FINS 2624 Problem Set 2 Solutions PDF

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FINS2624 PROBLEM SET 2 SOLUTIONS Question 1. Using Bond A, we know that the yield to maturity for a 1-year investment, y1, satisfies: 95.24 

100 100 →    1  5% 1   95.24

Using Bond B, we solve for the quantity y2 that satisfies: 110 10  1   󰇛1   󰇜

107.42 

110 10  󰇛1   󰇜 1.05

107.42 

→   6% Using Bond C, 140.51 

20 20 120   1   󰇛1   󰇜 󰇛1   󰇜

140.51 

20

1.05



120 20  󰇛1.06󰇜 󰇛1   󰇜

  5%

Finally, using Bond D, 85.48 

100 →   4% 󰇛1   󰇜

Graphically, the yield curve is as follows:

TermStructure 7 6

Yield(%)

5 4 3 2 1 0 0

1

2

3

Year

4

5

Question 2. The no-arbitrage price of Bond E, by the present value equation, is: 25 125  1   󰇛1   󰇜 Using y1 = 5% and y2 = 6% from Q1, this no-arbitrage price is:  

125 25   135.059   󰇛1.06󰇜 1.05

which is lower than the given market price of $136. Bond E is therefore overpriced. Thus, to exploit the arbitrage opportunity, we must short Bond E and buy some combination of Bonds A and B.

Shorting 1 unit of Bond E:

0

0

X*($10)

0

Total

X*($110) 2

1 Y*($100)

Y*(‐95.24)

Time Short 1 unit of Bond E Buy X units of Bond B Buy Y units of Bond A

2

1

X*(‐107.42)

Buying Y units of Bond A:

‐$125

‐$25

136

Buying X units of Bond B:

2

1

0

0 136 X*(-107.42) Y*(-95.24)

1 -25 X*(10) Y*(100)

2 -125.0 X*(110.0) 0

?

$0

$0

Match cash flow at time 2: 125    110 →  

125  1.136 110

Now, match cash flow at time 1: 25    10    100,   1.136 25  1.136  10    100 25  1.136  100 →  0.136 100  Thus, the arbitrage portfolio is as follows: - Short sell 1 unit of Bond E - Buy 1.136 units of Bond B (at a price of 1.136*107.42 = $122.03) - Buy 0.1364 units of Bond A (at a price of 0.136*95.24 = $12.95)

Updated cash flow timelines: Shorting 1 unit of Bond E:

1

0

0

0

Total

1.136*($10)

1.136*($110) 2

1

0.136*(‐95.24)

Time Short 1 unit of Bond E Buy X units of Bond B Buy Y units of Bond A

2

1

1.136*(‐107.42)

Buying 0.136 units of Bond A:

‐$125

‐$25

136

Buying 1.136 units of Bond B:

2

0.136*($100)

0

0 136 -122.03 -12.95

1 -25 11.36 13.64

2 -125.0 125 0

$1.02

$0

$0

Question 3. Method 1 Using Bond X, we have: 105.6 

110 10 … … … 󰇛1󰇜  1   󰇛1   󰇜

and similarly, using Bond Y, 123.86 

120 20  … … … 󰇛2󰇜 1   󰇛1   󰇜

Multiplying both sides of (2) by 110/120 gives: 123.86 

110

120



2200 1 110 … … … 󰇛3󰇜   120 1   󰇛1   󰇜

Subtracting (1) from (3) gives: 123.86 

2200 1 110  10  105.6   120 120 1  

This equation has only one unknown y1. We can solve that y1 ≈ 0.04975856 ≈ 0.05 Substituting the solution of y1 into (1): 105.6 

110 10 →   0.07  1.05 󰇛1   󰇜

Method 2 Buy 1 unit of Bond Y and 120/110 units of Bond X: Time Buy 1 unit of Bond Y Sell (120/110) units of Bond X Total

0 -123.86 115.2 -8.66

1 20 -10.91 9.09

Calculate y1: 8.66 

9.09 →   4.97% 1  

Next, calculate y2 using Bond X’s price:   105.6 

     1   󰇛1   󰇜

110 10  →   0.07 1.05 󰇛1   󰇜

2 120 120 0

Question 4. [a] Liquidity risk If you intend to invest for only 1 year but hold a 2-year bond, you have to sell the bond in 1 year. So, your return in one year will be: 1-year HPR = (P1 + c1)/P0 = (P1 + 10)/P0 (you should be able to calculate P0 given the information). And P1 = (c2 + FV)/(1 + 1y2) = 110/(1 + 1y2) is unknown at t = 0, so is P1 and HPR. Hence holding 2-year bond is risky here – you are exposed to liquidity risk (you find it difficult to sell when you want to).

1y2

[b] Reinvestment risk If you intend to invest for only 3 years but hold a 2-year bond, you have to reinvest the proceedings from this bond at the end of year 2 for another year. Let’s assume your 2-year HPR of investing in that 2-year bond is HRP2 = [c1*(1 + 1y2) + c2 + FV]/P0 – 1. Now at the end of year 2, you have to reinvest the total cash flows from this 2-year bond (i.e., c1*(1 + 1y2) + c2 + FV) to the end of year 3 at the rate 2y3. And your 3-year HPR = (1+ HRP2)*(1+2y3) – 1. Since 2y3 will be unknown until end of year 2, so is your 3-year HPR – you are exposed to reinvestment risk. (Note:to calculate 2-year HPR, you have to calculate your aggregate cash flows at the end of year 2, which requires the information of 1y2 to reinvest the year-1 coupon to the end of year 2. So 2-year HPR of the bond is also unknown at time 0 - but will surely be known at the end of year 2. So technically speaking, even for an investor with 2-year horizon, she is still exposed to reinvestment risk if she invests in a 2-year COUPON bond – and even worse if she has a 3-year investment horizon in this question. An investor with 2-year horizon will not bear the reinvestment risk if she invests in a 2-year zero-coupon bond.)

Question 5. Suppose we have $1 in period 0, given the spot rates ys, yt and the forward rate sft, s...