Problem Set 4 Fin 7000 Problem 1 Calculate The Annual Intern

Problem Set 4 Fin 7000problem 1calculate The Annual Internal Rates Of

Calculate the annual internal rates of return (IRR) for various investments with specific cash flows at different time periods. Additionally, analyze bond prices based on yield to maturity (YTM), and understand the impact of different coupon payment frequencies and bond holding periods on yields and pricing.

Paper For Above instruction

The calculation of internal rates of return (IRR) remains a fundamental aspect of investment analysis, providing investors with a measure of an investment's profitability. The process involves solving for the discount rate that equates the present value of cash inflows to the initial investment outlay. This paper explores the calculation of IRRs for various cash flow scenarios, evaluates bond pricing based on different yield environments, and examines the implications of coupon payment frequency and holding periods on bond yields and prices.

Calculating the Internal Rate of Return (IRR) for Multiple Investment Scenarios

The first task involves determining the IRR for four different investment projects. Each project involves an initial cost at t=0 followed by cash inflows at subsequent periods. The projects differ in their cash flow timelines and amounts, thus affecting their IRRs.

For the first project, an initial outlay of $100 at t=0, with cash inflows of $100 at t=1 and $250 at t=3, the IRR solution involves solving the equation:

-100 + 100/(1 + IRR) + 250/(1 + IRR)^3 = 0

Similarly, for the second project with an initial cost of $150, the equation is:

-150 + 100/(1 + IRR) + 250/(1 + IRR)^3 = 0

The third project involves cash inflows of $100 at t=1 and $250 at t=2, requiring solving:

-100 + 100/(1 + IRR) + 250/(1 + IRR)^2 = 0

Lastly, the fourth project with cash inflows of $250 at t=1 and $100 at t=3, involves solving:

-100 + 250/(1 + IRR) + 100/(1 + IRR)^3 = 0

By calculating the IRRs for each project using financial calculator or Excel IRR functions, the project with the highest IRR indicates the most profitable investment relative to its initial cost. Typically, the project with earlier and larger cash flows tends to have a higher IRR, but this must be verified numerically.

Bond Pricing Under Different Yield to Maturity (YTM) Scenarios

The second set of problems examines the price of a 10-year bond paying a 5% coupon semi-annually with a face value of $1000 under various YTM assumptions.

Given the semi-annual coupon payment is calculated as:

Coupon payment = (0.05 * 1000) / 2 = $25

for different yield scenarios, the bond price is determined by summing the present value of all coupon payments and the present value of the face value, discounted at the semi-annual YTM.

For example, at a YTM of 4% (0.02 semi-annually), the price is:

Price = ∑_{t=1}^{20} 25 / (1 + 0.02)^t + 1000 / (1 + 0.02)^20

The same method applies for YTMs of 5% and 6%. When considering the effective annual rate (EAR) of 5%, the semi-annual discount rate is derived from:

(1 + EAR)^{1/2} - 1

and used to discount cash flows accordingly, providing a more accurate bond valuation under the effective rate scenario.

Effects of Coupon Payment Frequency on Bond Valuation

The third problem shifts to bonds that pay coupons annually rather than semi-annually. The valuation process involves similar calculations, but the discounting is adjusted for annual payments, which influences the bond's price and yield calculations.

Zero-Coupon Bond Pricing and Yield Analysis

The final set of problems focuses on a zero-coupon bond priced at $670 for a $1000 face value. Its YTM is calculated by solving the equation:

670 = 1000 / (1 + YTM)^10

which yields the annualized YTM. The subsequent analysis considers holding the bond for five years and selling it at a higher price ($850), calculating the holding period yield (HPY) as:

HPY = (Selling Price / Purchase Price)^{1 / holding period} - 1

It is critical to understand that as the bond approaches maturity, its YTM tends to align with the bond's current yield. The question arises about the relationship between the initial YTM and the YTM at sell time, which depends on market interest rate movements during the holding period. To attain the same YTM after five years, the bond would need to sell at a price consistent with the remaining cash flows discounted at the original YTM.

Conclusion

The calculations of IRR and bond prices based on different interest rate scenarios underscore the importance of timing, cash flow structure, and interest compounding assumptions in investment decision-making. Investors should consider these factors to evaluate potential investments accurately, assess bond valuations, and understand how market interest rates influence bond prices and yields over time.

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