of the individual observations in a “normal” distribution.
For wider range of possible outcomes two standard deviations could be use. This covers approximately 95% of possible returns. At two standard deviations the range of possible returns widens dramatically to -20.54% on the low end and 43.90% on the high end (See Figure 2). The return on the S&P 500 fell in the one standard deviation range only 3 times in the past 10 years. In every other instance it fell outside the range, including a -22.10% return in 2002 and a 37.58% return in 1995.
Figure 1 (1 Standard Deviation)
Figure 2 (2 Standard Deviations)
Value At Risk
Another popular measure of risk in a portfolio that is relatively easy to understand is the Value At Risk (VAR). It measures how much you can lose under different possible scenarios. By assuming investors care about the odds of a really big loss, VAR answers the question, “What is my worst-case scenario?” or “How much could I lose in a really bad month?” But Value at Risk is only concerned with downside volatility. VAR also is measured in dollars. This is an absolute measure, not a relative one.
VaR, with the parameters: holding period x days; confidence level y%, measures what will be the maximum loss (i. e. decrease in portfolio market value) over x days, if one assumes that the x-days period will not be one of the (100 − y)% x-days periods that are the worst under normal conditions. One can also define VaR as a lower y% quantile of a profit/loss probability distribution, i.e., it is a best outcome from a set of bad outcomes on a bad day ( 2006).
For example, consider a trading portfolio. Its market value in US dollars today is known, but its market value tomorrow is not known. The investment bank holding that portfolio might report that its portfolio has a 1-day VaR of million at the 95% confidence level. This implies that (provided usual conditions will prevail over the 1 day) the bank can expect that, with a probability of 95%, the value of its portfolio will decrease by 5 million or less during 1 day, or in other words: it can expect that with a probability of 5% (i. e. 100%-95%) the value of its portfolio will decrease by more than 5 million during 1 day. Stated yet differently, the bank can expect that the value of its portfolio will decrease by 5 million or less on 95 out of 100 usual trading days, in other words by more than 5 million on 5 out of every 100 usual trading days.
There are three methods of calculating VAR. These are the historical method, the variance-covariance method and the Monte Carlo simulation.
Historical Method
The historical method simply re-organizes actual historical returns, putting them in order from worst to best. It then assumes that history will repeat itself, from a risk perspective.
Take the example of the Nasdaq 100 Index which trades under the ticker QQQ. The QQQ is a very popular index of the largest non-financial stocks that trade on the Nasdaq exchange.
The QQQ started trading in Mar 1999, and if we calculate each daily return, we produce a rich data set of almost 1,400 points. Put them in a histogram that compares the frequency of return buckets. At the highest point of the histogram (the highest bar), there were more than 250 days when the daily return was between 0% and 1%. At the far right, you can barely see a tiny bar at 13%; it represents the one single day (in Jan 2000) within a period of five-plus years when the daily return for the QQQ was a stunning 12.4%.
The red bars that compose the “left tail” of the histogram. These are the lowest 5% of daily returns. The red bars run from daily losses of 4% to 8%. With 95% confidence, we expect that our worst daily loss will not exceed 4%. If we invest 0, we are 95% confident that our worst daily loss will not exceed (0 x -4%).
The Variance-Covariance Method
In this method, it is assumed that stock returns are normally distributed. It requires the two factors – an expected (or average) return and a standard deviation - which allow us to plot a normal distribution curve.
Below, the normal curve plotted against the same actual return data:
The idea behind the variance-covariance is similar to the ideas behind the historical method – except that normal curve is used in which the worst 5% and 1% is automatically known on the curve. They are a function of our desired confidence and the standard deviation ():
The blue curve above plotted is based on the actual daily standard deviation of the QQQ, which is 2.64%. The average daily return is close to zero, so it is assumed that zero is the average return for illustrative purposes. Below are the results of plugging the actual standard deviation into the formulas above:
Monte Carlo Simulation
This method involves in developing a model for future stock price returns and running multiple hypothetical trials through the model. A Monte Carlo simulation refers to any method that randomly generates trials, but by itself does not tell us anything about the underlying methodology.
Monte Carlo simulation was run on the QQQ based on its historical trading pattern. 100 trials were conducted. Below is the result arranged into a histogram. In this graph, monthly returns are being displayed.
100 hypothetical trials of monthly returns was run for the QQQ. Among them, two outcomes were between -15% and -20%; and three were between 20% and 25%. That means the worst five outcomes (that is, the worst 5%) were less than -15%. The Monte Carlo simulation therefore leads to the following VAR-type conclusion: with 95% confidence, we do not expect to lose more than 15% during any given time.
Common Stock Valuation
A share of common stock is quite literally a share in the business, a partial claim to ownership of the firm. Owning a share of common stock provides a number of rights and privileges. These include sharing in the income of the firm, exercising a voice in the management of the firm, and holding a claim on the assets of the firm.
Gordon Dividend Valuation Model (Constant Growth Stock Valuation)
Common stock is not so easy to value. The cash flows are not stable or easily identified. One simple model that is sometimes used to value common stock is the Gordon Dividend Valuation Model
The Gordon Growth Formula, also known as The Constant Growth Formula assumes that a company grows at a constant rate forever. This, by the way, is impossible. I mean, it can’t grow forever. You know, if a company doubles in size every 5 years, pretty soon every single person in the world is their customer and then they can’t grow at that rate anymore. (because the world population isn’t doubling ever 5 years).
A constant growth stock is a stock whose dividends are expected to grow at a constant rate in the foreseeable future. This condition fits many established firms, which tend to grow over the long run at the same rate as the economy, fairly well. Assuming that a company has a constant growth rate, we can use the following equation to get its value.
where
- P0 = the stock price at time 0,
- D0 = the current dividend,
- D1 = the next dividend (i.e., at time 1),
- g = the growth rate in dividends, and
- r = the required return on the stock, and
- g < r.
Constant Growth Stock Valuation Example
Find the stock price given that the current dividend is per share, dividends are expected to grow at a rate of 6% in the foreseeable future, and the required return is 12%.
Solution:
To get a 12% rate of return on our money assuming that the company will grow forever at 6% per year, then we would be willing to pay .33 for this stock.
Constant Growth presents a more general approach which allows for the dividends/growth rates during the period of rapid growth to be forecast. Then, it assumes that dividends will grow from that point on at a constant rate which reflects the long-term growth rate in the economy.
Nonconstant Growth Stock Valuation
Stocks which are experiencing the above pattern of growth are called nonconstant, supernormal, or erratic growth stocks.
The value of a nonconstant growth stock can be determined using the following equation:
where
- P0 = the stock price at time 0,
- Dt = the expected dividend at time t,
- T = the number of years of nonconstant growth,
- gc = the long-term constant growth rate in dividends, and
- r = the required return on the stock, and
- gc < r.
Nonconstant Growth Stock Valuation Example
The current dividend on a stock is per share and investors require a rate of return of 12%. Dividends are expected to grow at a rate of 20% per year over the next three years and then at a rate of 5% per year from that point on. Find the price of the stock.
Solution:
There are 3 years of nonconstant growth, thus, T = 3. Before substituting into the formula given above it is necessary to calculate the expected dividends for years 1 through 4 using the provided growth rates.
Common Stock Valuation—Single Holding Period
For an investor holding a common stock for only 1 year, the value of the stock should equal the present value of both the expected dividend to be received in 1 year, D1, and the anticipated market price of the share at year end, P1. If kcs represents a common stockholder’s required rate of return, the value of the security, Vcs, would be
Example
Suppose an investor is contemplating the purchase of RMI common stock at the beginning of this year. The dividend at year end is expected to be .64, and the market price by the end of the year is projected to be . If the investor’s required rate of return is 18 percent, the value of the security would be
Valuation is a three-step process. First, estimate the expected future cash flows from common stock ownership (a .64 dividend and a end-of-year expected share price). Second, determine the investor’s required rate of return based on the riskiness of the expected cash flows (assumed to be 18 percent). Finally, discount the expected dividend and end-of-year share price back to the present at the investor’s required rate of return. The result is .03.
Therefore, the intrinsic value of a common stock, like preferred stock, is the present value of all future dividends. And the same problem with preferred stock: It is hard to value cash flows that continue in perpetuity. Thus, we must make some assumptions about the expected growth of future dividends.
Common Stock Valuation—Multiple Holding Periods
This model is an equation used to value stock that has no maturity date, but continues in perpetuity (or as long as the firm exists). Common stock has no maturity date and is frequently held for many years, a multiple-holding-period valuation model is needed. The general common stock valuation model can be defined in the equation as follows:
The equation indicates that discounting the dividend at the end of the first year, D1, back 1 year; the dividend in the second year, D2, back 2 years; the dividend in the nth year back n years; and the dividend in infinity back an infinite number of years. The required rate of return is kcs. In using this equation, note that the value of the stock is established at the beginning of the year, for instance January 1, 2000. The most recent past dividend, D0, would have been paid the previous day, December 31, 1999. Thus, if we purchased the stock on January 1, the first dividend would be received in 12 months, on December 31, 2000, which is represented by D1.
This equation can be reduced to a much more manageable form if dividends grow each year at a constant rate, g. The constant-growth common stock valuation equation may be presented as follows.
The intrinsic value (present value) of a share of common stock whose dividends grow at a constant annual rate can be calculated using equation above. Although the interpretation of this equation may not be intuitively obvious, simply remember that it solves for the present value of the future dividend stream growing at a rate, g, to infinity, assuming that kcs is greater than g.
Example:
The valuation of a share of common stock that paid a dividend at the end of the last year and is expected to pay a cash dividend every year from now to infinity. Each year, the dividends are expected to grow at a rate of 10 percent. Based on an assessment of the riskiness of the common stock, the investor’s required rate of return is 15 percent. Using this information, we would compute the value of the common stock as follows:
Because the dividend was paid last year (actually, yesterday), we must compute the next dividend to be received, that is, D1, where
Ways of Calculating the Cost of Capital for a Business Organization
Weighted Average Cost of Capital
A calculation of a firm’s cost of capital in which each category of capital is proportionately weighted. All capital sources - common stock, preferred stock, bonds and any other long-term debt - are included in a WACC calculation.
WACC is calculated by multiplying the cost of each capital component by its proportional weight and then summing:
Where:
Re = cost of equity
Rd = cost of debt
E = market value of the firm’s equity
D = market value of the firm’s debt
V = E + D
E/V = percentage of financing that is equity
D/V = percentage of financing that is debt
Tc = corporate tax rate
Example
You may own ,000 in a money market fund that has an expected yearly return of 6%. You also may own ,000 of a preferred stock with an expected return of 8%. And you also may own ,000 market value of a common stock with an expected return of 10%. The expected weighted average return of your 0,000 (in total) investment portfolio equals:
Exp. Port. = (,000 x .06) + (,000 x .08) + (,000 x .10) = ,400 = 9.4%
Return (0,000) 0,000
Capital Pricing Asset Model
A model that describes the relationship between risk and expected return and that is used in the pricing of risky securities.
The general idea behind CAPM is that investors need to be compensated in two ways: time value of money and risk. The time value of money is represented by the risk-free (rf) rate in the formula and compensates the investors for placing money in any investment over a period of time. The other half of the formula represents risk and calculates the amount of compensation the investor needs for taking on additional risk. This is calculated by taking a risk measure (beta) that compares the returns of the asset to the market over a period of time and to the market premium (Rm-rf).
For example, the current risk free-rate is 5%, and the S&P 500 is expected to return to 12% next year. You are interested in determining the return that Joe’s Oyster Bar Inc (JOB) will have next year. You have determined that its beta value is 1.9. The overall stock market has a beta of 1.0, so JOB’s beta of 1.9 tells us that it carries more risk than the overall market; this extra risk means that we should expect a higher potential return than the 12% of the S&P 500. We can calculate this as the following:
Required (or expected) Return =
5% + (12% – 5%)*1.9
Required (or expected) Return =
18.3%
What CAPM tells us is that Joe’s Oyster Bar has a required rate of return of 18.3%. So, if you invest in JOB, you should be getting at least 18.3% return on your investment. If you don’t think that JOB will produce those kinds of returns for you, then you should consider investing in a different company.
Discounted Cash Flow Model (DCF)
Discounted Cash Flow (DCF implies that the rate of return is a function of the growth of expected cash flows from dividends, which are considered to have a constant rate of growth.
The required rate of return (k) under the DCF model is calculated as follows:
k= (D/P)*(1+g)+ g
where
D is the dividend paid in the current year
P is the current share price
g is the expected dividend growth rate
Now we apply the formula to Peter’s Shortline Rail Company:
D = .96
P = .75
g = 3.00%
Therefore,
k = (1.96/29.75)*(1+0.0300) + 0.0300
k = 0.0679 + 0.0300
k = 0.0979
k = 9.79%
The return on common equity for Peter’s Shortline Rail Company is calculated to be 9.79%.
Credit:ivythesis.typepad.com
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