Final Exam 1: Calculate The Future Value Of An Annuity

Final Exam1 Calculate The Future Value Of An Annuity That Has The Fol

Calculate the future value of an annuity with the following characteristics: (a) PMT: $1,505, (b) RATE: 10%, and (c) NPER: 25.

Determine the maximum amount you would be willing to pay for an annuity due with the following characteristics: (a) PMT: $5,500, (b) RATE: 8%, and (c) NPER: 15.

Calculate how much you would be willing to pay for a bond that pays semi-annual coupon payments with these characteristics: (a) NPER: 18, (b) YTM: 8%, and (c) Coupon: $35.80.

Determine the maximum price for a no-growth stock with the following characteristics: (a) Dividend (Has Paid): $4.00, and (b) Required Rate of Return: 12%.

Calculate the maximum price for a constant growth stock with these characteristics: (a) Dividend (Has Paid): $3.25, (b) Growth: 7%, and (c) Required Rate of Return: 12%.

Estimate the maximum price you would pay for a non-constant growth stock with these characteristics: (a) Non-Constant Growth Rate: 20%, (b) Constant Growth Rate: 5%, (c) Dividend (Will Pay): $4.50, and (d) Required Rate of Return: 12%.

Calculate the current yield on a bond with the following: (a) Price: $1,055, (b) Coupon Rate: 5%, (c) YTM: 4.6%, and (d) NPER: 22.

Using CAPM, find the Expected Rate of Return on Stock XYZ with these parameters: (a) Expected Return on the Risk-Free Asset: 3%, (b) Expected Return on the Market: 9.5%, and (c) Beta for XYZ: 1.32.

Determine the Beta for XYZ Company given these details: (a) Expected Return on XYZ’s Stock: 9%, (b) Risk-Free Rate: 3%, and (c) Market Return: 9.5%.

Calculate the Yield to Maturity (YTM) on a bond with these characteristics: (a) Price: $884, (b) Coupon: $50.00, and (c) NPER: 24.

Calculate Company A’s weighted average cost of debt given these details: (a) Tax Rate: 20%, (b) Average Price of Outstanding Bonds: $1,120, (c) Coupon Rate: 5%, (d) NPER: 27, (e) Debt: $33,000,000, (f) Equity: $24,000,000, and (g) Preferred Stock: $5,000,000.

Estimate Company B’s weighted average cost of equity using these inputs: (a) Dividend: $1.50, (b) Growth Rate: 4.5%, (c) Price: $21.50, (d) Debt: $33,000,000, (e) Equity: $24,000,000, and (f) Preferred Stock: $5,000,000.

Calculate Company C’s weighted average cost of preferred stock with the following: (a) Coupon Payments: $6.00, (b) Price of Preferred Stock: $50.00, (c) Debt: $33,000,000, (d) Equity: $24,000,000, and (e) Preferred Stock: $5,000,000.

Determine Company D’s weighted average cost of capital (WACC) given: (a) Tax Rate: 22%, (b) Average Price of Bonds: $1,280, (c) Coupon Rate on Debt: 7%, (d) NPER: 10, (e) Dividend: $4.60, (f) Growth Rate: 6%, (g) Price: $40.50, (h) Preferred Stock Coupon: $4.00, (i) Preferred Stock Price: $45.60, (j) Debt: $10,000,000, (k) Equity: $15,000,000, and (l) Preferred Stock: $2,000,000.

Calculate Company E’s weighted average cost of equity with these parameters: (a) Expected Market Return: 14%, (b) Beta: 1.34, (c) Risk-Free Rate: 4%, (d) Debt: $33,000,000, (e) Equity: $24,000,000, and (f) Preferred Stock: $5,000,000.

Based on the provided data in Table 1, if Company XYZ has a WACC of 7% and the projects are independent, which project would you accept based on NPV rules?

If Company XYZ has a WACC of 26% and the projects are mutually exclusive, which project would you accept based on NPV?

Calculate the Internal Rate of Return (IRR) for Project A.

Calculate the Profitability Index (PI) for Project B.

Calculate the Discounted Profitability Index for Project A with a WACC of 8%.

Determine the Payback Period for Project B.

Calculate the Discounted Payback Period for Project A with a WACC of 8%.

Find the Crossover Rate between Projects A and B.

Calculate the difference between daily and annual compounding with these details: (a) PV: $52,000, (b) NPER: 30, and (c) RATE: 10%.

Compute the PMT on a mortgage with these parameters: (a) PV: $439,000, (b) RATE: 4%, and NPER: 30.

Calculate the present value of a lump sum payment with: (a) RATE: 5%, (b) NPER: 22, and (c) FV: $75,230.

Determine the RATE given: (a) PV: $29,325, (b) FV: $54,000, and (c) NPER: 15.

Calculate the NPER based on: (a) PV: $100,000, (b) FV: $134,000, and (c) RATE: 5%.

Find the RATE given: (a) PMT: $20,000, (b) FV: $134,000, and (c) NPER: 5.

Calculate the required rate of return for a stock with these features: (a) Constant Growth Rate: 5%, (b) Price: $50.00, and (c) Dividend (Has Been Paid): $5.00.

Paper For Above instruction

The comprehensive analysis of financial decision-making tools covers a broad spectrum ranging from time value of money calculations, bond and stock valuation, cost of capital estimations, to project appraisal measures. This paper elucidates the methodologies used in determining the future value of an annuity, bond valuation, stock price appraisal, and key financial metrics such as WACC, IRR, and CAPM-based expected returns to inform investment choices.

Future Value of an Annuity

The future value (FV) of an ordinary annuity can be computed using the formula FP = P \times \left( \frac{(1 + r)^n - 1}{r} \right), where P is the periodic payment, r is the interest rate per period, and n is the number of periods. For the given parameters, P = $1,505, r = 10\% = 0.10, and n = 25. Substituting these values yields FV = $1,505 \times \frac{(1 + 0.10)^{25} - 1}{0.10}. Calculating this provides a future value of approximately $97,850. This figure indicates the accumulated value after 25 periods, considering the interest compounding. Such calculations assist investors and savers in planning for future financial needs, demonstrating the power of compound interest over extended periods.

Valuation of Annuities Due

The present value (PV) of an annuity due, where payments occur at the beginning of each period, is calculated as PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r). Using the parameters: P = $5,500, r = 8\% = 0.08, n = 15, the PV becomes approximately $62,150. This reflects what an individual would be willing to pay today for a series of payments starting immediately, considering the discount rate. Annuities due are common in lease agreements and pensions, and understanding their valuation aids in accurate investment decision-making.

Bond Valuation with Semi-Annual Payments

Bond valuation integrates the present value of future coupon payments and the redemption value. The semi-annual coupon payment is given as $35.80, and since the bond pays semi-annually over 18 periods at an 8% YTM, the periodic discount rate is 4%. The present value of coupons (PVC) is calculated as PVC = C \times \left(1 - (1 + r)^{-n} \right)/ r, and the present value of face value (PVF) is calculated separately, summing these components yields the bond's price. Such valuations are critical for investors assessing bond attractiveness relative to market price, and for issuers to gauge the cost of debt issuance.

Valuation of No-Growth Stock

The maximum price for a no-growth stock is derived from the dividend discount model (DDM): P = D / r. With a dividend D = $4.00 and a required rate of return r = 12%, the price becomes P = $4.00 / 0.12 = $33.33. This reflects the intrinsic value of a stock expecting fixed dividends indefinitely. Investors utilize this model to determine whether stocks are overvalued or undervalued based on current market prices relative to foundational valuation metrics.

Constant Growth Stock Valuation

The Gordon Growth Model estimates stock value as P = D \times (1 + g) / (r - g), where D is the dividend, g the growth rate, and r the required rate. Given D = $3.25, g = 7%, and r = 12%, the stock's intrinsic value is P = $3.25 \times 1.07 / (0.12 - 0.07) = $69.55. This model assumes dividends grow at a constant rate and is widely used in mature companies with stable growth trajectories. Accurate estimation of P helps investors align their buying and selling strategies with intrinsic value assessments.

Valuation of Non-Constant Growth Stock

For stocks with non-constant (initial) growth, the valuation involves calculating the present value of dividends expected during the high-growth phase and appraising the stock's value at the end of that period, accounting for the subsequent perpetual growth rate. Using the given data: dividend: $4.50, non-constant growth 20%, and constant growth 5%, with a required return of 12%, the valuation process entails discounting projected dividends, then adding the present value of the perpetuity starting at the end of the high-growth phase. Accurate modeling of non-constant growth is vital for startups and high-growth firms, where initial earnings may substantially differ from mature phases.

Bond Current Yield Calculation

The current yield is defined as Coupon Payment / Market Price. Given a bond priced at $1,055 with a 5% coupon rate, the annual coupon payment is 5% of face value; assuming a standard face value of $1,000, the coupon payment is $50. Therefore, current yield = $50 / $1,055 ≈ 4.73%. This ratio provides a quick measure of income relative to price, though it does not account for capital gains or losses if held to maturity.

Expected Return on Stock Using CAPM

The Capital Asset Pricing Model (CAPM) states: E(r) = Rf + β \times (Rm - Rf). Substituting the values: Rf = 3%, β = 1.32, Rm = 9.5%, we obtain E(r) = 3% + 1.32 \times (9.5% - 3%) = 3% + 1.32 \times 6.5% = 3% + 8.58% = 11.58%. This expected return guides investors in assessing whether a stock's return exceeds the risk-adjusted market expectations, influencing portfolio choices.

Beta Calculation from Expected Return

Rearranging CAPM: β = (E(r) - Rf) / (Rm - Rf). Using E(r) = 9%, Rf = 3%, Rm = 9.5%, we find β = (9% - 3%) / (9.5% - 3%) = 6% / 6.5% ≈ 0.923. Beta indicates the stock's sensitivity to market movements, with values above 1 generally indicating higher volatility.

Yield to Maturity (YTM) Calculation

YTM can be numerically approximated or calculated via financial calculator or software. For the bond with price $884, coupon $50, and 24 periods, iterative methods or approximation formulas suggest a YTM around 6.5%. Precise calculation involves solving for rate r in the present value equation equating bond price to discounted cash flows.

Weighted Average Cost of Debt

The after-tax cost of debt is computed as y_d(1 - tax rate). Using the bond price ($1,120), coupon ($5%), and NPER (27), the yield y_d can be estimated, leading to an after-tax cost of debt. The weighted average considers the proportion of debt relative to total capital structure, reflecting the company's overall cost of debt after tax benefits.

Weighted Average Cost of Equity

The cost of equity employs the CAPM: r_e = Rf + β \times (Rm - Rf). Given the data: Rf = 4%, β = 1.34, Rm = 14%, the cost of equity is r_e = 4% + 1.34 \times (14% - 4%) = 4% + 1.34 \times 10% = 4% + 13.4% = 17.4%. The WACC then blends this with the cost of debt proportional to the firm's capital structure to find the overall required return.

Cost of Preferred Stock

The cost of preferred stock is calculated as the dividend divided by the current market price: r_ps = D / P. With dividend $6.00 and price $50, r_ps = $6.00 / $50 = 12%. This rate is used in WACC calculations to account for the cost of financing through preferred equity.

Weighted Average Cost of Capital (WACC)

For Company D, the WACC involves calculating the weighted costs of debt, equity, and preferred stock considering tax advantages and proportions in capital structure. Using the provided data, the WACC can be derived by summing the weighted after-tax cost of debt, the weighted cost of equity, and the cost of preferred stock, providing a comprehensive measure used in investment appraisal and corporate finance decisions.

Weighted Average Cost of Equity for Company E

Applying CAPM yields: r_e = Rf + β \times (Rm - Rf) = 4% + 1.34 \times (14% - 4%) = 4% + 1.34 \times 10% = 4% + 13.4% = 17.4%. This rate reflects the expected return required by shareholders given the company's risk profile, informing dividend policy and stock valuation.

Project Evaluation: NPV, IRR, PI, Payback

For the independent projects, accept those with positive NPVs at a WACC of 7%. For mutually exclusive projects at a 26% WACC, choose the project with the higher NPV. The IRR for Project A is obtained by solving discounted cash flow equations, typically via financial calculator or software, revealing the internal rate of return at which NPV equals zero. The profitability index (PI) for Project B, calculated as the present value of future cash flows divided by initial investment, guides investment efficiency. Discounted payback period involves calculating the time needed for discounted cash flows to cover initial investment, considering WACC as the discount rate. The crossover rate is the discount rate where NPVs of two projects are equal, aiding in analyzing project preferences at different discount rates.

Impact of Compounding Frequency

The difference between daily and annual compounding over 30 periods at 10% involves calculating the effective interest rates. Daily compounding yields a slightly higher effective rate due to more frequent compounding within the same nominal rate. Calculations involve using (1 + r/n)^{nt} formulas, emphasizing the importance of compounding frequency in time value of money calculations.

Mortgage Payment Calculation

The mortgage PMT is derived from the amortization formula, considering the PV, interest rate, and NPER. Using the provided data, the approximate monthly payment is around $2,096, illustrating the application of the mortgage formula to real-world borrowing scenarios.

Present Value of a Lump Sum

The PV of a lump sum: PV = FV / (1 + r)^n. Substituting: PV = $75,230 / (1 + 0.05)^22 ≈ $29,861. This calculation determines the current worth of a future sum at a given discount rate, which is fundamental in