Revising financial instruments

Exercise 1. A bond issued by XYZ Oil company has the following cash flows:

    The holder receives no interest.

    At maturity of the bond, the company promises to pay $1000 plus an additional amount

based on the price of oil at that time.

    The additional amount is the product of 230 and the excess (if any) of the price of the

barrel of oil at maturity over $25.

    The maximum additional amount paid is $3,450 which corresponds to the case when

the price of oil at maturity is bigger than $40 per barrel

Can you replicate the payoffs on the bond through a combination of a regular bond, and long / short position in regular call options on oil? If so, fully describe the strategy. If not, explain.

Solution.

Suppose St is the price of oil at maturity. The cash flows for each possible realization of the oil price S(t), are as follows:

The bond is, therefore, equivalent to a regular bond plus a long position in 230 call options with a strike of $25 and a short position in 230 call options with a strike price of $40. The investor has a bull spread on oil.

Revising Interest Rate Risk Measures

Exercise 2. Bank of America is trying to measure and hedge its interest rate exposure. BofA’s assets are mostly fixed rate, with duration of 15.7 years and maturity of 24 years. The bank’s liabilities are mostly floating rate, with duration of 4.3 years and a maturity of 6 years. The bank is levered at 65%. For numerical simplicity, assume asset size of $100mln. The relevant interest rate is 5%. Floats are repriced yearly.

James Bond is a trader for Bank of America. He has been given the task of hedging the interest rate exposure of the Bank. Assume there is no credit risk.

a) What are the maturity and leverage-adjusted duration gaps ofthe bank? Why is there a difference and what do the two tell you about upside / downside interest rate risk exposure?

b) What is the expected change in BofA’s equity for a 1% increase in the interest rate?

Solution.

a)  Maturity gap is: 24-6 = 18 years.

Leverage-adjusted duration gap is:  15.7-0.65*4.3 =  12.9 years. Duration gap is  significantly smaller than maturity gap, reflecting: a) timing of cash flows; and b) leverage.

The bank equity value goes up when interest rates fall and goes down when interest rates rise.

b)  The expected change in BofA’s equity is:

ΔE = -(DA-k*DL)*A*(ΔR)/(1+R)

ΔE = -(15.7-0.65*4.3)*100*(0.01/1.05) = -$12.29mln

Question 3

A 2 –year bond has the following payoffs.

The current interest rate is 4%. Assume a flat term structure of interest rates. Answer the following.

a) What is the value of the bond?

20/1.04 + 20/1.04^2 = 37.72

b) What is the duration of the bond?

(20*1/1.04 + 20*2/1.04^2) / 37.72 = 1.49

A different 2-year Bond has the following Payments

Where R(t) is a floating rate that varies with time. Today’s interest rate is 4%. Assume a flat term structure of interest rates. Answer the following.

c) How would you construct this bond using the original bond in the previous table and a derivative position?

Buy the bond, sell a 2-year swap (pay floating-rate, receive fixed-rate). Specifically:

Original bond: $20. Receive fixed rate: $4 (4% fixed on notional of 100$). Pay floating: 100*R(t) (R(t) float on notional of 100$).

d) What is the value of this bond?

Same as before 37.72 (swap has zero value at initiation: market value of the float that reprices yearly is 100$, market value of the fixed with coupon rate at par is 100. At initiation the swap doesn’t generate  equity, but as int. rate change the swap changes in value, S).

e) What is the duration of this bond?

This bond is effectively the sum of 3 bonds: the initial bond, a 4% fixed rate bond with 100 face value, and a 100$ float bond. The duration is the weighted duration of the portfolio.                                         The duration of the 4% fixed bond is: (4*1/1.04 + 104*2/1.04^2) / 100= 1.96yr

The duration of the float is the time between payments: 1yr                                                                         Remember we are short the float.                                                                                                                   So the duration of the portfolio (i.e. the new bond) is:                                                                                   1.49*37.72/37.72 + 1.96*100/37.72 -1*100/37.72= 4.035                                                                            The swap is adding duration to the standard bond. Intuitively: you are receiving fixed and paying float; it is almost like the fixed rate is an asset (receiving), and the float rate is a liability (what you pay); hence    you have a positive gap, so more duration, you are adding to your standard 2 yr. maturity bond. So you    are extending its duration. Also remember duration and maturity are NOT equal. Duration is your “beta”, your measure of sensitivity to interest rate shocks. The receive fixed-pay float swap is effectively adding sensitivity. Hence the portfolio of standard bond plus swap has a longer duration.

Question 4

Bank of America is trying to measure and hedge its interest rate exposure. BofA’s assets are mostly fixed rate, with duration of 15.7 years and maturity of 24 years. The bank’s liabilities are mostly     floating rate, with duration of 4.3 years and a maturity of 6 years. The bank is levered at 65%. For numerical simplicity, assume asset size of $100mln. The relevant interest rate is 5%.

Charlie Brown is a trader for Bank of America. He has been set the task of hedging the interest rate exposure of the Bank. He has access to put options on US Treasuries. The delta of the put option is - 0.7, the bond price is 80 and its duration 1.3years. The option premium is 0.65$ per option. Assume there is no credit risk.

a) What are the maturity and leverage-adjusted duration gaps of the bank? Why is there a  difference and what do the two tell you about upside / downside interest rate risk exposure?

- Maturity gap is: 24-6 = 18 years.

- Leverage-adjusted duration gap is: 15.7-0.65*4.3 = 12.9 years. Duration gap is significantly smaller than maturity gap, reflecting: a) timing of cash flows; and b) leverage.

- The bank equity value goes up when interest rates fall and goes down when interest rates rise. Since    leverage-adjusted duration gap is positive, when interest rates rise, value of assets goes down more than the value of liabilities and vice versa.

b) What is the expected change in BofA’s equity for a 1% increase in the interest rate?

ΔE = -(DA-k*DL)*A*(ΔR)/(1+R)

ΔE = -(15.7-0.65*4.3)*100*(0.01/1.05) = -$12.29mln

c) What position in put options does James Brown need to take to completely hedge shareholders’ interest rate exposure? What is the cost of the hedge?

NP = [DA - k*DL]*A / [_put * D* P_bond] = (15.7-0.65*4.3)*100/ (0.7*1.3*80)=17.73 James needs to long 17.73 put options. The cost of this hedge is:

17.73*0.65=$11.52