# 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 following table to price the some ﬁxed income securities:

1

Maturity T

0.50

1.00

1.5

2

Yield r2 (0, T )

6.49%

6.71%

6.84%

6.88%

(a) 1.5-year zero coupon bond

Answer: For the pricing question, we will rely on discount factors. We are given the semi-annual compounded interest rates, so let’s convert these into discount factors. the same set of discount factors will be used to price all ﬁxed income securities. 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 1.5-year zero coupon bond has time to maturity 1.5 years (and face value $100 which is the common assumption in the text), hence its price today should be: Pz (0, 1.5) = 100 × Z (0, 1.5) = 100 × 0.904 037 4 = $90.4 037 4 (b) 2-year coupon bond paying 15% semiannually

Answer: Note that the number 2 means the bond has time to maturity of 2 years. A coupon bond is a portfolio of zero-coupon bonds, with face value equal to the coupon payments (100 × 0.15 = 7. 5) before maturity, and coupon + face value of 100 = 107.5 at 2

maturity. Hence its price should be:

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

(d) 1.5-year coupon bond paying 9% annually

Answer: Similar to (b), just that the coupon bond is in between coupon payments. Draw the timeline (!!!) to see the timing of the cash ﬂows: Year

0.5

1.5

Cash Flow 0.09 × 100 = 9 0.09 × 100 + 100 = 109

Hence:

Pc (0, 1.5) = 9 × Z (0, 0.5) + 109 × Z (0, 1.5)

= 9 × 0.968 569 91 + 109 × 0.904 037 4

= $107. 257 21

(e) 2-year ﬂoating rate bond with zero spread and semiannual payments Answer: The ﬂoating rate bond is right on the reset date. Right after the coupon payment the ﬂoating...

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