微信客服：xiaoxionga100
微信客服：ITCS521
无代写-FIN430
时间：2021-04-02
University of Rochester, Simon Risk Management
PRACTICE QUESTIONS FIN430
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.
University of Rochester, Simon Risk Management
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 of the 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
University of Rochester, Simon Risk Management
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$).
University of Rochester, Simon Risk Management
d) What is the value of this bond?
Same as before 37.72 (swap has zero value at initiation: market value of the float that reprices yearly is
100$, 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.
University of Rochester, Simon Risk Management
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
University of Rochester, Simon Risk Management
d) Charlie Brown can enter a 6 year interest rate swap agreement to hedge BofA’s risk. Which side
of the IRS should he take? What should be the notional amount of the IRS so that BofA’s risk is
fully hedged? Suppose the 6 year T-bond paying the same amount as the fixed-payment in the IRS
has 5 year duration.
-he should take the fixed-payment side of the swap (receiving floating).
NS = [DA - k*DL]*A / [Dfixed-Dfloat]
NS = [15.7 – 0.65*4.3]*100 / [5-1]=$322.63mln.
Question 5
Genessee Light & Power is seeking to raise capital for a new gas fired power plant (gas is
the input). They are considering an equity offering but want to sweeten the deal for
potential investors. Their CFO suggests the following: The equity shares will entitle the
owner to a piece of the company as usual, however the owner of a share will also, at the end
of two years, receive a one time payment if the price of natural gas is worth more than $20
a barrel and the payment will max out at $3 per share.
a) Break this contract down into its component parts. How can you construct this hybrid
out of vanilla instruments. (Be sure to specify who is long and short each contract).
Issue equity. Buyer also gets (long) a call option on natural gas with a strike of 20 dollars. Buyer
then sells (short) a call option on natural gas at $23.
b) What is the effect of the sweetener on the amount of money raised given the same
number of shares issued?
For a given number of shares of ownership issued, the value of the “equity” is higher to the
buyer with a sweetener, so GLP will raise more money.
As the chief risk officer you point out that, this contract may amplify the downside
exposure that would occur if gas prices rise, since gas is an input to the firm.
c) How could you structure a similar hybrid that would still have some sweetener but
address this concern.
To address the risk manager’s concern, we can design the sweetener to pay out when gas price is
low (or when GLP would probably have more cash). For instance offer to pay out $3 if the price
of gas stays below $20. If the price of gas is between $20 and $23, pay off $23 – S. If the price of
gas rises above $23 pay nothing.
This contract is for the buyer to long a put option with strike of $23 and short a put option with
strike of $20.
学霸联盟
PRACTICE QUESTIONS FIN430
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.
University of Rochester, Simon Risk Management
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 of the 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
University of Rochester, Simon Risk Management
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$).
University of Rochester, Simon Risk Management
d) What is the value of this bond?
Same as before 37.72 (swap has zero value at initiation: market value of the float that reprices yearly is
100$, 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.
University of Rochester, Simon Risk Management
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
University of Rochester, Simon Risk Management
d) Charlie Brown can enter a 6 year interest rate swap agreement to hedge BofA’s risk. Which side
of the IRS should he take? What should be the notional amount of the IRS so that BofA’s risk is
fully hedged? Suppose the 6 year T-bond paying the same amount as the fixed-payment in the IRS
has 5 year duration.
-he should take the fixed-payment side of the swap (receiving floating).
NS = [DA - k*DL]*A / [Dfixed-Dfloat]
NS = [15.7 – 0.65*4.3]*100 / [5-1]=$322.63mln.
Question 5
Genessee Light & Power is seeking to raise capital for a new gas fired power plant (gas is
the input). They are considering an equity offering but want to sweeten the deal for
potential investors. Their CFO suggests the following: The equity shares will entitle the
owner to a piece of the company as usual, however the owner of a share will also, at the end
of two years, receive a one time payment if the price of natural gas is worth more than $20
a barrel and the payment will max out at $3 per share.
a) Break this contract down into its component parts. How can you construct this hybrid
out of vanilla instruments. (Be sure to specify who is long and short each contract).
Issue equity. Buyer also gets (long) a call option on natural gas with a strike of 20 dollars. Buyer
then sells (short) a call option on natural gas at $23.
b) What is the effect of the sweetener on the amount of money raised given the same
number of shares issued?
For a given number of shares of ownership issued, the value of the “equity” is higher to the
buyer with a sweetener, so GLP will raise more money.
As the chief risk officer you point out that, this contract may amplify the downside
exposure that would occur if gas prices rise, since gas is an input to the firm.
c) How could you structure a similar hybrid that would still have some sweetener but
address this concern.
To address the risk manager’s concern, we can design the sweetener to pay out when gas price is
low (or when GLP would probably have more cash). For instance offer to pay out $3 if the price
of gas stays below $20. If the price of gas is between $20 and $23, pay off $23 – S. If the price of
gas rises above $23 pay nothing.
This contract is for the buyer to long a put option with strike of $23 and short a put option with
strike of $20.
学霸联盟
