Calculate the price and yield of bonds, crucial for investors to determine the return on investment in fixed-income securities.
The Yield to Maturity (YTM) of a bond can be approximated with the following formula:
\[ \text{YTM} \approx \frac{C + \frac{F - P}{n}}{\frac{F + P}{2}} \]
In this formula:
- The numerator, \(C + \frac{F - P}{n}\), adds the annual coupon payment to the average annual price gain (or loss) if the bond is held to maturity.
- The denominator, \(\frac{F + P}{2}\), is the average of the bond's current price and its face value.
A bond has a face value of $1,000, a coupon rate of 5%, and is currently priced at $950. Calculate the current yield.
Solution:
\[ \text{Current Yield} = \frac{\text{Coupon Payment}}{\text{Current Price}} \] \[ = \frac{1000 \times 0.05}{950} = 5.26\% \]
A bond with a face value of $1,000, a coupon rate of 6%, and a current price of $920 will mature in 5 years. Calculate the approximate Yield to Maturity.
Solution:
\[ \text{YTM} \approx \frac{\text{Coupon Payment} + \frac{\text{Face Value} - \text{Current Price}}{\text{Years to Maturity}}}{\frac{\text{Face Value} + \text{Current Price}}{2}} \] \[ = \frac{60 + \frac{1000 - 920}{5}}{\frac{1000 + 920}{2}} = 7.19\% \]
A bond with a $1,000 face value and a 5% coupon rate can be called in 3 years for $1,050. It is currently priced at $1,020. Calculate the yield to call (YTC).
Solution:
\[ \text{YTC} \approx \frac{\text{Coupon Payment} + \frac{\text{Call Price} - \text{Current Price}}{\text{Years to Call}}}{\frac{\text{Call Price} + \text{Current Price}}{2}} \] \[ = \frac{50 + \frac{1050 - 1020}{3}}{\frac{1050 + 1020}{2}} = 4.86\% \]
A bond with a 6% coupon rate is compounded semiannually. Calculate the effective annual yield.
Solution:
\[ \text{Effective Yield} = \left(1 + \frac{\text{Coupon Rate}}{\text{Compounding Periods}}\right)^{\text{Compounding Periods}} - 1 \] \[ = \left(1 + \frac{0.06}{2}\right)^2 - 1 = 6.09\% \]
A zero-coupon bond has a face value of $1,000, and 5 years to maturity. The required yield is 4%. Calculate the price of the bond.
Solution:
\[ \text{Price} = \frac{\text{Face Value}}{(1 + \text{Yield})^{\text{Years to Maturity}}} \] \[ = \frac{1000}{(1 + 0.04)^5} = 821.93 \]
A corporate bond has a yield of 6.5%, while a government bond has a yield of 4%. Calculate the yield spread.
Solution:
\[ \text{Yield Spread} = \text{Corporate Bond Yield} - \text{Government Bond Yield} \] \[ = 6.5\% - 4\% = 2.5\% \]
A bond with a face value of $1,000, a coupon rate of 5%, and a yield to maturity of 4% has 10 years left to maturity. Calculate its price.
Solution:
\[ \text{Price} = \sum_{t=1}^{10} \frac{\text{Coupon Payment}}{(1 + \text{Yield})^t} + \frac{\text{Face Value}}{(1 + \text{Yield})^{10}} \] \[ = \sum_{t=1}^{10} \frac{50}{(1 + 0.04)^t} + \frac{1000}{(1 + 0.04)^{10}} = 1081.11 \]
Calculate the duration of a 5-year bond with a face value of $1,000, a 5% coupon rate, and a yield of 4%.
Solution:
The duration is a weighted average of the time until each cash flow, discounted by the yield. This calculation is complex and typically requires detailed steps using financial software.
A bond has a duration of 4.5 years and a yield of 5%. Calculate its modified duration.
Solution:
\[ \text{Modified Duration} = \frac{\text{Duration}}{1 + \text{Yield}} \] \[ = \frac{4.5}{1 + 0.05} = 4.29 \text{ years} \]
A bond with a modified duration of 4.5 and convexity of 30 has a yield change of 0.01. Calculate the convexity adjustment.
Solution:
\[ \text{Convexity Adjustment} = \frac{1}{2} \times \text{Convexity} \times (\Delta \text{Yield})^2 \] \[ = \frac{1}{2} \times 30 \times (0.01)^2 = 0.0015 \text{ or } 0.15\% \]
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