Problem Set 7 Fall 2020 Solutions PDF

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Problem Set Solutions for Econ 136 Fall 2020...

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Econ 136: Financial Economics Problem Set #7 – Solutions Due Date: October 30, 2020 General Instructions: • Please upload a PDF of your problem set to Gradescope by 11:00 pm. • Late homework will not be accepted. • Please put your name, student ID & your GSI’s name at the upper right corner of the front page.

1. Calculate the DV01 of a 34-year annual-pay floating-rate annuity in a 7% yield environment. Solution: Since a floating-rate annuity is the coupon portion of a floating-rate bond, and since a floating-rate bonds has zero dollar duration, it follows that the dollar duration of a floating-rate annuity is the negative of the dollar duration of the associated principal payment. This, together with the fact that the DV01 is 1/10,000 the dollar duration allows us to write   1 100 d (1) DV01 = 10, 000 dY (1 + Y )34   3400 1 = −0.03185 (2) =− 10, 000 1.0735 which one would interpret as -$0.03185 per $100 of face value. 2. Calculate the 4-year swap rate (par coupon) on August 15, 2020 given your results for question 4 in problem set 5. Solution: The par coupon C for this semi-annual-pay bond is the solution to 1=

8 X C n=1

or

2

DFi + DF8

(3)

1 − DF8 C = 2 × P8 n=1 DFi

(4)

which, given the results from problem set 5, becomes C = 2×

1 − 0.990114 0.999079 + 0.998471 + . . . + 0.992545 + 0.990114

= 0.002483 or 0.2483%

(5) (6)

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3. Use Excel or your favorite software to reproduce the table given in slide 16 of lecture 12 and then calculate the change in the swap price if all of the forward rates increase by 235 basis points. In this exercise assume that you have just entered into the swap (so the fixed leg payments are all 10% as indicated in the slide). Briefly explain the swap price change. Solution: Payments Year Forward DF Fixed Float 1 11.10 0.9001 10.0000 11.1000 2 11.62 0.8064 10.0000 11.6232 3 12.17 0.7188 10.0000 12.1746 4 12.76 0.6375 10.0000 12.7604 5 13.39 0.5622 10.0000 13.3885 6 14.07 0.4929 10.0000 14.0688

Present Values Fixed 9.0009 8.0636 7.1885 6.3750 5.6223 4.9288

Float 9.9910 9.3725 8.7517 8.1348 7.5274 6.9343

Sum of PV Payments = 41.1791 50.7116 PV of Principal = 49.2884 49.2884 PV of Payments plus PV of Principal = 90.4675 100.0000 The swap price has changed by 9.5325 (= 50.7116 - 41.1791 or 100.0000 - 90.4675). If it was a receiver swap the price would have decreased because one is receiving fixed and the present value of those payment has decreased relative to that of the floating leg; if it was a payer swap the price would have increased because one is receiving floating and the present value of those payment has increased relative to that of the fixed leg. A few things to note: (a) On both legs (fixed and floating) of the swap the PV of the principal remained the same: these cash flows always cancel as long as both legs of the swap are in the same currency. (b) The change in swap value comes from the change in the PV of the coupons. (c) The PV of the floating rate bond (PV of floating coupons and principal) is unchanged! The price of a the floating-rate bond is unchanged at par while the price of the fixed-rate bond has changed. (d) For the fixed rate bond, when rates went up the price went down.

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4. A client of yours wants to buy USD 25 million of the 30-year Treasury bond at the next Treasury auction, but they are concerned about a decline in the price of the bond if interest rates increase. (a) Design a swap hedge to meet your client’s needs and specify the following terms of the swap: • The swap rate. Solution: The swap rate is a par rate. Since bonds are issued at par the swap rate would be the fixed-coupon rate of the 30-year Treasury bond. • Whether the swap is a payer or a receiver. Solution: Since the client is worried about the price risk of the bond, the bond should be swapped to floating. To this end the client would need to receive floating and pay fixed, the fixed payments would come from the Treasury bond leaving the client with a net floating-rate coupon. Since the client would be paying fixed they would be entering into a payer swap. • The notional amount of the swap. Solution: For the fixed swap payments to cancel the fixed coupons as described above, they need to be exactly the same amount as those of the fixed coupons. Thus, the notional amount would be USD 25 million. • The tenor of the swap. Solution: For the fixed swap payments to cancel the fixed coupons as described above, they need to be made at the same time as the fixed coupons. Thus, the tenor of the swap would be 30 years. • The payment frequency of the swap. Solution: For the fixed swap payments to cancel the fixed coupons as described above, they need to be made at the same time as the fixed coupons. Thus, the payment frequency would be semiannual. Note that the answer to some of these terms may depend on the terms of the bond when it is issued.

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(b) Briefly explain the mechanics of how the swap hedges the bond to your client. Include the dollar duration and convexity of the bond plus swap portfolio in your explanation. Solution: The client pays the dealer the fixed coupon that they will receive from the bond and the dealer will pay the client a floating-rate coupon. The fixed-coupons are, thus, transformed (swapped) into floating-rate coupons. The fixed-rate bond is now a floating-rate bond with zero duration and zero convexity. Price risk has been exchanged for coupon-amount risk. 5. In 2000 Standard and Poor’s reported the default probabilities for CCC bonds shown in the table below. Assuming a recovery rate of 47% calculate the price (as a percent of par) of a 3-year 6% annual-pay bond in a 10% yield environment. Year: Marginal default probability (%):

1 2 3 4 5 20.93 8.99 7.38 5.22 6.08

Solution: Considering defaults at the end of each year: • Default at the end of year 1: P1 = P1

0.47 × 106 rrec (100 + C ) = 9.47939 = 0.2093 1.10 1+Y

(7)

• Default at the end of year 2: 

 C rrec (100 + C ) P2 = (1 − P1 ) P2 + 1+Y (1 + Y )2   0.47 × 106 6 + = 3.31451 = 0.7907 × 0.0899 1.10 1.102

(8) (9)

• Default at the end of year 3: C rrec (100 + C ) C + + P3 = (1 − P1 ) (1 − P2 ) P3 2 1+Y (1 + Y ) (1 + Y )3   6 0.47 × 106 6 + + = 0.7907 × 0.9101 × 0.0738 1.10 1.102 1.103 = 2.54087 

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

(10) (11) (12)

• No default: C (100 + C ) C + + P4 = (1 − P1 ) (1 − P2 ) (1 − P3 ) 2 1+Y (1 + Y ) (1 + Y )3   6 106 6 + + = 0.7907 × 0.9101 × 0.9262 1.10 1.102 1.103 = 60.02081 



(13) (14) (15)

Summing these, the price is P =

6 X

Pi

(16)

i=1

= 9.47939 + 3.31451 + 2.54087 + 60.02081

(17)

= 75.35558

(18)

Alternatively, the terms the sum of the expression on slide 8 of lecture 10 are given below together with the price as a percent of par.

t

Pt

St C+St−1 Pt rrec (100+C ) (1+r)t

St

0 0 1 1 0.2093 0.79070 2 0.0899 0.71962 3 0.0738 0.66651 100S3 (1+r)3

Price

5

13.79230 6.49512 4.99239 60.02081 75.35558

6. In the Merton model the “distance to default” is given by   V 1 ln σ B where σ is the asset volatility, V is the asset value and B is the amount of debt used to buy the asset. Explain briefly why that name applies to this term. Solution: This term represents the “distance to default” because at the default boundary (V /B = 1) we have that ln (V /B) = 0. When V /B > 1 the firm is not in default. By normalizing with respect to asset volatility the “distance to default” becomes a measure of risk.

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