Homework 2 Possible Solutions

 

K Foster, Options & Futures, Eco 275, CCNY, Spring 2010

 

 

  1. How did your portfolio from last week do?  What is it worth now?  How much profit/loss did you make?  Can you plot its payoff graph?  Discuss.

Answers will vary.

  1. Complete Assignment Question 3.23 in Hull.

See Excel sheet: h*=0.95

  1. Complete Assignment Question 3.24 in Hull.

Current portfolio of $100m has beta=1.2; want beta of 0.5.  Index price is 1000; each contract is 250*index.  The beta change portion of the hedge is (β - β*) = (1.2 - .5) = 0.7.  The hedge part is .  So to lower the beta requires the company to be short 400*.7 = 280 contracts.  To instead raise the beta to 1.5 requires  (β - β*) = (1.2 - 1.5) = -.3 so the company must be long 400*.3 = 120.

  1. Complete Assignment Question 4.24 in Hull.

A 5% annual rate with semiannual compounding is 5.0625%; see Excel sheet.  At monthly compounding, this is  so solve for R= 4.9487%.  At continuous compounding, exp(Rt) = 1.050625 so R=4.9385.  Somewhat lower interest rates, compounded more frequently, give the same return as a 5% rate compounded semiannually.

  1. Complete Assignment Question 4.25 in Hull.

The semiannual zero rates, at different maturities, are:

6-mo

4%

12-m0

4.5%

18-mo

4.75%

24-mo

5%

Find the continuous equivalent rates.  The 6mo sets  so with Rsa = 0.04 then the continuous rate is 3.96%.  At 12 mo set  so a 4.5 sa rate means 4.45 continuously compounded.  At 18mo,  so 4.75 implies 4.69.  At 2 years  so 5% is 4.94%.  To find the forward rate for 18mo-24mo, use the semiannual rates so that gives 1.5 years at 4.75% plus .5 years at the forward rate equals 2 years at 5%, or , where R18 is 4.75 and R24 is 5, so RF is 5.75.  We want to find the value of a FRA paying 6% on $1m for the 18mo-24mo period.  Since 6% is better than the 5.75 implied by present market prices, the bond will trade at a premium  although, since the money comes in the future, it still comes at (PDV) discount.  This FRA basically costs money ($1m) in 18 months and then pays $1m plus $60,000 (i.e. 6% of 1m) in 24 months.  So the value is .  What is PDV(.0465,1.5) at semiannual compounding?   = .932.  What is PDV(.05,2) at sa?   =.90595.  So the value of the FRA is $28,302.65.

 

  1. Complete Assignment Question 4.26 in Hull.

To find the par yield, find the coupon (c) such that  

So c=5.045.  If this were compounded semiannually then a bond would be worth  

109.6405, so find its yield by finding y such that

(continuous):  

(or semiannual):  

Solve to get continuous yield as 4.9%; semiannual yield as 4.9675.

  1. Complete Assignment Question 4.27 in Hull.

Given table:

Principal

t

coupon (annual, half paid every 6 mo)

bond price

100

.5

0

98

100

1

0

95

100

1.5

6.2

101

100

2

8

104

To find zero rates at 6, 12, 18, 24 mo.  The zero rate (continuously compounded) at 6mo is that rate that sets  so R6=4.04%.  At 12mo the zero rate solves  so R12 = 5.13%.  At 18mo the zero rate is that which makes the PDV of the stream of payments from the bond on line 3 equal to 101  the stream of payments is half of 6.2 paid in 6mo, then half of 6.2 paid at 12mo, then half of 6.2 paid at 18mo, then the principal at 18mo, so .  We know R6 and R12 so substitute these in,  and solve to get R18 = 0.0544.  The same procedure for the two-year zero rate: the stream of payments is half of 8 paid in 6mo, half of 8 in 12mo, half of 8 in 18mo, then half of 8 plus principal in 24mo.  So  and now we know R6, R12, R18, and just solve for R24.  So R24 is 0.0581.

To find forward rates for 6-12, 12-18, and 18-24.  So if 0-6 rate is 4.04, 0-12 is 5.13, 0-18 is 5.44, and 0-24 is 5.81, then the 6-12 rate must be 6.22 (because the average of 6.22 and 4.04 is 5.13).  Then the 12-18 rate is 6.07 (again because the average of 4.04, 6.22, and 6.07 is 5.44); the 18-24 rate is 6.91 (because average of 4.04, 6.22, 6.07, and 6.91 is 5.81).

To find 6, 12, 18, 24 par yields.  The par yields on the latter two bonds are 6.2% and 8%.

To find price and yield of 2-yr bond providing semiannual coupon of 7%.  Assuming principal is 100, this is  = 115.226.

  1. Complete Assignment Question 4.28 in Hull.

Portfolio A has a one-yr bond with face 2000 and a 10-yr bond with face 6000.  Portfolio B has a 5.95-yr bond with face 5000.  Current yield on all bonds is 10%.  To find the duration of the two portfolios, note that the duration of B is exactly its term, 5.95 (since  ).  The duration of A is .  We need to find the value of B, the portfolio with the two bonds.  This is  =4016.94.  Substitute these values in for the duration of A to find DA = (1809.67 + 22072.77)/4016.94 = 5.95.  The spreadsheet shows that, for small changes in interest rates, the approximation to the bond valuation works well, but is a poor approximation for large changes.

  1. What are the LIBOR rates currently?  Using these rates calculate the value (under continuous interest) of a bond paying $1m in 3 months, of a bond paying $3m in 6 months, and of a bond paying $10m in a year.  Compare these with Treasury rates (calculate the three bond prices again).

The website federalreserve.gov has all of this data; these are as of Feb 11.  I used Eurodollar rates for LIBOR 3 and 6 mo; Swap rates for 1-year.  The data and calc's are on the spreadsheet. 

  1. What are current fixed rates on 30-year home mortgages?  Suppose you want to buy a property for $300,000.  What series of 30 equal payments would have a present value of $300,000 (assume annual compounding)?  Suppose compounding were monthly (12 per year so 360 payments over 30 years)
     what level payments would have a $300,000 present value?

The Fed reports current fixed rates are 4.97%.  We want to find what coupons, c, solve  so solve  with Excel so c=$19,449 each year.  If instead it's monthly then  so Excel gives c = $1493 per month (or $17,914 per year).

  1. Suppose a mortgage lender offered an initial 'teaser' where the payments for the first 5 years would be 150 bp less than the fixed rate but then jumped up to a higher rate for the remaining 25 years.  What must this higher forward rate be, to have the same $300,000 present value?

Now we want to find what values of coupons, c1 and c2, solve .  Of course a variety of different payment options work.   First suppose that the initial coupon was made at a rate as if the whole loan were 150 bps lower, i.e. $16,250.  So with Excel find that later payments (c2) of $20,698 would do it.

  1. A bond pays $7m in ten years.  Its current price is $4.89m.  What continuously compounded 10-year zero rate is implied?  What semiannual 10-year zero rate is implied?  A different bond (paid by the same entity with identical risk characteristics) pays $4m in five years.  Its current price is $3.27m.  What continuously compounded 5-year zero rate is implied?  What semiannually compounded 5-year zero rate is implied?  What is the year-5-to-year-10 forward rate (continuously and semiannually)?

From the basic definitions, at continuous compounding,  so r = 3.59%.  At semiannual compounding, we would have  so R = 3.62%.  (Again, see the Excel sheet for details.)  For the other bond,  so r = 4.03%.  At semiannual compounding,  so R = 4.07%.  The forward rate from 5 to 10 years is therefore the rate that, for continuous compounding, makes the average of 4.03 and rf equal to 3.59  clearly it must be lower and in fact must be 4.47%.  For semiannual compounding, the forward rate must make the average of 4.07 and it equal to 3.62, so the forward rate is 4.52%.

  1. The interest rate jumps by 15 basis points.  What would be the new prices on the two bonds from the problem above?

Now the bond prices are  = 4.816 and  4.818.