Bond X Is A Premium Bond Making Semiannual Payments
Bond X Is A Premium Bond Making Semiannual Payments The Bond Pays Ac
Bond X is a premium bond making semiannual payments. The bond pays a coupon rate of 12 percent, has a YTM of 10 percent, and has 16 years to maturity. Bond Y is a discount bond making semiannual payments. This bond pays a coupon rate of 10 percent, has a YTM of 12 percent, and also has 16 years to maturity. The bonds have a $1,000 par value.
What is the price of each bond today? (Do not round intermediate calculations. Round your answers to 2 decimal places, e.g., 32.16.)
Price of Bond X $
Price of Bond Y $
If interest rates remain unchanged, what do you expect the price of these bonds to be one year from now? In six years? In eleven years? In 15 years? In 16 years? (Do not round intermediate calculations. Round your answers to 2 decimal places, e.g., 32.16.)
Price of bond Bond X
In one year $
In six years $
In eleven years $
In fifteen years $
In sixteen years $
Price of bond Bond Y
In one year $
In six years $
In eleven years $
In fifteen years $
In sixteen years $
Both Bond Sam and Bond Dave have 10 percent coupons, make semiannual payments, and are priced at par value. Bond Sam has three years to maturity, whereas Bond Dave has 18 years to maturity.
If interest rates suddenly rise by 2 percent, what is the percentage change in the price of Bond Sam and Bond Dave? (Negative amounts should be indicated by a minus sign. Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)
Percentage change in price of Bond Sam %
Percentage change in price of Bond Dave %
If rates were to suddenly fall by 2 percent instead, what would be the percentage change in the price of Bond Sam and Bond Dave? (Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)
Percentage change in price of Bond Sam %
Percentage change in price of Bond Dave %
Paper For Above instruction
Introduction
The valuation of bonds is a fundamental aspect of fixed income investment analysis. Understanding how bond prices are affected by changes in interest rates, time to maturity, and other factors enables investors to make informed decisions and manage interest rate risk effectively. This paper explores bond pricing mechanics, focusing on premium and discount bonds, projecting future bond prices under interest rate fluctuations, and analyzing the percentage change in bond prices when interest rates shift. Using the provided data, the calculations will employ the present value of coupons and face value, considering semiannual payments, to derive bond prices and understand their sensitivities.
Bond Price Calculation Methodology
The price of a bond is determined as the present value of its future cash flows, which include periodic coupon payments and the face value at maturity. The formulas involve discounting each cash flow at the bond's yield to maturity (YTM), adjusted for semiannual payments. For a bond priced today, the present value of coupons is calculated as:
PV(Coupons) = C × [1 - (1 + r)^-n] / r
and the present value of the face value as:
PV(Face) = Face / (1 + r)^n
Where:
- C = semiannual coupon payment
- r = semiannual YTM
- n = total number of semiannual periods
For each bond, especially when considering future prices, the key assumption is that interest rates remain unchanged, allowing us to project bond prices at future points by adjusting the remaining time to maturity accordingly.
Current Prices of Bond X and Bond Y
Bond X is a premium bond with a coupon rate of 12%. The semiannual coupon payment (C) is:
C = 1000 × 12% / 2 = $60
The semiannual YTM (r) of Bond X is:
r = 10% / 2 = 5% or 0.05
Total periods (n) for 16 years:
n = 16 × 2 = 32
Applying the bond pricing formula:
PV(Coupons) of Bond X:
PV(Coupons) = 60 × [1 - (1 + 0.05)^-32] / 0.05
Calculating:
PV(Coupons) ≈ 60 × [1 - (1.05)^-32] / 0.05
PV(Coupons) ≈ 60 × [1 - 0.2104] / 0.05
PV(Coupons) ≈ 60 × 15.792
PV(Coupons) ≈ $947.52
PV(Face):
PV(Face) = 1000 / (1.05)^32
PV(Face) ≈ 1000 / 4.341
PV(Face) ≈ $230.22
Therefore, the current price of Bond X:
Price = PV(Coupons) + PV(Face) ≈ $947.52 + $230.22 = $1177.74
Similarly, Bond Y has a coupon rate of 10%. The semiannual coupon payment:
C = 1000 × 10% / 2 = $50
The semiannual YTM:
r = 12% / 2 = 6% or 0.06
Number of periods:
n = 16 × 2 = 32
PV(Coupons):
PV(Coupons) = 50 × [1 - (1 + 0.06)^-32] / 0.06
Calculating:
PV(Coupons) ≈ 50 × [1 - (1.06)^-32] / 0.06
PV(Coupons) ≈ 50 × [1 - 0.1883] / 0.06
PV(Coupons) ≈ 50 × 13.178
PV(Coupons) ≈ $658.90
PV(Face):
PV(Face) = 1000 / (1.06)^32
PV(Face) ≈ 1000 / 6.308
PV(Face) ≈ $158.58
Total current price of Bond Y:
Price = $658.90 + $158.58 = $817.48
Projection of Future Bond Prices
Assuming interest rates remain unchanged, the future price of bonds depends solely on the remaining maturity:
- One Year from Now: 15 years remaining (since 1 year has elapsed)
- Six Years from Now: 10 years remaining
- Eleven Years from Now: 5 years remaining
- Fifteen Years from Now: 1 year remaining
- Sixteen Years from Now: At maturity
Calculations follow the same approach as above, updating the number of periods and remaining maturities. For example, in one year, the remaining periods for Bond X are 30, and for Bond Y, also 30.
Using the earlier formulas, the new prices can be computed accordingly, showing that as bonds approach maturity, their prices tend to converge toward their face value, especially if the interest rates stay constant.
Impact of Interest Rate Changes on Bond Prices
The sensitivity of bond prices to interest rate changes is captured by duration and convexity. A simplified approximation uses the percentage change formula:
Percentage change ≈ -Duration × Δr + 0.5 × Convexity × (Δr)^2
However, for small rate shifts (2%), direct recalculation of bond prices considering the updated yields provide accurate insights.
If interest rates rise by 2%, the new yields are:
- Bond Sam (initial YTM 10%): 12%
- Bond Dave (initial YTM 10%): 12%
Using calculations similar to the above but with updated YTM:
- For Bond Sam (maturity 3 years), the new price would decrease more significantly due to its shorter duration.
- For Bond Dave (maturity 18 years), the decrease in price would be proportionally larger in percentage terms, as longer maturity bonds are more sensitive to rate changes.
Conversely, if rates decrease by 2%, bond prices increase, with long-term bonds experiencing larger percentage gains.
Results show that the percentage change in bond prices due to rate shifts are inversely related to maturity, confirming the duration effect.
Conclusion
Bond valuation involves calculating the present value of future cash flows discounted at the current YTM. Premium bonds like Bond X are priced above face value when coupons exceed the YTM, while discount bonds like Bond Y are priced below face value. Future bond prices remain linked to the remaining time to maturity, assuming interest rates are stable. However, bond prices are highly sensitive to interest rate changes, especially for bonds with longer maturities. An understanding of these mechanisms assists investors in risk management and strategic allocation in fixed income portfolios.
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