Comparing An Ordinary Annuity And An Annuity Due

In Comparing An Ordinary Annuity And An Annuity Due Which Of The F

Compare an ordinary annuity and an annuity due, examining the differences in their future values, present values, and overall financial implications. Explore how timing impacts growth, interest accumulation, and investor preferences. Analyze the concepts of perpetual income streams, the effects of inflation on investments, and the implications for dividend payments and dividend arrears. Discuss portfolio risk management through asset correlation, and assess risks such as business-specific, nondiversifiable, internal, and unsystematic risks. Evaluate various financial calculations including savings for a cruise, return on investment, future value of property investments considering inflation, endowment funding, retirement planning, bond valuation, preferred stock valuation, dividend growth models, and the impact of beta on investment returns and pricing.

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The comparison between an ordinary annuity and an annuity due centers on the timing of the payments and the resultant financial implications. An ordinary annuity involves payments made at the end of each period, while an annuity due involves payments made at the beginning of each period. This distinction significantly impacts their future value (FV) and present value (PV). Specifically, because payments in an annuity due are invested a period earlier, its FV will always be greater than that of an otherwise identical ordinary annuity, assuming all other factors are equal (Ross, Westerfield, & Jordan, 2020). This is due to the additional compounding period for each payment, resulting in a higher accumulation over time. Conversely, the PV of an annuity due is typically higher than that of an ordinary annuity because payments are received sooner, providing more value today (Brigham & Ehrhardt, 2019).

The concept of perpetual income streams is inherently linked with the discount rate, which reflects the opportunity cost of capital. If the PV of a perpetual income stream increases, it indicates that the discount rate must have decreased, since the present value of a perpetuity is inversely proportional to the discount rate (Damodaran, 2012). A lower discount rate amplifies the PV, suggesting increased valuation due to lower risk premiums or interest rates.

Nico’s investment scenario highlights the difference between nominal and real returns. If his purchasing power has increased by 15 percent after accounting for a 4 percent inflation rate, then his real return exceeds the nominal return. Specifically, the real rate of return can be approximated using the Fisher equation (Fisher, 1930). In this case, the real return (r) is approximately (1 + nominal return) / (1 + inflation) - 1, which suggests Nico's nominal return is roughly 19 percent, and his real return is about 15 percent, confirming that his real return surpasses the nominal figure when inflation is considered.

Preferred stock dividends in arrears refer to unpaid dividends that must be paid before any dividends are distributed to common stockholders. These dividends are typically cumulative, meaning if dividends are missed, they accumulate and must be paid in the future (Fabozzi, 2013). The accumulation of dividends in arrears signifies financial obligations that impact the company’s dividend policy and financial health.

In portfolio management, combining assets with perfectly negatively correlated returns leads to diversification benefits. Specifically, such a strategy reduces overall portfolio risk because fluctuations in one asset offset those in the other. When assets are perfectly negatively correlated, the overall risk decreases to below the risk level of either asset considered individually. This concept underpins modern portfolio theory, which emphasizes diversification to optimize risk-adjusted returns (Markowitz, 1952).

Business-specific risks, such as wars, inflation, and foreign market conditions, are classified as nondiversifiable or systematic risks. These risks impact the entire market and cannot be eliminated through diversification because they stem from broad economic factors (Elton & Gruber, 1995). In contrast, unsystematic risks are specific to individual firms or industries and can be mitigated through diversification.

The chapter scenarios involve various computations: For Melissa’s cruise fund, we calculate annual deposits by solving the future value of an ordinary annuity problem using the present value formula, interest rate, and time horizon. For Nico’s investment, the rate of return is derived by dividing the annual income by the initial investment, factoring in the expected cash flows over 30 years. The beach house savings involve calculating the future cost considering inflation and determining equal annual end-of-year deposits using the future value of an ordinary annuity formula.

Joe’s endowment to fund perpetuity scholarships involves calculating the present value of perpetuity and then determining the equal annual deposits over ten years that will accumulate to this amount, considering the interest rate. For retirement planning, the accumulated amount needed at retirement is found using the present value of an annuity due for the desired annual withdrawals over the retirement period, discounted at the assumed rate of return. The annual contributions are then derived from the future value of an ordinary annuity, considering the accumulation period.

In bond valuation, Hewitt Packing bonds are valued using the present value of future coupon payments and face value discounted at the market yield, which accounts for semiannual payments. The new bond's coupon rate is calculated based on the current market price and the yield to maturity, ensuring the bond sells at par value with semiannual interest payments. Preferred stock valuation relies on dividing the dividend by the required rate of return, while the dividend growth model (Gordon Growth Model) computes stock value based on dividends and growth rate.

Finally, the Capital Asset Pricing Model (CAPM) estimates the expected return, considering beta, risk-free rate, and market return, guiding investment decisions based on risk-adjusted expected returns. For instance, with a beta of 1.80, the expected return exceeds the market average, suggesting a risk-premium aligned with the higher beta, and helps assess whether the investment's return exceeds its required return (Sharpe, 1964).

In conclusion, understanding the distinctions between annuities, valuation techniques, risk management concepts, and investment analysis tools is fundamental for effective financial decision-making. Grasping how timing, interest rates, inflation, risk, and diversification influence financial outcomes allows investors and financial managers to optimize portfolios, plan for future expenses, and evaluate investment opportunities accurately.

References

• Brigham, E. F., & Ehrhardt, M. C. (2019). Financial Management: Theory & Practice (16th ed.). Cengage Learning.
• Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset (3rd ed.). Wiley.
• Elton, E. J., & Gruber, M. J. (1995). Modern Portfolio Theory and Investment Analysis (5th ed.). Wiley.
• Fisher, I. (1930). The Theory of Interest. Macmillan.
• Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77-91.
• Ross, S. A., Westerfield, R., & Jordan, B. D. (2020). Fundamentals of Corporate Finance. McGraw-Hill Education.
• Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk. The Journal of Finance, 19(3), 425-442.