Time Value Of Money And Cost Of Capital Analysis
Time Value of Money and Cost of Capital Analysis
Assignment Instructions:
Christy and Michael are assessing their retirement savings and future retirement income. Their current assets include $250,000 in retirement plans and $90,000 in other investments. They contribute $30,000 annually to their retirement plans and $6,000 annually to other investments.
a. If their assets grow at an annual rate of 9%, how much money will they accumulate by the time they are 60?
b. After retirement, they plan to invest more conservatively at an annual return of 6%. What annual payment can they make if they expect to live for 30 years in retirement?
In addition, a company is analyzing projects using the Weighted Average Cost of Capital (WACC). Given two divisions, A and B, with respective betas of 0.5 and 1.5, and each financed with 50% debt and 50% equity, determine their WACCs. The company's current cost of equity is 16%, based on an average firm beta of 1.0 and a risk-free rate of 5%. The after-tax yield on bonds is 6%. Calculate the Market Risk Premium (MRP) and then determine the WACCs for divisions A and B.**
Paper For Above instruction
Introduction
The concepts of time value of money (TVM) and cost of capital are fundamental in financial decision-making, enabling individuals and firms to evaluate investment opportunities and plan for future financial needs. This paper explores Christy and Michael’s retirement savings plan, calculating their projected future wealth and retirement income, and analyzes how a company's divisions can determine their respective WACCs based on risk profiles. These analyses highlight the importance of understanding interest growth over time and the appropriate weighted cost of financing for investment decisions.
Future Value of Retirement Assets
Christy and Michael’s current combined assets comprise retirement plans valued at $250,000 and other investments worth $90,000. Their annual contributions are $30,000 to the retirement plans and $6,000 to their other investments. Assuming a growth rate of 9% per annum, the future value (FV) of their current assets and contributions after 15 years can be calculated using the formula for the future value of a lump sum plus an ordinary annuity.
The future value of their existing assets is calculated as:
\[
FV_{lump} = PV \times (1 + r)^n
\]
where \(PV = 340,000\), \(r=0.09\), and \(n=15\).
Applying the formula:
\[
FV_{assets} = 340,000 \times (1.09)^{15} \approx 340,000 \times 3.6425 \approx 1,240,650
\]
The future value of their annual contributions is computed using the future value of an ordinary annuity:
\[
FV_{annuity} = P \times \frac{(1 + r)^n - 1}{r}
\]
where \(P = 36,000\), \(r=0.09\), \(n=15\).
Calculating:
\[
FV_{contributions} = 36,000 \times \frac{(1.09)^{15} - 1}{0.09} \approx 36,000 \times \frac{3.6425 - 1}{0.09} \approx 36,000 \times 29.361 \approx 1,056,796
\]
Adding these together gives the total projected wealth at age 60:
\[
FV_{total} = 1,240,650 + 1,056,796 \approx 2,297,446
\]
Thus, Christy and Michael will have approximately $2,297,446 when they turn 60.
Retirement Income Planning
Post-retirement, assuming a more conservative investment return of 6%, they seek to determine the annual payments they can make to sustain themselves over a 30-year life expectancy.
The present value of their accumulated wealth is:
\[
PV = 2,297,446
\]
Using the annuity payout formula for a retirement period:
\[
P = PV \times \frac{r}{1 - (1 + r)^{-n}}
\]
where \(r = 0.06\) and \(n = 30\).
Calculating:
\[
P = 2,297,446 \times \frac{0.06}{1 - (1.06)^{-30}}
\]
Evaluate the denominator:
\[
(1.06)^{-30} \approx 0.174
\]
\[
1 - 0.174 = 0.826
\]
Thus,
\[
P = 2,297,446 \times \frac{0.06}{0.826} \approx 2,297,446 \times 0.0726 \approx 166,710
\]
They can expect to receive approximately $166,710 annually during their 30-year retirement.
Cost of Capital (WACC) Analysis
To determine the WACC for divisions A and B, the company uses a divisional approach, considering their respective betas and the overall cost structure.
Given:
- Risk-free rate (\(R_f\)) = 5%
- Current market risk premium (\(MRP\)) = ?
- Cost of equity (\(R_e\)) = 16%
- After-tax cost of debt (\(R_d\)) = 6%
- Beta of division A (\(\beta_A\)) = 0.5
- Beta of division B (\(\beta_B\)) = 1.5
- Debt and equity weights = 50% each
First, calculating the Market Risk Premium (\(MRP\)):
\[
R_e = R_f + \beta \times MRP
\]
Using the average beta of 1.0:
\[
16\% = 5\% + 1.0 \times MRP
\]
\[
MRP = 16\% - 5\% = 11\%
\]
Next, calculating the required return for each division using CAPM:
\[
R_{div} = R_f + \beta_{div} \times MRP
\]
- Division A:
\[
R_A = 5\% + 0.5 \times 11\% = 5\% + 5.5\% = 10.5\%
\]
- Division B:
\[
R_B = 5\% + 1.5 \times 11\% = 5\% + 16.5\% = 21.5\%
\]
Since each division's WACC is a weighted average of the cost of equity and the cost of debt, with weights of 50% each:
\[
WACC_{div} = 0.5 \times R_{div} + 0.5 \times R_d
\]
- Division A:
\[
WACC_A = 0.5 \times 10.5\% + 0.5 \times 6\% = 5.25\% + 3\% = 8.25\%
\]
- Division B:
\[
WACC_B = 0.5 \times 21.5\% + 0.5 \times 6\% = 10.75\% + 3\% = 13.75\%
\]
Conclusion
The analysis shows that Christy and Michael are on track to amass substantial savings for retirement, and their projected income will sustain their lifestyle. The WACC calculations highlight the risk premiums associated with divisions with different betas, informing investment and financing decisions. Both personal financial planning and corporate capital structure analysis rely heavily on PV, FV, and WACC concepts to optimize outcomes.
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