Fixed income securities
Fixed Income Securities
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 rates, we can compute the discount factors correspondingly.
From continuously compounded rate: 2.00% annualized rate:
Z (0, 1.5) = exp (−r (0, 1.5) × (T − t))
= exp (−0.02 × 1.5) = 0.970 45
From continuously compounded return on maturity: 3.00% (we did not cover this in class, but it means the unannualized interest rate)
Z (0, 1.5) = exp (−0.03) = 0.970 45
From annually compounded rate: 2.10% annualized rate:
Z (0, 1.5) =
1
(T −t)
(1 + r1 (0, 1.5))
=
1
= 0.969 307 08
(1 + 0.021)1.5
From semi-annually compounded rate: 2.01% annualized rate,
1
Z (0, 1.5) =
1+
r2 (0,1.5)
2
2×(T −t)
1
=
1+
0.0201 2×1.5
2
= 0.970 45
We can see that the third discount factor implied by r1 (0, 1.5) is different from the rest.
Therefore either this rate is wrong or the other 3 rates are wrong. The reason is that discount factor must be unique, otherwise there is an arbitrage opportunity which is to borrow at the relatively low interest rate (say r (0, 1.5) = 2% since its corresponding discount factor is relative high) and lend at the relatively high interest rate (r1 (0, 1.5) = 2.10% since its corresponding discount factor is relatively low).
Q4. Use the semi-annually compounded yield curve in the
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 rates, we can compute the discount factors correspondingly.
From continuously compounded rate: 2.00% annualized rate:
Z (0, 1.5) = exp (−r (0, 1.5) × (T − t))
= exp (−0.02 × 1.5) = 0.970 45
From continuously compounded return on maturity: 3.00% (we did not cover this in class, but it means the unannualized interest rate)
Z (0, 1.5) = exp (−0.03) = 0.970 45
From annually compounded rate: 2.10% annualized rate:
Z (0, 1.5) =
1
(T −t)
(1 + r1 (0, 1.5))
=
1
= 0.969 307 08
(1 + 0.021)1.5
From semi-annually compounded rate: 2.01% annualized rate,
1
Z (0, 1.5) =
1+
r2 (0,1.5)
2
2×(T −t)
1
=
1+
0.0201 2×1.5
2
= 0.970 45
We can see that the third discount factor implied by r1 (0, 1.5) is different from the rest.
Therefore either this rate is wrong or the other 3 rates are wrong. The reason is that discount factor must be unique, otherwise there is an arbitrage opportunity which is to borrow at the relatively low interest rate (say r (0, 1.5) = 2% since its corresponding discount factor is relative high) and lend at the relatively high interest rate (r1 (0, 1.5) = 2.10% since its corresponding discount factor is relatively low).
Q4. Use the semi-annually compounded yield curve in the