Chapter 2 Basics of Fixed Income Securities

Problem Set

(light version of the exercises in the text)

Q3.

You are given the following data on diﬀerent rates with the same maturity (1.5 years), but quoted on a diﬀerent basis and diﬀerent compounding frequencies:

• Continuously compounded rate: 2.00% annualized rate

• Continuously compounded return on maturity: 3.00%

• Annually compounded rate: 2.10% annualized rate

• Semi-annually compounded rate: 2.01% annualized rate

You want to ﬁnd an arbitrage opportunity among these rates. Is there any one that seems to be mispriced?

Answer: This exercise tests your knowledge of dealing with interest rates with diﬀerent compounding frequency.

Given the interest

*…show more content…*

= $103. 417 45

Q6. Consider the data in the following table:

Maturity T

0.50

1.00

1.5

2

Yield r2 (0, T )

6.49%

6.71%

6.84%

6.88%

3

Consider two bonds, both with 2 years to maturity, semiannual payments, but with diﬀerent coupon rates. Let the two coupon rates be 15% and 3%.

(a) Compute the prices and the yields to maturity of these coupon bonds.

Answer: The term structure of interest rate is the same as in Q4. We can just copy the discount factor table here:

Maturity T Yield r2 (0, T ) Discount Factor Z (0, T ) = 1/ (1 + r2 (0, T ) /2)2×(T −0)

0.50

6.49%

1/ (1 + 0.0649/2)2×0.5 = 0.968 569 91

1.00

6.71%

1/ (1 + 0.0671/2)2×1 = 0.936 131 84

1.5

6.84%

1/ (1 + 0.0684/2)2×1.5 = 0.904 037 4

2

6.88%

1/ (1 + 0.0688/2)2×2 = 0.873 465 90

The 15% 2-year coupon bond with semiannual payments has been priced in Q4, so we can just copy the results here:

Pc (0, 2) = 7.5 × Z (0, 0.5) + 7.5 × Z (0, 1) + 7.5 × Z (0, 1.5) + 107.5 × Z (0, 2)

= 7.5 × 0.968 569 91 + 7.5 × 0.936 131 84 + 7.5 × 0.904 037 4 + 107.5 × 0.873 465 90

= $114. 963 13

The yield to maturity (YTM) on this bond is deﬁned through the equation:

Pc (0, 2) = 114.96313 =

7.5

7.5

107.5

7.5

2×0.5 +

2×1 +

2×1.5 +

(1 + y/2)

(1 + y/2)

(1 + y/2)

(1 + y/2)2×2

Solution is: y = 6. 865 540 8%. (You won’t have Excel to help you ﬁnd the root in the exam.

However, I expect you to be able to write down the deﬁning equation for YTM.)