Interest Rates And Security Prices Problem Set 3

Interest Rates And Security Prices Problem Set 3 interest Rates And Security

Analyze various financial problems involving bond and stock valuation, duration, and sensitivity to yield changes. The problems include calculating bond prices based on coupon rates and maturity, evaluating stock values with growth assumptions, determining bond duration, and assessing how market yield fluctuations impact bond prices. The tasks involve understanding present value calculations, dividend growth models, duration concepts, and the effect of yield changes on bond valuations.

Paper For Above instruction

Financial markets fundamentally rely on the valuation of bonds and stocks, which are influenced by prevailing interest rates, expected dividend growth, time to maturity, and market conditions. Understanding these valuation techniques is crucial for investors, financial analysts, and portfolio managers to make informed investment decisions. This paper explores the core principles behind bond and stock valuation, the concept of duration and its application, and the sensitivity of bond prices to interest rate changes, illustrating these with practical examples.

Bond Valuation and Market Price Calculation

Western Enterprises' bonds, with a 10-year maturity, coupon rate of 9 percent, and a yield to maturity (YTM) of 7 percent, are valued using the present value formula of bond pricing. This involves discounting the future coupon payments and the face value at the YTM. Since interest is paid annually, the bond price is calculated as the sum of the present value of annuity (coupons) and the present value of a lump sum (par value). The formula is expressed as:

Bond Price = C * [1 - (1 + r)^(-n)] / r + F / (1 + r)^n

Where C is the annual coupon payment, r is the YTM, n is the number of years, and F is the face value. Plugging in the values:

C = $1,000 * 9% = $90


r = 7% or 0.07


n = 10 years


F = $1,000

This yields a bond price approximately higher than its face value, reflecting the lower YTM relative to the coupon rate, indicating the bond is trading at a premium. Calculations show the bond's market price is around $1,082. Based on this, investors are willing to pay more than the face value for a bond offering a coupon rate higher than the yield environment.

Present Value of Bonds with Different Maturities

Calculating the fair value of bonds with a 10% semiannual coupon rate, face value of $1,000, and a required return of 8% involves similar present value calculations, but with semiannual periods. The periodic coupon payment becomes $50 (half of annual $100), and the market rate per period is 4%. The value of bonds with varying maturities (10, 15, 20 years) increases with longer maturities because they provide more cash flows, assuming constant coupon and discount rates. The calculations reveal that as maturity extends, present value increases, showing an inverse relationship between maturity length and discounting effect, emphasizing the time value of money in bond valuation.

Stock Valuation through Growth Models

Stock valuation typically employs the Gordon Growth Model (Dividend Discount Model) for stocks with constant growth, articulated as:

P = D1 / (r - g)

where P is the stock's fair value, D1 is the next expected dividend, r is the required rate of return, and g is the growth rate. For Safeco Corp., forecasted to grow at 10%, with a recent dividend of $1.20 and a required return of 12%, the stock's fair value is calculated as:

D1 = $1.20 * (1 + 10%) = $1.32


P = $1.32 / (12% - 10%) = $66

This indicates the stock’s highly sensitive value to the assumptions about growth and required returns. When dividend growth is at 1.5% and the required rate of return varies, the valuation responds accordingly, emphasizing the importance of accurate growth estimates for stock valuation.

Supernormal Growth and Terminal Value

For stocks experiencing supernormal growth over a finite period, valuation involves calculating the present value of dividends during the high-growth period and the perpetuity of dividends thereafter. The dividend modeling considers a dividend of $5.50 last year, with 8% supernormal growth for six years, and a constant 3% growth afterward. The valuation sums the discounted dividends during the supernormal growth phase and the terminal value computed at the end of this phase, using the Gordon Model for the perpetual stage. This comprehensive approach provides an accurate estimate of stock value, accounting for both high-growth and stable-growth periods.

Bond Duration and Interest Rate Sensitivity

Bond duration measures the sensitivity of a bond’s price to changes in interest rates. The duration is the weighted average time until cash flows are received, weighted by the present value of each cash flow divided by the total bond price. For a 4-year bond with an 8% coupon and a face value of $1,000, the duration can be calculated by summing the present value-weighted times. Reinvestment rates influence the future cash flows, but the core duration calculation remains based on cash flow timing and magnitude.

For a Treasury bond with semiannual coupons, the duration varies inversely with the yield to maturity. A higher YTM shortens the duration because cash flows are discounted more heavily and received sooner relative to the present. Conversely, when yields decrease, the duration lengthens. Empirical calculations show that duration decreases as the yield increases, confirming the inverse relationship. The relationship occurs because higher yields diminish the present value of distant cash flows more rapidly, shifting the weighted average closer to the current date.

Impact of Yield Changes on Bond Prices

Assessing the impact of yield changes involves applying duration and convexity concepts. Using the duration approximation, a 0.10% yield change results in a proportionate price change, calculated as:

% Price Change ≈ -Duration × Change in Yield

For larger yield swings, the approximation becomes less accurate, and convexity adjustments are necessary. For the portfolio of bonds with a duration of 12.1608 years, an immediate yield increase of 2% would reduce the bond price significantly, while a decrease of 2% would increase it correspondingly. Calculations reveal that using duration alone tends to underestimate the price decrease or overestimate the increase for large yield shifts, underscoring the importance of considering convexity for precise valuation.

Conclusion

In conclusion, bond and stock valuations are central to investment decision-making and rely heavily on the understanding of present value concepts, dividend growth models, and sensitivity measures like duration. Longer maturities generally lead to higher bond prices but also increase interest rate risk. The relationship of duration with yield to maturity highlights the importance of interest rate expectations in fixed-income investing. Accurate valuation and risk assessment are integral to portfolio management, emphasizing the need for a thorough grasp of these fundamental financial principles.

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