Homework 3 Possible Solutions

 

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

 

 

  1. Suppose a financial institution is creating synthetic Treasury "strips".  You set up a Special Purpose Vehicle that buys 100 Treasury bonds paying for 5 year s.  Each bond has $1m face value and pays $30,000 every 6 months.  This allows you to create 10 synthetic bonds paying $(3,000,000
     p) at some date in the next five years (and a zero-rate principal payment)
     each synthetic makes only one payment, making calculation of the zero rate simple.  The amount "p" is the price that you charge for these transactions.  If you charge 1bp what is the present discounted value of your fees for the transaction?

Each bond will pay 0.03 every 6 months for the next 5 years and then 100 after 5 years.  So the current value of each bond is .  The strips pay  and of course one big one of .  So the difference in value between the original bonds and the strips, if there are 10 payments of p=0.0001, is .  (Given interest rates you could get a number but for now it's just a formula.)

  1. Consider a very simple example of a FRA between a country (call it Zembla, "Z") and "Stib" (Sharp-Toothed Investment Bank).  Assume current LIBOR forward for the period from 5 to 6 years into the future is 5%.  Both sides expect that the actual LIBOR in that same period will be 5.5% (assume neither side is fooled).  Assume that the principal is $1m.  Assume that money flows from Zembla to Stib after 6 years but Stib pays Zembla now.  What range of values for "RK," the rate of interest agreed in the FRA, would pay Zembla money now?  How much?  Explain the way(s) in which this is like and/or unlike a loan.

Stib could pay Z a certain amount of money now, in return guaranteeing that it will allow Z to invest $1m at 5% in the future while, for itself, locking in an investment of $1m at 5.5% for that same future period.  So Z gets $1.05m after 6 years but Stib gets $1.055m after six years, a difference of $5000 (50bps).  So the present value of this is 5000exp{-.055*5} = 3798.  The upfront payment makes it somewhat like a loan but where the payback period is stretched into the future.

  1. Please complete Assignment Question 5.27 in Hull.

A trader owns gold; can buy at $550 & sell at $549.  Can borrow at 6% or invest at 5.5% (annual compounding).  What 1-yr forward leaves no arbitrage?  If buying gold forward returns more than 6% then the trader would want to borrow at 6% and invest in gold, so the forward price must be F0 such that , so, with the rate at which she can borrow, rb=0.06, F0 < 583.  If the forward price of gold returned less than leaving money in the bank then the trader would sell short at $549 and deliver at the forward rate after investing her money at 5.5%, so the forward price must be such that  where ri is the rate at which she can invest, 0.055, so F0 > 579.  So the forward price must be within this $4 range.

  1. Please complete Assignment Question 6.23 in Hull.

Bank can borrow/lend at LIBOR; 90-day is 10% and 180-day is 10.2%, both continuous with actual/actual day counts.  Eurodollar futures in 91 days is 89.5  what arbitrage opportunities are open?  The Eurodollar futures contract means that I can, in 91 days, buy a bond paying 100 in 180 days, for 89.5.  This implies an interest rate of  (annualized) so r=11%.  The LIBOR rates imply that the forward rate from 90  180 days is 10.4%, so it would be better to invest for 90 days and buy the forward.

  1. Please complete Assignment Question 7.20 in Hull.

Company A wants to borrow dollars at fixed rate; Company B wants to borrow sterling at fixed rate.  Co A is quoted r£ = 11%, r$ = 7; Co B r£ = 10.6% and r$ = 6.2%.  Design a swap that will net a bank 10bps and gain each company 15bps. 

Consider Scenario I (no swap) on 100m of each currency:

A borrows $ at 7%

each year A pays $7m

B borrows £ at 10.6%

each year B pays £10.6m

Versus Scenario II where they borrow in the opposite currency and swap,

A borrows £ at 11%

each year A pays $6.2m

B borrows $ at 6.2%

each year B pays £11m

Clearly Scenario II suffers from the fact that B is worse off  so B wouldn't take that deal since it is worse off by 40 bps.  But there's room for negotiation since A improves its rate by 80 bps.  So they can work out Scenario III, where they borrow and swap so that:

A borrows £ at 11%

each year A pays $6.85m

B borrows $ at 6.2%

each year B pays £10.45m

and the i-bank in the middle gets 10 bps.

  1. Two riskless bonds for the same company are valued in the market (assume they are now riskless because the US government guarantees them!).  The first, with face of 10,000, pays this principal in six months.  It is currently worth 9875.78.  The second bond is a strip of the coupons, paying 500 in three months and then another 500 in six months.  It is currently worth 990.06.
    1. What is the six-month zero rate?
    2. What is the three-month zero rate?
    3. What is the three-to-six month forward rate?

The 3-mo zero rate is found by setting 9875.78 = 10000exp{-rt} where t=3/12, so rt = ln[9875.78/10000] so r=5%.  then the second bond is worth the PDV of its payments, so 990.06 = 500 exp{-r3t} + 500 exp{-r6t}, where r3 is the 3-month zero rate and r6 is the 6-month zero rate and the respective t's are 3/12 and 6/12, so .  We just solved to find r3=0.05 so then solve to find r6 is 1.5%.  the forward rate went negative 2%!

  1. A portfolio of bonds includes the following:

§             Bond ZZZ pays a $4000 coupon every six months, including 6 months from today, 12 months from today, and 18 months from today.  It also pays its principal of $200,000 in 18 months at the same time as its last coupon.  Its current market value is $201,359.31.

§             Bond YYY pays its $8500 coupon in 6 months and then that coupon again in 12 months along with its $500,000 principal.  Its current market value is $499,342.04.

§             Bond VVV pays its principal of $600,000 plus its $8000 semi-annual coupon in 6 months.  Its current market value is $595,960.79.

a.      Find the six month zero rate (continuously compounded).

b.      Find the six-to-twelve month forward rate and the twelve month zero rate (continuously compounded).

c.       Find the twelve-to-eighteen month forward rate and the eighteen month zero rate (continuously compounded).

d.      Find the par yield for each bond.

e.      Find the semiannually-compounded (discrete time every 6 months) zero rates.

f.        What is the duration of bond VVV?

Continuous

Work backwards to get zero rates, from VVV.   We know that the value of a bond is the present discounted value of its future cash flows, so BVVV = $608,000*PDV(r, t), where t=6 months = 0.5 years.  In continuous time the PDV(r,t) function is e-rt.  The interest rate to be used is the zero rate corresponding to that time, so call it r6.  Substitute these in so  .  We are given that BVVV = 595,960.79; solve as follows:

 

So r6 = 4%.

 

Next use this information on r6 to find r12, using the price of bond YYY.  (If you noticed, I skipped Bond XXX  not going there!)  The present discounted value of bond YYY is 8500PDV(r6,t=6mo) + 508500PDV(r12,t=12mo).  So the equation is

, where we found r6 above so put that in and solve:

 

So r12= 3.4% and this means that the forward rate is that rate which averages with 4% to produce 3.4%; so 2.8%.

 

Finally use these first 2 zero rates to figure out the r18 rate.  The present discounted value of the bond cashflows give 201,359.31 = 4000PDV(r,t=6mo) + 4000PDV(r,t=12mo) + 204,000PDV(r,t=18mo).  So the equation is:

.

Substitute in the values already found for r6 and r12 and solve:

 

So r18 is 3.5%.  So the forward rate is that rate which makes the average of 4%, 3.4%, and x% equal to 3.5%; so this is 3.1%.

 

Par Yield

The par yield is the coupon rate that makes the bond price equal its par value.  For bond VVV, this is the c such that  (where we normalize the par value to 100) so solve to get  and c=4.04.  The par yield for bond YYY is c such that .  Solve so c=3.43.  For bond ZZZ,  so c=3.53.

 

Semiannual

From VVV we know that (using semiannual compounding, we just use a different expression for PDV(r,t,m)) , where t is 0.5 and m=2, so get r6=4%. 

 

Then use YYY to find the next zero rate.  Thus  where we know m=2 and r6 is 4% so just solve for r12.  So , , then ,  and r12 = 3.53%.  So the forward rate to get the average down from 4% to 3.53% must be 3.015%.

 

Then use ZZZ to find the final zero rate from  and r18 = 0.0353.

 

Duration

This one is easy: the duration of a bond with only one payment is the time to that one payment, so 6 months.