AFF811/FIN801: Financial Risk Management Eric Terry AFF811/FIN801: Financial Risk Management Eric Terry
1. Introduction Answers
Additional Practice Exercises
Question 1.
Question 1. (a) The one-day 99.5% VaR is
(a) Hy Marx has an investment portfolio with a daily mean return of $500 and a daily standard 2.57583 × $18,000 − $500 = $45,864.94
deviation of $18,000. If returns for this portfolio are i.i.d. normal through time, what is the and thus the 250-day 99.5% VaR is
250-day 99.5% VaR of this portfolio? 250 $45,864.94 = $725,188.
(b) Suppose instead that this portfolio had a 10-day mean return of 0.04%, a 10-day return
standard deviation of 3.0%, and a current value of $800,000. If returns for this portfolio are (b) The 10-day dollar mean return and dollar return standard deviation are
i.i.d. normal through time, what would be the 40-day 97.5% ES of this portfolio? = $800, 000 .0004 = $320
SD = $800, 000 .03 = $24, 000
Question 2*. Therefore, the 10-day 97.5% ES is
(a) The daily return distribution for the investment holdings of Hy Raitt Funds has fat tails that
2.33780×$24,000 − $320 = $55,787.20.
can be approximated by a non-standard t-distribution with a mean of 0.05%, standard Consequently, the 40-day 97.5% ES would be
deviation of 1.0%, and 10 degrees of freedom. The current value of the investment holdings
is $12.5 million. What is the one-day 99% VaR of this portfolio? 4 $55, 787.20 = $111,574.
(b) Annual operational losses (in $ million) for this firm can be approximated by a chi-square
distribution with 15 degrees of freedom. What is the one-year 99.5% VaR for the firm’s Question 2.
operational losses? (a) The 99% cut-off for a standard t-distribution with 10 degrees of freedom is 2.76377. Also,
Note: A standard t-distribution table and chi-square table have been posted on D2L and can be the daily dollar mean return and return standard deviation are
used to get approximate answers to this question. = $12.5M .0005 = $6, 250
SD = $12.5M .01 = $125, 000
Question 3. Therefore, the one-day 99% Var is:
I.N. Stein Financial is expected to have $5 million in earnings before taxes. The firm is taxed at 2.76377 × $125,000− $6,250 = $339,221.
a 40% rate. Credit losses are expected to be $500,000 but would reach $6 million in a worst-case
scenario. What is the firm’s after-tax RAROC? (b) The 99.5% cut-off for a chi-square distribution with 15 degrees of freedom is 32.801.
Therefore, the one-year 99.5% VaR for operational risk is $32.801M
Question 4.
Critically comment on the following statement: “Risk management is simple. You first identify Question 3.
the major risks your firm faces and then you determine the cheapest way to minimize these For this firm,
risks.” Expected profit = $5M (1 − .4) = $3M
Economic capital = $6M − 0.5M = $5.5M
Question 5.
and thus
Identify the type of risk management policy being followed in each situation. RAROC = 3M / 5.5M = 54.6%.
(a) Ample Computers reinforces its manufacturing plant that is near a major earthquake fault.
(b) I.B. Was locates its manufacturing plant away from this major earthquake fault. Question 4.
(c) Dull Computers buys earthquake insurance for its manufacturing plant. First, you need to measure these risks in order to determine the best way to manage them.
Second, good risk management should look not only at current risks but at probable future risks.
Most importantly, managing risks is not equivalent to minimizing them. Besides minimizing
* Advanced problem
them, a risk manager may decide to accept them, reduce them partially (to an acceptable level),
or transfer them partially.
Question 5.
(a) This is an example of reducing risk.
(b) This is an example of avoiding risk.
(c) This is an example of transferring risk.
1 2
◼
◼
◼
Measuring derivative risk
Measuring interest rate risk
Mathematical background
⚫
⚫ Swaps
⚫ Options and exotics
⚫ Forwards and futures
⚫
⚫ Dollar duration and convexity
Derivative portfolios
Interest rate deltas and gammas
Financial Risk Management
Derivatives and Interest Rate
2. Risk Mathematics:
AFF811/FIN801
Overview
Products
By Eric Terry
1
, Mathematical Background (2) Dollar Duration and Convexity (2)
◼ These formulas can be generalized to situations where the Dollar convexity
asset price depends on multiple risk factors ◼Measures the sensitivity of the bond’s dollar duration to
the bond’s yield
Asset Derivatives Bonds
◼ Graphically, it captures the curvature of the bond price
St Change in derivative Change in bond price with respect to its YTM
price
◼ Under continuous compounding, dollar convexity is given
xt Change in price of Change in interest
by tN
underlying asset rate
C$ = t 2
(Ct e − yt )
St Delta − Interest rate delta t =t1
St Gamma Interest rate gamma ⚫ You multiply the PV of each bond payment by the
square of the time until that payment occurs and then
sum these up
3 5
Mathematical Background (1) Dollar Duration and Convexity (1)
◼ Suppose the value of an asset depends on some factor x Dollar duration
⚫ The value of a bond depends on interest rates ◼Measures the dollar price sensitivity of a bond with respect
to the bond’s yield
⚫ The value of a derivative depends on the price of the
underlying asset ◼ Consider a bond with cash flows of Ct at time t{t1,…, tN }
◼ Let St be the asset value at time t, and St and St be the first ◼ The value of the bond is
and second derivatives of the asset value with respect to x
tN
B = C e t
− y t
◼ The first-order (i.e., linear) Taylor series approximation of t =t1
changes in the asset price is where y = the bond’s continuously compounded YTM
St St xt ◼ The dollar duration of the bond is
and the second-order (i.e., quadratic) Taylor series B tN
D$ = − = t (Ct e − yt )
approximation is y t =t1
St St xt + St ( xt )
1
⚫ You multiply the PV of each bond payment by the time
2
2 until that payment occurs and then sum these up
2 4
, Interest Rate Deltas & Gammas (2) Interest Rate Deltas & Gammas (4)
◼ Interest rate gammas measure the sensitivity of the ◼ Basel rules require interest rate risk to be measured using
interest rate delta to the interest rate at that point on the at least six points on the yield curve
yield curve ⚫ This applies to each currency in which interest-rate
⚫ Interest rate gammas sum to the dollar convexity of a products are denominated
bond and thus measure the contribution of each section ◼ For linear IR products (e.g., bonds, FRAs, IR futures, and
of the yield curve to the dollar convexity IR swaps), only interest rate deltas are required, resulting
◼ Under continuous compounding, the interest rate delta and in the simple first-order linear approximation
tN
gamma of a bond with respect to the t-year interest rate B − rt
are t =t1
t
t = t (Ct e − rt t ) and t = t 2 (Ct e − rt t ), ◼ For IR options and other non-linear IR products, both
where interest rate gamma and vega terms must be added
Ct = the cash flow from the bond at time t
◼ The impact of changing credit spreads must also be
rt = the t -year continuously compounded interest rate incorporated into market risk for all IR products
7 9
Interest Rate Deltas & Gammas (1) Interest Rate Deltas & Gammas (3)
◼ The yield curve often does not move in parallel shifts ◼ The price impact of a change in the yield curve can be
◼ Interest rate deltas measure the price sensitivity of a bond second-order approximated by
to the interest rate at specified points on the yield curve tN
1 tN
B − t rt + t (rt ) 2
⚫ E.g., the effect of a change in the five-year interest rate t =t1 2 t =t1
6
◼ Interest rate deltas
where
5
sum to the dollar t1, ,t N = the selected points on the yield curve
4
duration of a bond rt = the change in the cont. comp. interest rate at t
Zero Rate (%)
3
and thus measure the
contribution of each ◼ For computational simplicity, risk managers often replace
the interest rate gammas with the dollar convexity
2
1 section of the yield tN
1
0 curve to the dollar B − rt + C $ (rt ) 2
duration
0 2 4 6 8 10 12 t
Maturity (yrs) t =t1 2
6 8
, Interest Rate Deltas & Gammas (6) Interest Rate Deltas & Gammas (7)
◼ The interest rate delta of a bond portfolio is the sum of the
◼ The T-note has interest rate deltas of 1,216.09 for six individual interest rate deltas at that point on the yield
months and 96,530.85 for one year, giving a dollar curve
duration of 97,746.95
⚫ Similarly, the interest rate gamma of a bond portfolio is
◼ It also has interest rate gammas of 608.05 for six months the sum of the individual interest rate gammas
and 96,530.85 for one year, giving a dollar convexity of Example 1
97,138.89
◼ Continuing the example on slides 10-11, suppose C. Shore
◼ For Basel reporting purposes, the firm could use the linear Resorts has also gone short five $10,000 risk-free zero-
approximation coupon bonds that mature in one year
B − 1,216.10 r.5 − 96,530.85 r1 ◼ The $10,000 payment in one year has a PV of
PV1 = 10,000 e -.0601 = 9,417.65
◼ Thus, each bond has an interest rate delta of 1 = 9,417.65
1 = 9,417.65 for one year and an interest rate delta of
11
zero at every other point on the yield curve 13
Interest Rate Deltas & Gammas (5) Review
◼ Myles Long Travel holds a $1,000 bond with a 3.5%
Example annual coupon and two years to maturity. One-year and
◼ C. Shore Resorts holds a $100,000 T-note that matures in two-year interest rates are 3.1% and 3.6%. What linear
one year and has a 5% coupon rate paid semi-annually model best describes the impact of interest rate changes
◼ The continuously compounded risk-free interest rate is on the value of this bond?
5.5% for six months and 6.0% for one year
◼ The interest rate deltas and gammas are found as follows:
Years CF PV Years × PV Years2 × PV
0.5 2,500 2,432.19* 1,216.09 608.05
1.0 102,500 96,530.85** 96,530.85 96,530.85
Total 98,963.05 97,746.95 97,138.89
* PV0.5 = 2,500 × e-.055×0.5 = 2,432.19
** PV1.0 = 102,500 × e-.060×1.0 = 96,530.85
10 12
1. Introduction Answers
Additional Practice Exercises
Question 1.
Question 1. (a) The one-day 99.5% VaR is
(a) Hy Marx has an investment portfolio with a daily mean return of $500 and a daily standard 2.57583 × $18,000 − $500 = $45,864.94
deviation of $18,000. If returns for this portfolio are i.i.d. normal through time, what is the and thus the 250-day 99.5% VaR is
250-day 99.5% VaR of this portfolio? 250 $45,864.94 = $725,188.
(b) Suppose instead that this portfolio had a 10-day mean return of 0.04%, a 10-day return
standard deviation of 3.0%, and a current value of $800,000. If returns for this portfolio are (b) The 10-day dollar mean return and dollar return standard deviation are
i.i.d. normal through time, what would be the 40-day 97.5% ES of this portfolio? = $800, 000 .0004 = $320
SD = $800, 000 .03 = $24, 000
Question 2*. Therefore, the 10-day 97.5% ES is
(a) The daily return distribution for the investment holdings of Hy Raitt Funds has fat tails that
2.33780×$24,000 − $320 = $55,787.20.
can be approximated by a non-standard t-distribution with a mean of 0.05%, standard Consequently, the 40-day 97.5% ES would be
deviation of 1.0%, and 10 degrees of freedom. The current value of the investment holdings
is $12.5 million. What is the one-day 99% VaR of this portfolio? 4 $55, 787.20 = $111,574.
(b) Annual operational losses (in $ million) for this firm can be approximated by a chi-square
distribution with 15 degrees of freedom. What is the one-year 99.5% VaR for the firm’s Question 2.
operational losses? (a) The 99% cut-off for a standard t-distribution with 10 degrees of freedom is 2.76377. Also,
Note: A standard t-distribution table and chi-square table have been posted on D2L and can be the daily dollar mean return and return standard deviation are
used to get approximate answers to this question. = $12.5M .0005 = $6, 250
SD = $12.5M .01 = $125, 000
Question 3. Therefore, the one-day 99% Var is:
I.N. Stein Financial is expected to have $5 million in earnings before taxes. The firm is taxed at 2.76377 × $125,000− $6,250 = $339,221.
a 40% rate. Credit losses are expected to be $500,000 but would reach $6 million in a worst-case
scenario. What is the firm’s after-tax RAROC? (b) The 99.5% cut-off for a chi-square distribution with 15 degrees of freedom is 32.801.
Therefore, the one-year 99.5% VaR for operational risk is $32.801M
Question 4.
Critically comment on the following statement: “Risk management is simple. You first identify Question 3.
the major risks your firm faces and then you determine the cheapest way to minimize these For this firm,
risks.” Expected profit = $5M (1 − .4) = $3M
Economic capital = $6M − 0.5M = $5.5M
Question 5.
and thus
Identify the type of risk management policy being followed in each situation. RAROC = 3M / 5.5M = 54.6%.
(a) Ample Computers reinforces its manufacturing plant that is near a major earthquake fault.
(b) I.B. Was locates its manufacturing plant away from this major earthquake fault. Question 4.
(c) Dull Computers buys earthquake insurance for its manufacturing plant. First, you need to measure these risks in order to determine the best way to manage them.
Second, good risk management should look not only at current risks but at probable future risks.
Most importantly, managing risks is not equivalent to minimizing them. Besides minimizing
* Advanced problem
them, a risk manager may decide to accept them, reduce them partially (to an acceptable level),
or transfer them partially.
Question 5.
(a) This is an example of reducing risk.
(b) This is an example of avoiding risk.
(c) This is an example of transferring risk.
1 2
◼
◼
◼
Measuring derivative risk
Measuring interest rate risk
Mathematical background
⚫
⚫ Swaps
⚫ Options and exotics
⚫ Forwards and futures
⚫
⚫ Dollar duration and convexity
Derivative portfolios
Interest rate deltas and gammas
Financial Risk Management
Derivatives and Interest Rate
2. Risk Mathematics:
AFF811/FIN801
Overview
Products
By Eric Terry
1
, Mathematical Background (2) Dollar Duration and Convexity (2)
◼ These formulas can be generalized to situations where the Dollar convexity
asset price depends on multiple risk factors ◼Measures the sensitivity of the bond’s dollar duration to
the bond’s yield
Asset Derivatives Bonds
◼ Graphically, it captures the curvature of the bond price
St Change in derivative Change in bond price with respect to its YTM
price
◼ Under continuous compounding, dollar convexity is given
xt Change in price of Change in interest
by tN
underlying asset rate
C$ = t 2
(Ct e − yt )
St Delta − Interest rate delta t =t1
St Gamma Interest rate gamma ⚫ You multiply the PV of each bond payment by the
square of the time until that payment occurs and then
sum these up
3 5
Mathematical Background (1) Dollar Duration and Convexity (1)
◼ Suppose the value of an asset depends on some factor x Dollar duration
⚫ The value of a bond depends on interest rates ◼Measures the dollar price sensitivity of a bond with respect
to the bond’s yield
⚫ The value of a derivative depends on the price of the
underlying asset ◼ Consider a bond with cash flows of Ct at time t{t1,…, tN }
◼ Let St be the asset value at time t, and St and St be the first ◼ The value of the bond is
and second derivatives of the asset value with respect to x
tN
B = C e t
− y t
◼ The first-order (i.e., linear) Taylor series approximation of t =t1
changes in the asset price is where y = the bond’s continuously compounded YTM
St St xt ◼ The dollar duration of the bond is
and the second-order (i.e., quadratic) Taylor series B tN
D$ = − = t (Ct e − yt )
approximation is y t =t1
St St xt + St ( xt )
1
⚫ You multiply the PV of each bond payment by the time
2
2 until that payment occurs and then sum these up
2 4
, Interest Rate Deltas & Gammas (2) Interest Rate Deltas & Gammas (4)
◼ Interest rate gammas measure the sensitivity of the ◼ Basel rules require interest rate risk to be measured using
interest rate delta to the interest rate at that point on the at least six points on the yield curve
yield curve ⚫ This applies to each currency in which interest-rate
⚫ Interest rate gammas sum to the dollar convexity of a products are denominated
bond and thus measure the contribution of each section ◼ For linear IR products (e.g., bonds, FRAs, IR futures, and
of the yield curve to the dollar convexity IR swaps), only interest rate deltas are required, resulting
◼ Under continuous compounding, the interest rate delta and in the simple first-order linear approximation
tN
gamma of a bond with respect to the t-year interest rate B − rt
are t =t1
t
t = t (Ct e − rt t ) and t = t 2 (Ct e − rt t ), ◼ For IR options and other non-linear IR products, both
where interest rate gamma and vega terms must be added
Ct = the cash flow from the bond at time t
◼ The impact of changing credit spreads must also be
rt = the t -year continuously compounded interest rate incorporated into market risk for all IR products
7 9
Interest Rate Deltas & Gammas (1) Interest Rate Deltas & Gammas (3)
◼ The yield curve often does not move in parallel shifts ◼ The price impact of a change in the yield curve can be
◼ Interest rate deltas measure the price sensitivity of a bond second-order approximated by
to the interest rate at specified points on the yield curve tN
1 tN
B − t rt + t (rt ) 2
⚫ E.g., the effect of a change in the five-year interest rate t =t1 2 t =t1
6
◼ Interest rate deltas
where
5
sum to the dollar t1, ,t N = the selected points on the yield curve
4
duration of a bond rt = the change in the cont. comp. interest rate at t
Zero Rate (%)
3
and thus measure the
contribution of each ◼ For computational simplicity, risk managers often replace
the interest rate gammas with the dollar convexity
2
1 section of the yield tN
1
0 curve to the dollar B − rt + C $ (rt ) 2
duration
0 2 4 6 8 10 12 t
Maturity (yrs) t =t1 2
6 8
, Interest Rate Deltas & Gammas (6) Interest Rate Deltas & Gammas (7)
◼ The interest rate delta of a bond portfolio is the sum of the
◼ The T-note has interest rate deltas of 1,216.09 for six individual interest rate deltas at that point on the yield
months and 96,530.85 for one year, giving a dollar curve
duration of 97,746.95
⚫ Similarly, the interest rate gamma of a bond portfolio is
◼ It also has interest rate gammas of 608.05 for six months the sum of the individual interest rate gammas
and 96,530.85 for one year, giving a dollar convexity of Example 1
97,138.89
◼ Continuing the example on slides 10-11, suppose C. Shore
◼ For Basel reporting purposes, the firm could use the linear Resorts has also gone short five $10,000 risk-free zero-
approximation coupon bonds that mature in one year
B − 1,216.10 r.5 − 96,530.85 r1 ◼ The $10,000 payment in one year has a PV of
PV1 = 10,000 e -.0601 = 9,417.65
◼ Thus, each bond has an interest rate delta of 1 = 9,417.65
1 = 9,417.65 for one year and an interest rate delta of
11
zero at every other point on the yield curve 13
Interest Rate Deltas & Gammas (5) Review
◼ Myles Long Travel holds a $1,000 bond with a 3.5%
Example annual coupon and two years to maturity. One-year and
◼ C. Shore Resorts holds a $100,000 T-note that matures in two-year interest rates are 3.1% and 3.6%. What linear
one year and has a 5% coupon rate paid semi-annually model best describes the impact of interest rate changes
◼ The continuously compounded risk-free interest rate is on the value of this bond?
5.5% for six months and 6.0% for one year
◼ The interest rate deltas and gammas are found as follows:
Years CF PV Years × PV Years2 × PV
0.5 2,500 2,432.19* 1,216.09 608.05
1.0 102,500 96,530.85** 96,530.85 96,530.85
Total 98,963.05 97,746.95 97,138.89
* PV0.5 = 2,500 × e-.055×0.5 = 2,432.19
** PV1.0 = 102,500 × e-.060×1.0 = 96,530.85
10 12
