Question 1
a) Explain with examples, how you would measure the risk of a single asset.
When an individual makes an investment, for example in a stock, no one knows accurately how much it would sell in the future so there is a certain risk accompanied when making an investment. Defining risk, it is the chance of an asset or a portfolio of assets to suffer a financial loss ( 2003). Risk can also be defined as the variability of returns associated with a given asset. Basing on most examples in financial books and lectures, return can be defined as the total gain or loss on an investment and can be calculated as:
(Present market value – value at the time of purchase) / (value at the time of purchase)
For example you invested $20,000 in a certificate of deposit. After 5 years, the money in which you invested now amounts to $25,000. Computing for the return: ($25,000 - $2000)/ ($20,000) x 100 = 25%
Assets may it be real or financial which have a greater chance of loss are considered more risky than those with a lower chance of loss (, 2003). Investing in a business, investing in stocks or bond poses greater risks than investments made in banks such as CDs. Because of these uncertainties, risk of an asset should be measured. One way of measuring risk is by using historical data or the past rate of return of an asset which represent the possible outcomes. Looking at historical data, one can assess which data are the most probable to occur which can be used in measuring risk. For example during the last ten years, a company’s asset has an average annual return of 13.1%. This figure can be the basis of measuring the risk of that asset which is commonly measured with the use of standard deviation.
Standard deviation, σk, is the common indicator of an asset’s risk whose formula can be written as:
(1)
Where: ki = return for the ith outcome
Ќ = the most likely return of an asset or the average return
Pi = probability of occurrence of the ith outcome
For example:
Based on the historical data of an asset, its highest rate of return was 17% while the lowest was 13% making an average return of 15% which is also the return most likely to occur. Calculating its standard deviation using the table below with the probability of the lowest and highest return being 25% each and 50% for the average return:
i
ki
Ќ
ki – Ќ
(ki – Ќ)2
Pi
Pi (ki – Ќ)2
1
13%
15%
-2%
4
0.25
1%
2
15%
15%
0
0
0.50
0
3
17%
15%
2%
4
0.25
1%
Using the formula (1) for standard deviation:
= = 1.41%
(There is a 1.41% chance that this asset will suffer financial loss)
One can also compare the risk of different assets by calculating their coefficient of variation, CV which is standard deviation divided by the expected or the most likely return of an asset: CV = σ/ Ќ. Coefficient of variation is a relative measure of dispersion or the measure of how risk will vary since the computed risk is only an assumption.
For example: Asset A has an expected return of 15% with a computed standard deviation of 9% while asset B has an expected return of 20% with a standard deviation or risk of 11%. Computing for each asset’s CV:
Asset A: CV = 9% / 15% = 0.6 Asset B: CV = 11% / 20% = 0.55
b) Explain with examples, how you would measure the risk of a portfolio.
A portfolio is basically a combination or collection of assets ( 2003) usually diversified. Most investors choose to invest in a portfolio than in a single asset alone in order to vary the risk inherent in holding a single asset. When an investor invested only in a single asset, once that asset suffer a loss, the investor has no investment left while when an investor has a portfolio, once one of his asset suffer a loss, he still has other assets that can still benefit him.
Standard deviation also indicates the risk of a portfolio by modifying the formula above. The formula below is used in measuring the risk posed in a portfolio:
(2)
Where: n= number of outcomes considered.
It is noticeable in the formula that Pi was eliminated when the probabilities of occurrence in all outcomes are assumed equal or is not given.
Example:
A portfolio consists of three assets, 50% of asset A; 25% of asset B; and 25% of asset C. Each asset is forecasted to have the returns in the next five years as shown on the table:
Year
A
B
C
Portfolio (50%A+25%B+25%C)
2006
8%
16%
8%
10%
2007
10
14
10
11
2008
12
12
12
12
2009
14
10
14
13
2010
16
8
16
14
Computed Standard Deviation of each asset
3.16%
3.16%
3.16%
Calculating for the expected value of return:
k = (10%+ 11 + 12+ 13 + 14)/ 5 = 10%
Calculating for the standard deviation using formula (2):
σ = = 2.45%
By diversifying the portfolio, the overall risk was reduced especially when a portfolio is combinations of more risky and less risky assets. However, there is what is called systematic risk (, 2003) which is the variability in stocks primarily due to interest rates, inflation or business cycle.
Question 2: Explain with examples, 4 ways of common stock valuation.
The primary reason why stock valuation is important is for stockholders to determine the dividend they will get from their investments. Common stocks are also valuated in order to determine when is the best time to buy or sell stocks. There are many ways of common stock valuation depending on different cases. One way of common stock valuation is with the used of the dividend discount model, DDM. DDM values a share of stock as the sum of all expected future dividend payments, where the dividends are adjusted for risk and the time value for money ( 2006). The formula for this method is:
(Equation 1) V = D1/(1+k) + D2/(1+k)2 + D3/(1+k)3 + .....+ Dt /(1+k)t
Where: V = the value of the future dividend or the value of the stock
Dn = dividend to be paid n years from now
k = the appropriate risk-adjusted discount rate or the required
return
t= the number of years the valuation is considered
In cases of zero growth stocks, dividend to be paid every year is constant thus: D1 = D2 = D3 and so on, and since the dividend is always the same, the stock can be viewed as an ordinary perpetuity if to be continued indefinitely given by V = D/k.
Examples:
1. A company pays three annual dividend of $10 per share for three consecutive years with a discount rate of 15%. Compute for the value of the stock.
Solution:
D= D1 = D2 = D3 = $10
Using equation 1:
V = $10/ 1.15 + $10/ (1.15) 2 + $10/ (1.15) 3
V = $22.83
2. Suppose a company has policy of paying a per-share dividend of $10 every year with a 15% required return, compute for the value of the stock.
Solution: using the formula V = D/k
V = $10/ 0.15 = $66.67
3. A company pays three annual dividends of $10, $20 and $30 for three consecutive years with an expected return of 15%. Compute for the value of the stock.
Solution:
V = $10/ (1.15) + $20/ (1.15) 2 + $30/ (1.15) 3
V = $43.54
Another way of stock valuation is by assuming that dividends will grow annually at a constant growth rate, g. This model is called the constant growth rate model. For example, a dividend just paid is D0= $100 has a steady growth rate of 10% will yield a dividend D1 of D0 x (1+g) after 1 year, and a dividend, D2 = D0 x (1+g) x (1+g) or D0 x (1+g) 2 , thus after t years into the future:
(Equation 2) Dt = D0 x (1+g) t
The value of the stock then can be given by the formula:
V = [D0 x (1+g)] / (1+k) + [D0 x (1+g) 2]/ (1+k) 2 + [D0 x (1+g) 3]/ (1+k) 3 ..+...
But as long as the growth rate, g, is less than the discount rate, k, stock value can be computed as:
V0= [D0 x (1+g)]/ (k-g) or simply
(Equation 3) V0 = D1 / (k-g)
To get the value of the stock with a constant dividend growth at any point in time the formula below is used:
(Equation 4) Vt = Dt x (1+g)/ (k-g) or Vt = Dt+1 / (k-g)
Example:
It is required to determine the value of the stock in five years, V5, given the dividend just paid which is $10 with a growth rate of 5% per year and an expected return of 15%.
Solution:
Determine first the dividend after 5 years which is denoted by D5 using equation 2.
D5 = $10 x (1 + .05)5 = $12.76
V5 = [$12.76 x (1+.05)]/ (0.15 – 0.05)
V5 = $133.98
It is noticeable in the formula that the growth rate should always be less than the expected return otherwise, the value of the stock would be negative or if they are equal, we can get an undefined value.
Another way of valuating stocks is by looking at the expected rate of return, k. Using equation 3 and rearranging it, we can get the formula for k which is:
(Equation 5) k = (D1 / V0) + g
The growth rate, g, is also the capital gains yield or the rate at which the value of the investment grow (Anonymous, 2006).
For example:
The present value, V0, of a stock is $10 and the next dividend, D1 will be $1 per share. The dividend is expected to grow by 20% indefinitely. Calculate for the value of the stock after one year, V1.
Solution:
Calculate first for the expected return, k using equation 5:
k = (1/ 10) + 0.20 = 0.30 = 30%
Then calculate for V1 using equation 4:
V1 = $1 x (1+ 0.20)/ (0.30- 0.20)
V1 = $12
The value of the stock after one year will gain $2 which is actually 20% (the growth rate) of the present value which is $10.
Question 3: Explain with examples, 4 ways of calculating the cost of capital for a business organization.
The cost of capital for a business is the rate of return demanded by the market for the portfolio of assets held by the business organization (, 2006). Most firms obtain capital from different types and sources of financing such as bond, equity, long-term debt and others.
There are basically three ways of calculating WACC. The first way is by measuring the capital structure which is basically the calculation of the proportion that debt, equity and other financing source from which the total capital was obtained. The capital structure can be measured by using either book value weights from the balance sheet, or by using market value weights (2006). Using the book value weight, which easier than the other method, the various equity accounts such as common stock, additional paid in capital and retained earnings, on the balance sheet were added.
For example a company has a book value of debt amounting to $25million and the book value of equity is $43million. Calculating for the capital structure:
$25million + $43million = $68 million
Debt: $25million/ $68 million = 36.76%
Equity: $43million/ $68 million = 63.24%
Calculation of the required rates of return is also needed to determine the cost of capital. As stated above, cost of capital is the rate of return demanded by the market. One way of determining required rates of return is by using the current rate of new debt and adjusting for taxes (, 2006). Another way is by using the capital asset pricing model, CAPM which estimates the cost of capital with the formula: r = rf + B( rm - rf )
where: r = the required rate of return - the cost of equity
rf = the risk free interest rate
B = the Beta for the investment or the systematic risk
rm = the expected rate of return on the market portfolio.
For example: a company has a beta coefficient of 1.5, risk-free rate of 10% and an expected return of 15%. Using the formula:
r = 10% + 1.5 (15% - 10%)
r = 17.5%
Calculating for the weighted average cost of capital, WACC, is another way of determining the cost of capital. WACC is given by:
WACC = (re x Pe) + (rd x Pd)
Where: re = the rate of equity or cost of equity
Pe = the proportion of equity
Rd = the rate or cost of debt
Pd = the proportion of debt
For example: the capital of a company is structured as follows given that the rate of return of equity is 17.5% and the rate of return of debt is 8%:
Debt: $25million/ $68 million = 36.76%
Equity: $43million/ $68 million = 63.24%
Solution:
WACC = 0.3676 x 8% + 0.6324 x 17.5%
WACC = 14%
References:
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