Corporate Finance: Essay Quiz Value & Beta for Common Stock and Annuity

The Innovate Corporation has just invented a cheap form of energy. Investors with a 20% required rate of return expect this company to grow at an accelerated rate of 40% per year for the next 3 years and then continue at a constant growth rate of 15% per year. What value would the investors be willing to pay for the common stock of the most recent annual dividend was RM3.00 per share?

Answer: 

To determine the value investors would be willing to pay for the common stock of Innovate Corporation, we will calculate the present value of future dividends based on the expected growth rates and the required rate of return.

Given Information

• Most Recent Annual Dividend (D0): RM3.00
• Growth Rate for the Next 3 Years (g1): 40% per year
• Constant Growth Rate After 3 Years (g2): 15% per year
• Required Rate of Return (r): 20%

Step-by-Step Calculation

1. Calculate Future Dividends:
   • Year 1 Dividend (D1):
     $$D1=D0\times (1+g1)=3.00\times (1+0.40)=3.00\times 1.40=RM4.20$$
   • Year 2 Dividend (D2):
     $$D2=D1\times (1+g1)=4.20\times (1+0.40)=4.20\times 1.40=RM5.88$$
   • Year 3 Dividend (D3):
     $$D3=D2\times (1+g1)=5.88\times (1+0.40)=5.88\times 1.40=RM8.23$$
2. Calculate Dividend at Year 4 (D4):
   • This dividend will grow at the constant rate of 15%:
   $$D4=D3\times (1+g2)=8.23\times (1+0.15)=8.23\times 1.15=RM9.46$$
3. Calculate Terminal Value at Year 3:
   • The terminal value represents the present value of all future dividends from Year 4 onward, growing at a constant rate.
   • Using the Gordon Growth Model:
   $$TV3=\frac{D4}{r-g2}=\frac{9.46}{0.20-0.15}=\frac{9.46}{0.05}=RM189.20$$
4. Calculate Present Values:
   • Now we discount the future dividends and terminal value back to present value using the required rate of return:
   • Present Value of D1:
   $$PV(D1)=\frac{D1}{(1+r{)}^1}=\frac{4.20}{(1+0.20{)}^1}=\frac{4.20}{1.20}=RM3.50$$
   • Present Value of D2:
   $$PV(D2)=\frac{D2}{(1+r{)}^2}=\frac{5.88}{(1+0.20{)}^2}=\frac{5.88}{1.44}=RM4.08$$
   • Present Value of D3:
   $$PV(D3)=\frac{D3}{(1+r{)}^3}=\frac{8.23}{(1+0.20{)}^3}=\frac{8.23}{1.728}=RM4.76$$
   • Present Value of Terminal Value:
   $$PV(TV3)=\frac{TV3}{(1+r{)}^3}=\frac{189.20}{(1+0.20{)}^3}=\frac{189.20}{1.728}=RM109.66$$
5. Total Present Value:
   • Finally, sum all present values to find the total value of the stock:
   Total\PV=PV(D1)+PV(D2)+PV(D3)+PV(TV3)
   • Total Present Value Calculation:
   Total\PV=RM3.50+RM4.08+RM4.76+RM109.66≈RM122.00

Investors would be willing to pay approximately RM122.00 for the common stock of Innovate Corporation based on the projected growth rates and required rate of return on investment.

You deposit RM10,000 today into a retirement account that earns 5% annual interest. You continue to deposit RM10,000 each year for a total of 40 payments. One year after the last deposit, you make your first annual withdrawal, and continue to receive level annual cash flows for 20 total payments. what is your annual withdrawal?

Answer: 

To determine the annual withdrawal amount from a retirement account after making a series of deposits, we will follow these steps:

1. Calculate the future value of the retirement account after all deposits are made.
2. Determine the annual withdrawal amount that can be sustained for 20 years based on the future value calculated.

Step 1: Calculate the Future Value of Deposits

You deposit RM10,000 today and continue to deposit RM10,000 each year for a total of 40 payments. The interest rate is 5% per year.

Future Value of Annuity Formula

The future value of an annuity can be calculated using the formula:

$$FV=P\times \frac{(1+r{)}^n-1}{r}$$

where:

• $FV$ = future value of the annuity
• $P$ = annual payment (RM10,000)
• $r$ = annual interest rate (0.05)
• $n$ = number of payments (40)

Calculation

1. Calculate the future value of the annuity (40 payments):
   • The first deposit grows for 40 years, and the last deposit grows for 1 year.
   • We can calculate the future value in two parts:
     • The future value of the first 39 deposits (which grow for 39 years).
     • The future value of the last deposit (which grows for 1 year).
   Future Value of First 39 Payments:
   $$F{V}_{39}=10,000\times \frac{(1+0.05{)}^{39}-1}{0.05}$$
   Future Value of Last Payment:
   $$F{V}_{last}=10,000\times (1+0.05{)}^1$$
2. Calculating Each Part:
   • For First 39 Payments:
     $$F{V}_{39}=10,000\times \frac{(1.05{)}^{39}-1}{0.05}$$
     Using a calculator:
     $$(1.05{)}^{39}\approx 5.1307$$
     So,
     $$F{V}_{39}\approx 10,000\times \frac{5.1307-1}{0.05}\approx 10,000\times 82.614=RM826,140$$
   • For Last Payment:
     $$F{V}_{last}=10,000\times (1+0.05{)}^1=RM10,500$$
3. Total Future Value at the End of Year 40:
   $$F{V}_{total}=F{V}_{39}+F{V}_{last}=RM826,140+RM10,500=RM836,640$$

Step 2: Calculate Annual Withdrawal Amount

After making your last deposit at the end of year 40, you will start withdrawing funds one year later (after one year). You want to withdraw a constant amount annually for a total of 20 years.

Present Value of Annuity Formula

To find the annual withdrawal amount $W$, we can use the present value formula for an annuity:

$$PV=W\times \frac{1-(1+r{)}^{-n}}{r}$$

Where:

• $PV$ = present value available for withdrawal (RM836,640)
• $W$ = annual withdrawal amount
• $r$ = interest rate (0.05)
• $n$ = number of withdrawals (20)

Rearranging this formula to solve for $W$:

$$W=PV\times \frac{r}{1-(1+r{)}^{-n}}$$

Calculation

Substituting in our values:

$$W=RM836,640\times \frac{0.05}{1-(1+0.05{)}^{-20}}$$

Calculating $(1+r{)}^{-n}$:

$$(1+0.05{)}^{-20}\approx (1.05{)}^{-20}\approx 0.3769$$

Now substituting back into our equation:

$$W=RM836,640\times \frac{0.05}{1-0.3769}$$

$$W=RM836,640\times \frac{0.05}{0.6231}$$

$$W=RM836,640\times 0.0804\approx RM67,174.78$$

The annual withdrawal amount that you can sustain for 20 years is approximately RM67,174.78 per year.

How do you compute the beta of common stock and what does it mean? Discuss the limitations of using beta to measure systematic risk of individual stocks.

Answer:

To compute the beta of common stock and understand its implications, we follow a defined process and acknowledge its limitations.

How to Compute Beta

Definition of Beta

Beta is a measure of a stock's volatility in relation to the overall market. It quantifies the sensitivity of a stock's returns to the movements of a market index, typically the S&P 500. A beta greater than 1 indicates that the stock is more volatile than the market, while a beta less than 1 indicates it is less volatile.

Calculation Process

Beta can be calculated using historical return data through the following steps:

1. Gather Data:
   • Collect historical price data for the stock and the market index over a specific time period (e.g., daily, weekly, or monthly returns for 1-5 years).
2. Calculate Returns:
   • Compute the periodic returns for both the stock and the market index using the formula:
     $$\text{Return}=\frac{{\text{Price}}_{\text{end}}-{\text{Price}}_{\text{start}}}{{\text{Price}}_{\text{start}}}$$
3. Perform Regression Analysis:
   • Use linear regression to analyze the relationship between the stock's returns and the market's returns. The formula for beta ($\beta$) is derived from:
     $$\beta =\frac{\text{Cov}(R_{\text{stock}},R_{\text{market}})}{\text{Var}(R_{\text{market}})}$$
   where:
   • $\text{Cov}(R_{\text{stock}},R_{\text{market}})$ is the covariance between the stock's returns and market returns.
   • $\text{Var}(R_{\text{market}})$ is the variance of market returns.
4. Interpret Beta:
   • A beta of 1 means the stock moves with the market.
   • A beta greater than 1 means higher volatility (more risk).
   • A beta less than 1 means lower volatility (less risk).
   • A negative beta indicates that the stock moves inversely to the market.

Meaning of Beta

Beta provides insights into systematic risk, which is risk inherent to the entire market that cannot be eliminated through diversification. Investors use beta to gauge how much risk they are taking on relative to market movements. A higher beta suggests that an investment may yield higher returns but also comes with greater risk during market fluctuations.

Limitations of Using Beta

While beta is a useful measure, it has several limitations:

1. Historical Data Dependency:
   • Beta is calculated using historical data, which may not accurately predict future performance. Market conditions can change significantly, making past correlations less relevant.
2. Assumption of Linear Relationship:
   • The calculation assumes a linear relationship between stock and market returns. However, actual price movements may not follow this linearity, especially during extreme market conditions.
3. Time Variation:
   • Beta can change over time due to shifts in company fundamentals or market dynamics. A company may have different betas during different growth phases or economic cycles.
4. Ignores Idiosyncratic Risk:
   • Beta only measures systematic risk and does not account for unsystematic risk (specific to individual stocks), which can be significant for investors holding concentrated positions.
5. Market Portfolio Selection:
   • The choice of benchmark index can affect beta calculations. Different indices might show different betas for the same stock due to variations in their composition and volatility.
6. Industry and Sector Bias:
   • Certain industries have inherent characteristics that affect their betas differently; thus, comparing betas across sectors without context can be misleading.
7. Limited Scope:
   • Beta does not provide a complete picture of an asset's total risk profile since it focuses solely on systematic risk.

While beta serves as a valuable tool for assessing systematic risk relative to market movements, investors should be cautious about its limitations and consider using additional metrics and analyses to gain a comprehensive understanding of an asset's risk profile.

Source: Corporate Finance First Semester Examination Academic Session 2011/2012, Master of Business Administration, Universiti Sains Malaysia