Application 22-27 Flashcards
solve problem questions
1
Q
Shares of closed-end fund ABC were selling at
a premium of 10% and then fell to $44 per share while ABC’s net asset value
held constant at $50 per share. What were the previous market price,
subsequent discount, NAV-based return, and market-price return for ABC?
A
previous market price:
1 + premium x net asset value
1 + 0.10 x 50
= 55
subsequent discount:
fall in price / net asset value (- 1)
44 / 55 (- 1)
= -0.12 (-12%)
market price return:
fall in price - net asset value / net asset value
44 - 55 / 55 = -0.20 (20%)
2
Q
Shares of closed-end fund ABC were selling at
a premium of 12% and then fell to $40 per share while ABC’s net asset value
held constant at $40 per share. What were the previous market price,
subsequent discount, NAV-based return, and market-price return for ABC?
A
previous market price return:
1 + 0.12 x 40
= 44.8
NAV-based return:
40 / 40 (-1) = 0
Subsequent discount:
40 - 44.80 / 44.80
= -0.1071 (-10.71%)
3
Q
Shares of closed-end fund ABC were selling at
a premium of 12% and then fell to $20 per share while ABC’s net asset value
held constant at $20 per share. What were the previous market price,
subsequent discount, NAV-based return, and market-price return for ABC?
A
?
4
Q
Shares of closed-end fund ABC were selling at
a premium of 5% and then fell to $16 per share while ABC’s net asset value
held constant at $20 per share. What were the previous market price,
subsequent discount, NAV-based return, and market-price return for ABC?
A
?
5
Q
Shares of closed-end fund ABC were selling at
a premium of 1% and then fell to $30 per share while ABC’s net asset value
held constant at $30 per share. What were the previous market price,
subsequent discount, NAV-based return, and market-price return for ABC?
A
?
6
Q
Shares of closed-end fund ABC were selling at
a premium of 3% and then fell to $18 per share while ABC’s net asset value
held constant at $20 per share. What were the previous market price,
subsequent discount, NAV-based return, and market-price return for ABC?
A
?
7
Q
Shares of closed-end fund ABC were selling at
a premium of 2% and then fell to $22 per share while ABC’s net asset value
held constant at $25 per share. What were the previous market price,
subsequent discount, NAV-based return, and market-price return for ABC?
A
?
8
Q
A convertible preferred stock with a par or face
value of $100 per share is convertible into four shares of common stock. What is
the conversion ratio, and what is the conversion price? What would be the
conversion ratio if the conversion price were $20?
A
conversion price:
four shares of common stock with value of $100
100 / 4 = $25
conversion ratio of $20:
100/20 = 5/1
9
Q
A convertible preferred stock with a par or face
value of $100 per share is convertible into four shares of common stock. What would be the
conversion ratio if the conversion price were $50?
A
?
10
Q
A convertible preferred stock with a par or face
value of $350 per share is convertible into four shares of common stock. What is
the conversion ratio, and what is the conversion price? What would be the
conversion ratio if the conversion price were $50?
A
?
11
Q
A convertible preferred stock with a par or face
value of $250 per share is convertible into four shares of common stock. What is
the conversion ratio, and what is the conversion price? What would be the
conversion ratio if the conversion price were $83.33?
A
?
12
Q
A convertible preferred stock with a par or face
value of $450 per share is convertible into four shares of common stock. What is
the conversion ratio, and what is the conversion price? What would be the
conversion ratio if the conversion price were $90?
A
?
13
Q
A convertible preferred stock with a par or face
value of $750 per share is convertible into four shares of common stock. What is
the conversion ratio, and what is the conversion price? What would be the
conversion ratio if the conversion price were $187.50?
A
?
14
Q
A VC fund manager raises $100 million in
committed capital for his VC fund. The management fee is 2.5%. To date, only
$50 million of the raised capital has been called and invested in start-ups. What
would be the annual management fee?
A
annual management fee:
capital committed capital for VC x management fee %
100,000,000 x 0.025
= 2,500,000 (even though not all capital has been invested)
15
Q
A VC fund manager raises $100 million in
committed capital for his VC fund. The management fee is 1%. To date, only
$70 million of the raised capital has been called and invested in start-ups. What
would be the annual management fee?
A
?
16
Q
A VC fund manager raises $500 million in
committed capital for his VC fund. The management fee is 2%. To date, only
$400 million of the raised capital has been called and invested in start-ups. What
would be the annual management fee?
A
?
17
Q
Suppose
that all other facts remain the same, initial investment $100 million valuation cash flow $120 million, growth rate 2%, except that the discount rate used at the end
of seven years is 15%.
A
the new rate of return is:
valuation cash flow / (discount rate - growth rate)
120,000,000 / (0.15 - 0.02)
= $923 million
923,000,000,000/ 100,000,000,000 ^(1/7) (-1)
= 0.374 (37.4%)
1/7 because it is over seven years.
18
Q
Suppose
that all other facts remain the same, initial investment $100 million valuation cash flow $120 million, growth rate 5%, except that the discount rate used at the end
of seven years is 12%.
A
(1,714,285,714,284,714/100,000,000,000)^1/7 (-1)
new valuation:
120,000,000,000 / (0.12-0.05)
= 1,714,285,714,284,714
19
Q
Suppose
that all other facts remain the same, initial investment $100 million valuation cash flow $120 million, growth rate 2%, except that the discount rate used at the end
of seven years is 12%.
A
?
20
Q
Suppose
that all other facts remain the same, initial investment $100 million valuation cash flow $120 million, growth rate 7%, except that the discount rate used at the end
of seven years is 12%.
A
?
21
Q
Suppose
that all other facts remain the same, initial investment $100 million valuation cash flow $120 million, growth rate 1%, except that the discount rate used at the end
of seven years is 14%.
A
?
22
Q
Suppose
that all other facts remain the same, initial investment $100 million valuation cash flow $120 million, growth rate 2%, except that the discount rate used at the end
of eight years is 15%.
A
?
23
Q
Suppose
that all other facts remain the same, initial investment $100 million valuation cash flow $120 million, growth rate 5%, except that the discount rate used at the end
of eight years is 12%.
A
?
24
Q
Suppose
that all other facts remain the same, initial investment $100 million valuation cash flow $120 million, growth rate 2%, except that the discount rate used at the end
of eight years is 12%.
A
?
25
Suppose
that all other facts remain the same, initial investment $100 million except that the $120 million cash flow estimate
given is a year 7 cash flow that is anticipated to grow by year 8.
120 (1.02) / 0.12 - 0.02)
= $1.224 billion
$1,224 billion / $100 million ^1/7 (- 1)
= 43%
26
Suppose that the structure on the right-hand side
of Exhibit 24.1 is changed such that the mezzanine debt rises to being 30% of
the capital structure, and the bank debt falls to being 50% of the capital structure.
If the costs of bank debt and equity remain the same (8% and 32%, respectively),
what must the new cost of mezzanine debt be such that the weighted average
cost of capital would be 15.8%?
Cost of equity 20.00%.
WACC = C(a) x P(a) + C(b) x P(b) + C(c) x P(c)
0.158 = C(a) x 0.30 + 0.32 x (-0.20) + (-0.08) x 0.50
manipulate equation
= 0.18
27
Suppose that the structure on the right-hand side
of Exhibit 24.1 is changed such that the mezzanine debt rises to being 25% of
the capital structure, and the bank debt falls to being 50% of the capital structure.
If the costs of bank debt and equity remain the same (8% and 30%, respectively),
what must the new cost of mezzanine debt be such that the weighted average
cost of capital would be 13.25%?
Cost of equity 22.00%.
?
28
Suppose that the structure on the right-hand side
of Exhibit 24.1 is changed such that the mezzanine debt rises to being 25% of
the capital structure, and the bank debt falls to being 50% of the capital structure.
If the costs of bank debt and equity remain the same (8% and 25%, respectively),
what must the new cost of mezzanine debt be such that the weighted average
cost of capital would be 12.5%?
Cost of equity 22.00%.
?
29
Suppose that the structure on the right-hand side
of Exhibit 24.1 is changed such that the mezzanine debt rises to being 15% of
the capital structure, and the bank debt falls to being 55% of the capital structure.
If the costs of bank debt and equity remain the same (8% and 30%, respectively),
what must the new cost of mezzanine debt be such that the weighted average
cost of capital would be 13.25%?
Cost of equity 23.00%.
?
30
Suppose that the structure on the right-hand side
of Exhibit 24.1 is changed such that the mezzanine debt rises to being 20% of
the capital structure, and the bank debt falls to being 55% of the capital structure.
If the costs of bank debt and equity remain the same (8% and 25%, respectively),
what must the new cost of mezzanine debt be such that the weighted average
cost of capital would be 13%?
Cost of equity 24.00%.
?
31
Suppose that the structure on the right-hand side
of Exhibit 24.1 is changed such that the mezzanine debt rises to being 15% of
the capital structure, and the bank debt falls to being 60% of the capital structure.
If the costs of bank debt and equity remain the same (8% and 25%, respectively),
what must the new cost of mezzanine debt be such that the weighted average
cost of capital would be 13.15%?
Cost of equity 25.00%.
?
32
Suppose that the structure on the right-hand side
of Exhibit 24.1 is changed such that the mezzanine debt rises to being 14% of
the capital structure, and the bank debt falls to being 60% of the capital structure.
If the costs of bank debt and equity remain the same (8% and 20%, respectively),
what must the new cost of mezzanine debt be such that the weighted average
cost of capital would be 13.20%?
Cost of equity 28.00%.
?
33
If 20% of the bonds in a portfolio default each
year and if 60% of the bonds' value is ultimately unrecovered (i.e., 40% of the
bonds’ cost is recovered), then the total loss due to default over that time period
is?
(Coupon rate) Total Loss Due to Default:
Annual Default x Loss Rate Given Default
0.20% x 0.40%
= 0.12%
34
If 30% of the bonds in a portfolio default each
year and if 80% of the bonds' value is ultimately unrecovered (i.e., 40% of the
bonds’ cost is recovered), then the total loss due to default over that time period
is?
coupon rate = bond default x bond value
| 0.30% x 0.80% = 0.24%
35
If 20% of the bonds in a portfolio default each
year and if 30% of the bonds' value is ultimately unrecovered (i.e., 40% of the
bonds’ cost is recovered), then the total loss due to default over that time period
is?
?
36
If 15% of the bonds in a portfolio default each
year and if 5% of the bonds' value is ultimately unrecovered (i.e., 40% of the
bonds’ cost is recovered), then the total loss due to default over that time period
is?
?
37
If 25% of the bonds in a portfolio default each
year and if 80% of the bonds' value is ultimately unrecovered (i.e., 40% of the
bonds’ cost is recovered), then the total loss due to default over that time period
is?
?
38
If 30% of the bonds in a portfolio default each
year and if 70% of the bonds' value is ultimately unrecovered (i.e., 40% of the
bonds’ cost is recovered), then the total loss due to default over that time period
is?
?
39
If 40% of the bonds in a portfolio default each
year and if 65% of the bonds' value is ultimately unrecovered (i.e., 40% of the
bonds’ cost is recovered), then the total loss due to default over that time period
is?
?
40
Consider a firm with $50 million in assets and
$25 million in equity value. The firm has one debt issue: a zero-coupon bond
maturing in one year with a face value of $30 million. A riskless zero-coupon
bond of the same maturity sells for 90% of its face value. What is the value of the
firm’s debt? What is the value of a one-year put option on the firm’s assets with a
strike price of $30 million?
Assets = (Call) + (Riskless Bond - Put)
Assets - Call = Debt = Riskless Bond - Put
50,000,000 - 25,000,000 = (30,000,000 x 0.9) - Put
25,000,000 = 27,000,000 - Put
Put = 2,000,000
41
Consider a firm with $25 million in assets and
$5 million in equity value. The firm has one debt issue: a zero-coupon bond
maturing in one year with a face value of $25 million. A riskless zero-coupon
bond of the same maturity sells for 85% of its face value. What is the value of the
firm’s debt? What is the value of a one-year put option on the firm’s assets with a
strike price of $25 million?
asset - call = face value (face value %) - Put
25 - 5 = 25 (0.85) - Put
20 = 21.2500,000 - Put
Put = 1,250,000
42
Consider a firm with $25 million in assets and
$10 million in equity value. The firm has one debt issue: a zero-coupon bond
maturing in one year with a face value of $20 million. A riskless zero-coupon
bond of the same maturity sells for 95% of its face value. What is the value of the
firm’s debt? What is the value of a one-year put option on the firm’s assets with a
strike price of $20 million?
?
43
Consider a firm with $50 million in assets and
$25 million in equity value. The firm has one debt issue: a zero-coupon bond
maturing in one year with a face value of $35 million. A riskless zero-coupon
bond of the same maturity sells for 80% of its face value. What is the value of the
firm’s debt? What is the value of a one-year put option on the firm’s assets with a
strike price of $35 million?
?
44
Consider a firm with $75 million in assets and
$30 million in equity value. The firm has one debt issue: a zero-coupon bond
maturing in one year with a face value of $48 million. A riskless zero-coupon
bond of the same maturity sells for 90% of its face value. What is the value of the
firm’s debt? What is the value of a one-year put option on the firm’s assets with a
strike price of $48 million?
?
45
Consider a firm with $55 million in assets and
$50 million in equity value. The firm has one debt issue: a zero-coupon bond
maturing in one year with a face value of $7 million. A riskless zero-coupon
bond of the same maturity sells for 75% of its face value. What is the value of the
firm’s debt? What is the value of a one-year put option on the firm’s assets with a
strike price of $7 million?
?
46
Consider a firm with $65 million in assets and
$55 million in equity value. The firm has one debt issue: a zero-coupon bond
maturing in one year with a face value of $20 million. A riskless zero-coupon
bond of the same maturity sells for 65% of its face value. What is the value of the
firm’s debt? What is the value of a one-year put option on the firm’s assets with a
strike price of $20 million?
?
47
Consider a firm with $100 million in assets and
$60 million in equity value. The firm’s debt has a face value of $50 million and a
maturity of one year. The volatility of the firm’s equity is estimated at 40%. How
would an analyst estimate the value of the firm’s equity if the volatility of the firm’s
assets doubled?
Find volatility of firms assets:
Ó(assets) = (wavy) Ó(equity) x (Equity / Assets)
60,000,000 / 100,000,000 (x 0.40) = 0.24
Find the approximate equity with expected asset volatility:
0.24 x 2
= 0.48
48
Consider a firm with $75 million in assets and
$45 million in equity value. The firm’s debt has a face value of $50 million and a
maturity of one year. The volatility of the firm’s equity is estimated at 35%. How
would an analyst estimate the value of the firm’s equity if the volatility of the firm’s
assets doubled?
45,000,000/75,000,000 (x 0.35)
| = 0.21 (21%)
49
Consider a firm with $50 million in assets and
$20 million in equity value. The firm’s debt has a face value of $50 million and a
maturity of one year. The volatility of the firm’s equity is estimated at 30%. How
would an analyst estimate the value of the firm’s equity if the volatility of the firm’s
assets doubled?
?
50
Consider a firm with $150 million in assets and
$75 million in equity value. The firm’s debt has a face value of $50 million and a
maturity of one year. The volatility of the firm’s equity is estimated at 25%. How
would an analyst estimate the value of the firm’s equity if the volatility of the firm’s
assets doubled?
?
51
Consider a firm with $200 million in assets and
$125 million in equity value. The firm’s debt has a face value of $50 million and a
maturity of one year. The volatility of the firm’s equity is estimated at 45%. How
would an analyst estimate the value of the firm’s equity if the volatility of the firm’s
assets doubled?
?
52
Consider a firm with $250 million in assets and
$200 million in equity value. The firm’s debt has a face value of $50 million and a
maturity of one year. The volatility of the firm’s equity is estimated at 50%. How
would an analyst estimate the value of the firm’s equity if the volatility of the firm’s
assets doubled?
?
53
Consider a firm with $300 million in assets and
$275 million in equity value. The firm’s debt has a face value of $50 million and a
maturity of one year. The volatility of the firm’s equity is estimated at 60%. How
would an analyst estimate the value of the firm’s equity if the volatility of the firm’s
assets doubled?
?
54
Suppose that the CDO depicted in Exhibit 25.5
alters its portfolio such that the average coupon on the assets is 6%. Ignoring
defaults, fees, and expenses, how much annual income should be available to
the equity tranche?
Yield on Assets 6.00%
Cash from Assets $6.00
Senior Tranche Coupon 3.00%
Mezzanine Tranche Coupon 6.00%
Value of Equity Tranche $10.00
Value of senior Tranche $70.00
Mezzanine Value $20.00
Total value of assets $100.00
Total value of assets x Yield on asset %
100 x 0.06
= 6
Value of senior tranche x senior tranche coupon
70 x 0.03%
= 2.1
Mezzanine tranche value x mezzanine tranche coupon
20 x 0.06
= 1.2
6 - 2.1 - 1.2
= 2.7
55
Suppose that the CDO depicted in Exhibit 25.5
alters its portfolio such that the average coupon on the assets is 7%. Ignoring
defaults, fees, and expenses, how much annual income should be available to
the equity tranche?
Yield on Assets 7.00%
Cash from Assets $7.00
Senior Tranche Coupon 2.50%
Mezzanine Tranche Coupon 6.50%
Value of Equity Tranche $15.00
Value of senior Tranche $15.00
Mezzanine Value $70.00
Total value of assets $100.00
assets:
0.07 x 100
= 7
senior:
0.025 x 15
= 0.375
mezzanine:
0.065 x 70
= 4.55
7 - 0.375 - 4.55
= 2.075
56
Suppose that the CDO depicted in Exhibit 25.5
alters its portfolio such that the average coupon on the assets is 8%. Ignoring
defaults, fees, and expenses, how much annual income should be available to
the equity tranche?
Yield on Assets 6.00%
Cash from Assets $8.00
Senior Tranche Coupon 2.00%
Mezzanine Tranche Coupon 7.50%
Value of Equity Tranche $20.00
Value of senior Tranche $75.00
Mezzanine Value $5.00
Total value of assets $100.00
?
57
Suppose that the CDO depicted in Exhibit 25.5
alters its portfolio such that the average coupon on the assets is 10%. Ignoring
defaults, fees, and expenses, how much annual income should be available to
the equity tranche?
Yield on Assets 10.00%
Cash from Assets $10.00
Senior Tranche Coupon 3.50%
Mezzanine Tranche Coupon 9.00%
Value of Equity Tranche $25.00
Value of senior Tranche $50.00
Mezzanine Value $25.00
Total value of assets $100.00
?
58
Suppose that the CDO depicted in Exhibit 25.5
alters its portfolio such that the average coupon on the assets is 9%. Ignoring
defaults, fees, and expenses, how much annual income should be available to
the equity tranche?
Yield on Assets 9.00%
Cash from Assets $9.00
Senior Tranche Coupon 4.00%
Mezzanine Tranche Coupon 8.00%
Value of Equity Tranche $33.00
Value of senior Tranche $33.00
Mezzanine Value $34.00
Total value of assets $100.00
?
59
Suppose that the CDO depicted in Exhibit 25.5
alters its portfolio such that the average coupon on the assets is 7.5%. Ignoring
defaults, fees, and expenses, how much annual income should be available to
the equity tranche?
Yield on Assets 7.50%
Cash from Assets $7.50
Senior Tranche Coupon 1.50%
Mezzanine Tranche Coupon 7.50%
Value of Equity Tranche $10.00
Value of senior Tranche $80.00
Mezzanine Value $10.00
Total value of assets $100.00
?
60
Suppose that the CDO depicted in Exhibit 25.5
alters its portfolio such that the average coupon on the assets is 5.5%. Ignoring
defaults, fees, and expenses, how much annual income should be available to
the equity tranche?
Yield on Assets 5.50%
Cash from Assets $5.50
Senior Tranche Coupon 6.00%
Mezzanine Tranche Coupon 8.00%
Value of Equity Tranche $15.00
Value of senior Tranche $10.00
Mezzanine Value $75.00
Total value of assets $100.00
?
61
Suppose that the CDO depicted in Exhibit 25.5
experiences defaults in $50 million of the assets with 30% recovery. What will
happen to the tranches?
$10 Equity tranche
$20 Mezzanine tranche
$70 Senior tranche
1 - recovery rate
1 - 0.30
x experiences defaults
x 50
= 35
35 - 10 (equity tranche)
= 25 - 20 (mezzanine tranche)
= 5
70 (Senior tranche) - 5
= 65
62
Suppose that the CDO depicted in Exhibit 25.5
experiences defaults in $25 million of the assets with 25% recovery. What will
happen to the tranches?
$15 Equity tranche
$70 Mezzanine tranche
$15 Senior tranche
1 - 0.25
x 25 = 18.75
18.75 - 15
= 3.75
- 70
= -66.25
15 - (-66.25)
= 81.25
63
Suppose that the CDO depicted in Exhibit 25.5
experiences defaults in $35 million of the assets with 15% recovery. What will
happen to the tranches?
$20 Equity tranche
$75 Mezzanine tranche
$5 Senior tranche
?
64
Suppose that the CDO depicted in Exhibit 25.5
experiences defaults in $45 million of the assets with 10% recovery. What will
happen to the tranches?
$25 Equity tranche
$50 Mezzanine tranche
$25 Senior tranche
?
65
Suppose that the CDO depicted in Exhibit 25.5
experiences defaults in $50 million of the assets with 50% recovery. What will
happen to the tranches?
$33 Equity tranche
$33 Mezzanine tranche
$34 Senior tranche
?
66
Suppose that the CDO depicted in Exhibit 25.5
experiences defaults in $55 million of the assets with 35% recovery. What will
happen to the tranches?
$10 Equity tranche
$80 Mezzanine tranche
$10 Senior tranche
?
67
Suppose that the CDO depicted in Exhibit 25.5
experiences defaults in $65 million of the assets with 45% recovery. What will
happen to the tranches?
$15 Equity tranche
$10 Mezzanine tranche
$75 Senior tranche
?
68
A bank has extended a $50 million one-year
loan at an interest rate of 14% to a client with a BBB credit rating. Suppose that
historical data indicate that the one-year probability of default for firms with a
BBB rating is 5% and that investors are typically able to recover 40% of the
notional value of an unsecured loan to such firms. What is the expected credit
loss?
expected credit loss = probability of default x estimated amount of default x (1 - recovery rate)
estimated amount of default: the loan x interest rate. = 50,000,000 x 0.14 = 57,000,000
0.05 x 57,000,000 (1 - 0.40)
1,710,000
69
A bank has extended a $100 million one-year
loan at an interest rate of 12% to a client with a BBB credit rating. Suppose that
historical data indicate that the one-year probability of default for firms with a
BBB rating is 6% and that investors are typically able to recover 40% of the
notional value of an unsecured loan to such firms. What is the expected credit
loss?
probability of default x amount of default (1 - recovery rate)
(estimated amount of default is the loan plus the interest rate)
= 0.06 (100,000,000x1.12) (1 - 0.40)
= 4,032,000
70
A bank has extended a $50 million one-year
loan at an interest rate of 13% to a client with a BBB credit rating. Suppose that
historical data indicate that the one-year probability of default for firms with a
BBB rating is 7% and that investors are typically able to recover 40% of the
notional value of an unsecured loan to such firms. What is the expected credit
loss?
?
71
A bank has extended a $65 million one-year
loan at an interest rate of 13% to a client with a BBB credit rating. Suppose that
historical data indicate that the one-year probability of default for firms with a
BBB rating is 8% and that investors are typically able to recover 40% of the
notional value of an unsecured loan to such firms. What is the expected credit
loss?
?
72
A bank has extended a $75 million one-year
loan at an interest rate of 11% to a client with a BBB credit rating. Suppose that
historical data indicate that the one-year probability of default for firms with a
BBB rating is 8% and that investors are typically able to recover 40% of the
notional value of an unsecured loan to such firms. What is the expected credit
loss?
?
73
A bank has extended a $85 million one-year
loan at an interest rate of 19% to a client with a BBB credit rating. Suppose that
historical data indicate that the one-year probability of default for firms with a
BBB rating is 6% and that investors are typically able to recover 40% of the
notional value of an unsecured loan to such firms. What is the expected credit
loss?
?
74
A bank has extended a $90 million one-year
loan at an interest rate of 20% to a client with a BBB credit rating. Suppose that
historical data indicate that the one-year probability of default for firms with a
BBB rating is 5% and that investors are typically able to recover 40% of the
notional value of an unsecured loan to such firms. What is the expected credit
loss?
?
75
Suppose that the risk-free rate is 5% per year
and that a one-year, zero-coupon corporate bond yields 6% per year. What are
the precise and approximate risk-neutral probabilities of default? Assuming a
recovery rate of 80% on the corporate bond.
```
RR = recovery rate
Y = Yields (bond yields)
RF = risk free rate
```
λ = (1 / 1 - RR) (Y - RF / (1 + RF + (Y - RF)
(1 / 1 - 0.80) (0.06 - 0.05 / (1 + 0.05 + (0.06 - 0.05)
= 0.04716981132
(4.72%)
76
Suppose that the risk-free rate is 8% per year
and that a one-year, zero-coupon corporate bond yields 10% per year. What are
the precise and approximate risk-neutral probabilities of default? Assuming a
recovery rate of 90% on the corporate bond.
(1 / 1-0.9) (0.10 - 0.08 / 1 + 0.08 + (0.10 - 0.08)
(10) (0.02 / 1.1)
= 0.1818181818182
(18.18%)
77
Suppose that the risk-free rate is 2.5% per year
and that a one-year, zero-coupon corporate bond yields 7% per year. What are
the precise and approximate risk-neutral probabilities of default? Assuming a
recovery rate of 85% on the corporate bond.
?
78
Suppose that the risk-free rate is 5.5% per year
and that a one-year, zero-coupon corporate bond yields 7.5% per year. What are
the precise and approximate risk-neutral probabilities of default? Assuming a
recovery rate of 85% on the corporate bond.
?
79
Suppose that the risk-free rate is 3% per year
and that a one-year, zero-coupon corporate bond yields 5.5% per year. What are
the precise and approximate risk-neutral probabilities of default? Assuming a
recovery rate of 95% on the corporate bond.
?
80
Suppose that the risk-free rate is 3.5% per year
and that a one-year, zero-coupon corporate bond yields 5% per year. What are
the precise and approximate risk-neutral probabilities of default? Assuming a
recovery rate of 65% on the corporate bond.
?
81
Suppose that the risk-free rate is 4% per year
and that a one-year, zero-coupon corporate bond yields 8% per year. What are
the precise and approximate risk-neutral probabilities of default? Assuming a
recovery rate of 60% on the corporate bond.
?
82
Suppose that the risk-neutral probability of
default for a bond is 5% per year and that the recovery rate of the bond is 70%.
What is the approximate spread by which the bond should trade relative to the
yield of a riskless bond?
(1 - recovery rate) x probability of default = yield of a riskless bond
(1-0.7) x 0.05
= 0.015 (1.5%)
83
Suppose that the risk-neutral probability of
default for a bond is 5.5% per year and that the recovery rate of the bond is 90%.
What is the approximate spread by which the bond should trade relative to the
yield of a riskless bond?
(1 - recovery rate) x probability of default = yield of a riskless bond
(1 - 0.9) x 0.055
= 0.0055 (0.55%)
84
Suppose that the risk-neutral probability of
default for a bond is 6.5% per year and that the recovery rate of the bond is 75%.
What is the approximate spread by which the bond should trade relative to the
yield of a riskless bond?
?
85
Suppose that the risk-neutral probability of
default for a bond is 6% per year and that the recovery rate of the bond is 85%.
What is the approximate spread by which the bond should trade relative to the
yield of a riskless bond?
?
86
Suppose that the risk-neutral probability of
default for a bond is 7% per year and that the recovery rate of the bond is 95%.
What is the approximate spread by which the bond should trade relative to the
yield of a riskless bond?
?
87
Suppose that the risk-neutral probability of
default for a bond is 4% per year and that the recovery rate of the bond is 65%.
What is the approximate spread by which the bond should trade relative to the
yield of a riskless bond?
?
88
Suppose that the risk-neutral probability of
default for a bond is 4.5% per year and that the recovery rate of the bond is 60%.
What is the approximate spread by which the bond should trade relative to the
yield of a riskless bond?
?
89
Suppose that the junior debt of XYZ
Corporation is frequently traded and currently trades at a credit spread of 2.50%
over riskless bonds of comparable maturity. The senior debt of the firm has not
been regularly traded because it was primarily held by a few institutions, and a
new issue of debt that is subordinated to all other debt has been rated as
speculative. The expected recovery rate of the senior debt is 80%, the old junior
debt is 50%, and the recently issued speculative debt is 20%. Using
approximation formulas, what arbitrage-free credit spreads should be expected
on the senior and speculative debt issues?
Risk-neutral default probability of 5.0%
λ = s / (1 - RR)
rearrange:
s = λ x (1 - RR)
arbitrage-free credit spread for Senior debt:
S = 0.05 (1-0.8)
= 0.01
arbitrage-free credit spread for speculative debt:
S = 0.05 x (1 - 0.20)
= 0.04
90
Suppose that the junior debt of XYZ
Corporation is frequently traded and currently trades at a credit spread of 3.50%
over riskless bonds of comparable maturity. The senior debt of the firm has not
been regularly traded because it was primarily held by a few institutions, and a
new issue of debt that is subordinated to all other debt has been rated as
speculative. The expected recovery rate of the senior debt is 75%, the old junior
debt is 60%, and the recently issued speculative debt is 15%. Using
approximation formulas, what arbitrage-free credit spreads should be expected
on the senior and speculative debt issues?
Risk-neutral default probability of 8.75%
senior debt:
0.0875 (1-0.75)
= 0.021875
speculative debt:
0.0875 (1-0.15)
= 0.074375
91
Suppose that the junior debt of XYZ
Corporation is frequently traded and currently trades at a credit spread of 2%
over riskless bonds of comparable maturity. The senior debt of the firm has not
been regularly traded because it was primarily held by a few institutions, and a
new issue of debt that is subordinated to all other debt has been rated as
speculative. The expected recovery rate of the senior debt is 70%, the old junior
debt is 45%, and the recently issued speculative debt is 25%. Using
approximation formulas, what arbitrage-free credit spreads should be expected
on the senior and speculative debt issues?
Risk-neutral default probability of 3.64%
?
92
Suppose that the junior debt of XYZ
Corporation is frequently traded and currently trades at a credit spread of 3.0%
over riskless bonds of comparable maturity. The senior debt of the firm has not
been regularly traded because it was primarily held by a few institutions, and a
new issue of debt that is subordinated to all other debt has been rated as
speculative. The expected recovery rate of the senior debt is 65%, the old junior
debt is 55%, and the recently issued speculative debt is 30%. Using
approximation formulas, what arbitrage-free credit spreads should be expected
on the senior and speculative debt issues?
Risk-neutral default probability of 6.67%
?
93
Suppose that the junior debt of XYZ
Corporation is frequently traded and currently trades at a credit spread of 4.0%
over riskless bonds of comparable maturity. The senior debt of the firm has not
been regularly traded because it was primarily held by a few institutions, and a
new issue of debt that is subordinated to all other debt has been rated as
speculative. The expected recovery rate of the senior debt is 60%, the old junior
debt is 45%, and the recently issued speculative debt is 35%. Using
approximation formulas, what arbitrage-free credit spreads should be expected
on the senior and speculative debt issues?
Risk-neutral default probability of 7.27%
?
94
Suppose that the junior debt of XYZ
Corporation is frequently traded and currently trades at a credit spread of 4.50%
over riskless bonds of comparable maturity. The senior debt of the firm has not
been regularly traded because it was primarily held by a few institutions, and a
new issue of debt that is subordinated to all other debt has been rated as
speculative. The expected recovery rate of the senior debt is 85%, the old junior
debt is 40%, and the recently issued speculative debt is 30%. Using
approximation formulas, what arbitrage-free credit spreads should be expected
on the senior and speculative debt issues?
Risk-neutral default probability of 7.5%
?
95
Suppose that the junior debt of XYZ
Corporation is frequently traded and currently trades at a credit spread of 1.50%
over riskless bonds of comparable maturity. The senior debt of the firm has not
been regularly traded because it was primarily held by a few institutions, and a
new issue of debt that is subordinated to all other debt has been rated as
speculative. The expected recovery rate of the senior debt is 90%, the old junior
debt is 70%, and the recently issued speculative debt is 45%. Using
approximation formulas, what arbitrage-free credit spreads should be expected
on the senior and speculative debt issues?
Risk-neutral default probability of 5.0%
?
96
In this example, a hypothetical transaction takes
place between a hedge fund (the Fund) as a credit protection seller and a
commercial bank (the Bank) as a credit protection buyer. The reference entity is
an airline company (the Firm). The referenced asset is $20 million of face value
debt. The term of the transaction is seven years. In exchange for the protection
provided over the next seven years, the Fund receives 2% of the notional amount
per year, payable quarterly. The contract will be settled physically. This means
that if a credit event takes place, the Bank will deliver $20 million in face value of
any qualifying senior unsecured paper issued by the Firm in return for a $20
million payment by the Fund. Further, the contract will be terminated, and no
further payments will be made by the Bank. Let's assume that default takes place
after exactly three years. What cash flows and exchanges take place?
annual premium x face value of debt x N = cash flow from bank in the first 3 years
($20,000.000 x 0.02%) x 3
= 1,200,000
97
In this example, a hypothetical transaction takes
place between a hedge fund (the Fund) as a credit protection seller and a
commercial bank (the Bank) as a credit protection buyer. The reference entity is
an airline company (the Firm). The referenced asset is $25 million of face value
debt. The term of the transaction is seven years. In exchange for the protection
provided over the next seven years, the Fund receives 2.5% of the notional amount
per year, payable quarterly. The contract will be settled physically. This means
that if a credit event takes place, the Bank will deliver $25 million in face value of
any qualifying senior unsecured paper issued by the Firm in return for a $25
million payment by the Fund. Further, the contract will be terminated, and no
further payments will be made by the Bank. Let's assume that default takes place
after exactly three years. What cash flows and exchanges take place?
?
98
In this example, a hypothetical transaction takes
place between a hedge fund (the Fund) as a credit protection seller and a
commercial bank (the Bank) as a credit protection buyer. The reference entity is
an airline company (the Firm). The referenced asset is $30 million of face value
debt. The term of the transaction is seven years. In exchange for the protection
provided over the next seven years, the Fund receives 3% of the notional amount
per year, payable quarterly. The contract will be settled physically. This means
that if a credit event takes place, the Bank will deliver $30 million in face value of
any qualifying senior unsecured paper issued by the Firm in return for a $30
million payment by the Fund. Further, the contract will be terminated, and no
further payments will be made by the Bank. Let's assume that default takes place
after exactly three years. What cash flows and exchanges take place?
?
99
In this example, a hypothetical transaction takes
place between a hedge fund (the Fund) as a credit protection seller and a
commercial bank (the Bank) as a credit protection buyer. The reference entity is
an airline company (the Firm). The referenced asset is $45 million of face value
debt. The term of the transaction is seven years. In exchange for the protection
provided over the next seven years, the Fund receives 3.5% of the notional amount
per year, payable quarterly. The contract will be settled physically. This means
that if a credit event takes place, the Bank will deliver $45 million in face value of
any qualifying senior unsecured paper issued by the Firm in return for a $45
million payment by the Fund. Further, the contract will be terminated, and no
further payments will be made by the Bank. Let's assume that default takes place
after exactly three years. What cash flows and exchanges take place?
?
100
In this example, a hypothetical transaction takes
place between a hedge fund (the Fund) as a credit protection seller and a
commercial bank (the Bank) as a credit protection buyer. The reference entity is
an airline company (the Firm). The referenced asset is $50 million of face value
debt. The term of the transaction is seven years. In exchange for the protection
provided over the next seven years, the Fund receives 4% of the notional amount
per year, payable quarterly. The contract will be settled physically. This means
that if a credit event takes place, the Bank will deliver $50 million in face value of
any qualifying senior unsecured paper issued by the Firm in return for a $50
million payment by the Fund. Further, the contract will be terminated, and no
further payments will be made by the Bank. Let's assume that default takes place
after exactly three years. What cash flows and exchanges take place?
?
101
In this example, a hypothetical transaction takes
place between a hedge fund (the Fund) as a credit protection seller and a
commercial bank (the Bank) as a credit protection buyer. The reference entity is
an airline company (the Firm). The referenced asset is $65 million of face value
debt. The term of the transaction is seven years. In exchange for the protection
provided over the next seven years, the Fund receives 4.5% of the notional amount
per year, payable quarterly. The contract will be settled physically. This means
that if a credit event takes place, the Bank will deliver $65 million in face value of
any qualifying senior unsecured paper issued by the Firm in return for a $65
million payment by the Fund. Further, the contract will be terminated, and no
further payments will be made by the Bank. Let's assume that default takes place
after exactly three years. What cash flows and exchanges take place?
?
102
In this example, a hypothetical transaction takes
place between a hedge fund (the Fund) as a credit protection seller and a
commercial bank (the Bank) as a credit protection buyer. The reference entity is
an airline company (the Firm). The referenced asset is $75 million of face value
debt. The term of the transaction is seven years. In exchange for the protection
provided over the next seven years, the Fund receives 1.5% of the notional amount
per year, payable quarterly. The contract will be settled physically. This means
that if a credit event takes place, the Bank will deliver $75 million in face value of
any qualifying senior unsecured paper issued by the Firm in return for a $75
million payment by the Fund. Further, the contract will be terminated, and no
further payments will be made by the Bank. Let's assume that default takes place
after exactly three years. What cash flows and exchanges take place?
?
103
Consider a bank with a $500 million loan portfolio
that it wishes to sell. It must hold risk-based capital equal to 8% to support these
loans. If the bank sponsors a CDO trust in which the trust purchases the $500
million loan portfolio from the bank for cash, how much reduction in risk-based
capital will the bank receive if it finds outside investors to purchase all of the CDO
securities?
value of loan x risk based capital = reduction in risk-based capital
$500,000,000 x 0.08%
= $40,000,000
104
Consider a bank with a $450 million loan portfolio
that it wishes to sell. It must hold risk-based capital equal to 75% to support these
loans. If the bank sponsors a CDO trust in which the trust purchases the $450
million loan portfolio from the bank for cash, how much reduction in risk-based
capital will the bank receive if it finds outside investors to purchase all of the CDO
securities?
450,000,000 x 0.75
= 337,500,00
105
Consider a bank with a $400 million loan portfolio
that it wishes to sell. It must hold risk-based capital equal to 9% to support these
loans. If the bank sponsors a CDO trust in which the trust purchases the $400
million loan portfolio from the bank for cash, how much reduction in risk-based
capital will the bank receive if it finds outside investors to purchase all of the CDO
securities?
?
106
Consider a bank with a $600 million loan portfolio
that it wishes to sell. It must hold risk-based capital equal to 10% to support these
loans. If the bank sponsors a CDO trust in which the trust purchases the $600
million loan portfolio from the bank for cash, how much reduction in risk-based
capital will the bank receive if it finds outside investors to purchase all of the CDO
securities?
?
107
Consider a bank with a $650 million loan portfolio
that it wishes to sell. It must hold risk-based capital equal to 11% to support these
loans. If the bank sponsors a CDO trust in which the trust purchases the $650
million loan portfolio from the bank for cash, how much reduction in risk-based
capital will the bank receive if it finds outside investors to purchase all of the CDO
securities?
?
108
Consider a bank with a $350 million loan portfolio
that it wishes to sell. It must hold risk-based capital equal to 12% to support these
loans. If the bank sponsors a CDO trust in which the trust purchases the $350
million loan portfolio from the bank for cash, how much reduction in risk-based
capital will the bank receive if it finds outside investors to purchase all of the CDO
securities?
?
109
Consider a bank with a $300 million loan portfolio
that it wishes to sell. It must hold risk-based capital equal to 20% to support these
loans. If the bank sponsors a CDO trust in which the trust purchases the $300
million loan portfolio from the bank for cash, how much reduction in risk-based
capital will the bank receive if it finds outside investors to purchase all of the CDO
securities?
?
110
Consider a bank with a $400 million loan
portfolio that it wishes to sell. It must hold risk-based capital equal to 8% to
support these loans. If the sponsoring bank has to retain a $10 million equity
piece in the CDO trust to attract other investors, how much reduction in
regulatory capital will result?
bank loan x risk based capital % - equity the bank needs to retain = reduction of risk-based capital
400,000,000 x 0.08 - 10,000,000
= 22,000,000
111
Consider a bank with a $350 million loan
portfolio that it wishes to sell. It must hold risk-based capital equal to 8.5% to
support these loans. If the sponsoring bank has to retain a $15 million equity
piece in the CDO trust to attract other investors, how much reduction in
regulatory capital will result?
350,000,000 x 0.085 - 15,000,000
| = 14,750,000
112
Consider a bank with a $450 million loan
portfolio that it wishes to sell. It must hold risk-based capital equal to 9% to
support these loans. If the sponsoring bank has to retain a $20 million equity
piece in the CDO trust to attract other investors, how much reduction in
regulatory capital will result?
?
113
Consider a bank with a $500 million loan
portfolio that it wishes to sell. It must hold risk-based capital equal to 10% to
support these loans. If the sponsoring bank has to retain a $25 million equity
piece in the CDO trust to attract other investors, how much reduction in
regulatory capital will result?
?
114
Consider a bank with a $600 million loan
portfolio that it wishes to sell. It must hold risk-based capital equal to 10% to
support these loans. If the sponsoring bank has to retain a $14 million equity
piece in the CDO trust to attract other investors, how much reduction in
regulatory capital will result?
?
115
Consider a bank with a $650 million loan
portfolio that it wishes to sell. It must hold risk-based capital equal to 7.5% to
support these loans. If the sponsoring bank has to retain a $13 million equity
piece in the CDO trust to attract other investors, how much reduction in
regulatory capital will result?
?
116
Consider a bank with a $700 million loan
portfolio that it wishes to sell. It must hold risk-based capital equal to 7% to
support these loans. If the sponsoring bank has to retain a $17 million equity
piece in the CDO trust to attract other investors, how much reduction in
regulatory capital will result?
?
