In: Finance
You own a $1,000-par zero-coupon bond that has six years of
remaining maturity. She
plans on selling the bond in one year and believes that the
required yield next year will
have the following probability distribution:
Probability | Required Yield |
0.1 | 0.067 |
0.4 | 0.0685 |
0.4 | 0.071 |
0.1 | 0.073 |
a. What is the expected price of the bond at the time of
sale?
b. What is the standard deviation of the bond price?
Price of a Zero Coupon Bond | |||||
= Face Value / (1+r)n | |||||
Where, | |||||
Face Value of Zero Coupon Bond = $1000 | |||||
The bond is selling in one year, so | |||||
n = No of years remaing maturity = 6 - 1 = 5years | |||||
r = Required Yield | |||||
So, | |||||
Required Yield | Price of Bond | ||||
0.067 | = 1000/(1+0.067)5 = 1000/(1.067)5 = 1000/1.3830 = $723.07 | ||||
0.0685 | = 1000/(1+0.0685)5 = 1000/(1.0685)5 = 1000/1.39275 = $718.00 | ||||
0.071 | = 1000/(1+0.071)5 = 1000/(1.071)5 = 1000/1.40912 = $709.66 | ||||
0.073 | = 1000/(1+0.073)5 = 1000/(1.073)5 = 1000/1.42232 = $703.08 | ||||
Now, | |||||
Probability | Reqired Yield | Price | Prob*Price | Variance = Prob*(Price - Expected Price)2 | |
0.1 | 0.067 | $723.07 | $72.31 | 0.1*(723.07-713.68)2=0.1(9.39)2=0.1*88.1721=8.81721 | |
0.4 | 0.0685 | $718.00 | $287.20 | 0.4*(718.00-713.68)2=0.4(4.32)2=0.4*18.6624=7.46496 | |
0.4 | 0.071 | $709.66 | $283.86 | 0.4*(709.66-713.68)2=0.4(-4.02)2=0.4*16.1604=6.46416 | |
0.1 | 0.073 | $703.08 | $70.31 | 0.1*(703.08-713.68)2=0.1(-10.6)2=0.1*112.36=11.236 | |
Expected Price | $713.68 | 33.98233 | |||
a) | Expected Price = $713.68 | ||||
b) | Standard Deviation = √Variance = √33.98233 = 5.83 | ||||