Download - Long-Term Capital Management Intro
![Page 1: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/1.jpg)
Risk Management Lessons from Long-Term Capital Management
簡報⼈人 / 吳培菱 吳岱恩 張博能 楊庭杰
Philippe JorionPublished in: European Financial Management 6(September 2000): 277 300
![Page 2: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/2.jpg)
論⽂文架構
Section1 Overview of the Long-term Capital Management (LTCM) saga
Section2 Review how to use the Value at Risk (VAR) to assess the capital
Section3 Discuss how the portfolio optimization(PO) went wrong
Section4 & 5 LTCM’s Risk Profile & Concluding comment
VARLeverage
Risk
LTCM
![Page 3: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/3.jpg)
本⽇日⼤大綱
21
3 4
LTCM 公司介紹LTCM 投資策略LTCM 如何隕落
VAR 介紹LTCM ⾵風險管理模型偏誤
LTCM 投資組合資產相關程度⾵風險估計偏誤
⾵風險概況
LTCM隕落結論整理
![Page 4: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/4.jpg)
曾經有⼀一家避險基⾦金公司只⽤用四年的時間,震撼了華爾街
但在第五年,卻突然⾯面臨了巨額虧損
倒閉前還把整個華爾街和美國政府⼀一起拖下⽔水。
![Page 5: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/5.jpg)
共同基⾦金 避險基⾦金
報酬⺫⽬目標 追求相對於指標指數的 相對報酬率 追求絕對報酬率
費⽤用 管理費 管理費(1%) 績效費(20%)
操作策略 較簡單 買進持有(buy and hold)
較具彈性 可放空、可使⽤用槓桿、 亦可交易衍⽣生性⾦金融商品
⾵風險較⼤大
無法使⽤用避險⼯工具, 或僅有少數避險⼯工具可供使⽤用
較⼩小 可使⽤用避險⼯工具降低⾵風險
透明度及 控制權
投資⼈人擁有較⾼高⾃自主權 與資訊取得權利
控制權主要在經理⼈人⼿手中, 投資⼈人的⾃自主空間相對縮⼩小。 亦不受主管機關監管。
報酬預測 預測共同基⾦金的報酬率 與預測市場⾛走勢⼀一樣困難
投資⼈人可預測某些策略的報酬區間、 可能之最⼤大獲利與最⼤大損失
![Page 6: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/6.jpg)
What happened ?
![Page 7: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/7.jpg)
LTCM’s Saga
![Page 8: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/8.jpg)
John Meriwether & Salomon Brothers1965 1970 1975
• 匯率固定
• 利率管制
• ⾦金價穩定 • 通貨膨脹
• ⽯石油危機
• 放鬆管制
1977
• 成⽴立套利部⾨門(Arbitrage Group)
John Meriwether
![Page 9: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/9.jpg)
1980 1985 1990
1977
(Arbitrage Group)
1995
1983
Eric Rosenfeld+Gregory Hawkins+William Krasker+Lawrence Hilibrand+
Scandal1991
✦ 成⽴立LTCM
+
+
+David Mullins
+USD$50 m/year
John Meriwether & Salomon Brothers
![Page 10: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/10.jpg)
Market is Efficient
![Page 11: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/11.jpg)
LTCM1990
1992✦ LTCM 成⽴立
融資⺫⽬目標:USD$2.5b
績效費: 25%
管理費: 2%
閉鎖期間: 3年
投資⾦金額下限: USD$10 m
不提供正式的投資報告
![Page 12: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/12.jpg)
1990 1995
1992✦ LTCM 成⽴立
+USD$1.25 b
• 義⼤大利銀⾏行• 瑞⼠士信貸• 瑞⼠士銀⾏行• 美林證券• ⾹香港地產開發局• 新加坡政府投資公司• 台灣銀⾏行• 曼⾕谷銀⾏行• HP CEO 唐納德·⾺馬⻯⿓龍
• 花旗集團 桑迪•威爾• ⿆麥肯錫合夥⼈人• ⾙貝爾斯登執⾏行⻑⾧長肯恩• 德國德利銀⾏行• ⽇日本住友銀⾏行• 英國保誠⼈人壽公司• 瑞⼠士寶盛銀⾏行• 共和國國家銀⾏行• 聖約翰⼤大學捐贈基⾦金
1994.02✦ LTCM 開張
LTCM
+ +Eric Rosenfeld
Gregory Hawkins
William Krasker
Lawrence Hilibrand
+
David Mullins
++++
+USD$50 m/year
![Page 13: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/13.jpg)
LTCM
0
1250
2500
3750
5000
0
15
30
45
60
1994 1995 1996 1997 1998
Annual return (%) Equity (million)
4,990.3
3,990.4
2,540.4
1,6001,250
25
5759
2828
59 57
25
+USD$ 3,740,260,000
![Page 14: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/14.jpg)
A 當期國債與⾮非當期國債交易 (On the run vs. Off the run bond)
B 信⽤用價差套利 (Credit Spresd trades)
C 出售波動度策略 (Selling volatility)
交易策略
![Page 15: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/15.jpg)
On the run vs.
Off the run bond
![Page 16: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/16.jpg)
6%
2013
6.1%
2013
6.0%
2014
on the runoff the run
賺得10bp/元
6.0%on the run
6.1%off the run
Converge
Tiny profit Big leverage
當期國債與⾮非當期國債交易 (On the run vs. Off the run bond)
shortlong
![Page 17: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/17.jpg)
• 1994:
美國當期債券(On the run)與⾮非當期債券(Off the
run)相差12 bp
• LTCM⼤大量買⼊入⾮非當期債券、賣出當期債券
當期國債與⾮非當期國債交易 (On the run vs. Off the run bond)
![Page 18: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/18.jpg)
LTCM Bank回購債券
⽀支付現⾦金+回購利息
LTCM Bank以1.25 b為擔保
借⼊入 2 b 現⾦金
LTCM Market2 b
買⼊入 2 b ⾮非當期國債
LTCM Market以 2 b 為擔保
放空 2 b 當期國債
LTCM Bank以2 b⾮非當期國債為擔保
借⼊入 2 b - Hair cut
Repos
![Page 19: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/19.jpg)
![Page 20: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/20.jpg)
1998
✦ 25% 1997年報酬率
1998.08.17
• 俄羅斯國債違約
• 信⽤用利差(credit spread)、
⾵風險貼⽔水(risk premia)、
流動性價差(liquidity spread)飆升
- USD$550 m /day x55
LTCM
![Page 21: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/21.jpg)
Credit Spread Trades
![Page 22: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/22.jpg)
信⽤用價差套利 (Credit Spresd trades)
1997. 亞洲⾦金融危機, 使俄羅斯國債與美國國債之間的yield spread 過⾼高
Converge
long
俄羅斯
high yield
short
美國
low yield
1998. 俄羅斯⾦金融危機
俄羅斯
long
high yield
Diverge 美國
short
low yield
![Page 23: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/23.jpg)
1998.09
- USD$550 m /day
1998.08.17
HELP!!
• 股市波動度劇增• 出售波動度交易虧損
1998.09.21
LTCM
![Page 24: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/24.jpg)
Selling volatility
![Page 25: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/25.jpg)
出售波動度 (Selling volatility)
1998. 市場剛經歷亞洲⾦金融⾵風暴,隱含波動度過⾼高
shortVIX
Option
![Page 26: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/26.jpg)
1998.09
- USD$550 m /day
1998.08.17
HELP!!
• 股市波動度劇增• 出售波動度交易虧損
1998.09.21
- USD$550 m /day
LTCM
• Margin call耗盡所有流動性資產
![Page 27: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/27.jpg)
- USD$5,500,000 /day
1998.09.23
Bailout!
x14 +USD$3.6 b
1998.09.21
LTCM
• Next-to-zero haircut• ⾦金融機構急欲變現
LTCM的Repo• 設⽴立於開曼群島
![Page 28: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/28.jpg)
1998.09.28
x14
+USD$3.6 b
1998.09.23
+• LTCM已損失83%資產,共損失USD$4.4 b• 避險基⾦金價值僅存USD$400 m• 投資⼈人損失92%原始投資⾦金額
40.9%
15.9%
43.2%
PartnersUnion Bank of SwitzerlandOther Investors
LTCM
![Page 29: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/29.jpg)
1999.121998.12
• Oversight Partners I LLC• portfolio gained 13%
• LTCM 解散 • 還清負債
未完待續
LTCM
![Page 30: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/30.jpg)
VAR & Equity Capital
![Page 31: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/31.jpg)
釐清風險來源 衡量風險程度 控制風險
![Page 32: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/32.jpg)
► Introduction of VAR
► Equity coverage
![Page 33: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/33.jpg)
What is VAR ?Variance ?
Vector AutoRegression?
![Page 34: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/34.jpg)
Value At Risk 風險值 u Definition 定義
- The worst loss that can happen under normal market conditions over a specified horizon at a specified confidence level.
- 在某一特定時間內、某一信賴水準下,金融商品由於價格變動所產生
的最大損失。
u Basel Committee on Banking Supervision
- Horizon : 10 day
- Confidence level : 99%
Value At Risk
![Page 35: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/35.jpg)
u Application
- 與市場⾵風險(market risk) 作連接,將⾵風險化為具體數據,作為解析和判斷相對⾵風險
- 交易若發⽣生突發損失,作為緩衝,所需資本的量
Value At Risk
![Page 36: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/36.jpg)
Equity coverage
![Page 37: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/37.jpg)
Desired rating� 1-year probability of default�
Equity coverage�normal� t(6)� t(4)�
Aaa 0.009� 3.75� 9.26� 15.96�Aa1� 0015� 3.62� 8.45� 14.03�Aa2� 0.022� 3.51� 7.89� 12.72�Aa3� 0.035� 3.39� 7.25� 11.30�A1� 0.05� 3.29� 6.79� 10.31�A2� 0.06� 3.24� 6.56� 9.83�A3� 0.07� 3.19� 6.37� 9.45�
Baa1� 0.13� 3.01� 5.67� 8.04�Baa2 0.16� 2.95� 5.44� 7.61�Baa3 0.7� 2.46� 4.01� 5.10�Ba1 1.25� 2.24� 3.52� 4.31�Ba2 1.79� 2.10� 3.23� 3.88�Ba3 3.96� 1.76� 2.62� 3.01�B1 6.14� 1.54� 2.30� 2.58�B2 8.31� 1.38� 2.08� 2.30�B3 15.08� 1.03� 1.65� 1.77�
Equity coverage(權益覆蓋率)
The number of annual S.D. necessary to achieve desired default probability.
VAR 協助訂定信賴⽔水準
![Page 38: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/38.jpg)
Desired rating1-year
probability of default
Equity coveragenormal t(6) t(4)
Aa2 0.022 3.51 7.89 12.72
![Page 39: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/39.jpg)
LTCM’s Risk Management Story
![Page 40: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/40.jpg)
How to apply VAR
to the LTCM story ?
![Page 41: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/41.jpg)
u 1978-1997 S&P 500 annual average volatility 15%
u σ daily $ =4.7b x 15% ÷ √(252)= 44m
u In May 1998, LTCM’s target daily volatility was 45m
![Page 42: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/42.jpg)
Heroic assumption
{ 1 波動度是固定的
2 報酬分配為對稱
3 報酬符合常態分配
![Page 43: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/43.jpg)
波動度是固定的
事實上,在發⽣生狀況時,波動度會飆升!
1
![Page 44: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/44.jpg)
Merton(1974)
選擇權賣⽅方-
上⽅方⾵風險有限
下⽅方⾵風險無限
short call
short put
2 報酬分配是對稱
![Page 45: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/45.jpg)
3 報酬符合 常態分配
實際上發⽣生極端情況的機率, ⽐比常態分配⾼高, 具有厚尾(Fat Tail)現象
![Page 46: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/46.jpg)
Biases show upreturn annually = 18%
return monthly = 4.7b x 18% ÷12 = 71 m
VAR daily, 99% = 105m
VAR monthly, 99% = 105m x √21 =480 m
480 m - 71 m =409 m
May, 1998 June, 1998
310 m 450m 409 m 71 m
Largest Acceptable Loss
0
![Page 47: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/47.jpg)
u 移除投資組合中獲利較少的投資
u 然⽽而此舉也移除了投資組合中流
動性較佳的投資
u 因此,造成LTCM更無⼒力應付之後
的保證⾦金催繳
Adjustment
![Page 48: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/48.jpg)
u 每⽇日的計算波動,仍與模型計算的相異
u 可能有些錯誤
After adjustment
市場的波動度增加1模型有偏誤2
![Page 49: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/49.jpg)
♦ 1995-1997
平均波動度低於歷史⽔水準
♦ 1998.08
波動度飆升
0.019
Above 0.04
市場的波動度增加1
![Page 50: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/50.jpg)
u 1998. 8 lost 1710m
u σ monthly,$ = 45m x √21 = 206m
u 1710m ÷ 206m = 8.3 σ
u Occur once every 800 trillion
years
- Thus, normal assumption was
surely wrong.
u If assume with fatter tails, such as
Student t(4)
- Occur once every 8 years
- ➜ make sense
模型有偏誤2
![Page 51: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/51.jpg)
u 依據歷史資料去評估 - 低估⾵風險
模型有偏誤2
![Page 52: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/52.jpg)
Some lessons1 壓⼒力測試的重要性 ➜估計最⼤大損失為 2.3b ➔1998年實際最⼤大損失為4.4b
2 歸因於兩種因素造成更⼤大損失
{ 投資部位規模
資產的流動性
![Page 53: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/53.jpg)
Portfolio & Leverage
![Page 54: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/54.jpg)
債券
公司債券
房屋抵押債券
避險基⾦金
利率交換
套利策略
過期公債
當期公債
出售波動度策略
價差策略
低估⾵風險值
波動度變化
⾵風險值
模型估計偏誤
Value At Risk回購協議
操作槓桿收斂套利
MBS
流動性溢酬 信⽤用⾵風險
違約⾵風險
市場中⽴立策略
⻑⾧長期資本管理公司
![Page 55: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/55.jpg)
“
”
隨著越來越多的⼈人進⼊入這個市場,
要想發現新的市場不均衡,
並且從中獲利變得很困難。
![Page 56: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/56.jpg)
加⼤大投資槓桿“”
![Page 57: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/57.jpg)
Risk
Return
Efficient frontier
Different Utility function
Risk-free rate
Market Optimal Portfolio
High leverage !
![Page 58: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/58.jpg)
Corporate Bond
預期報酬: 7.28%
波動度: 1.58%
BAA-rated
Treasure Bond
預期報酬率: 5.75%
波動度: 1.90 %
公司債 vs. 國債
相對利差= 7.28% - 5.75% = 1.53%
買 賣
![Page 59: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/59.jpg)
建構⼀一組投資組合
效⽤用函數
找尋效⽤用最佳資產配置權重{ }
![Page 60: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/60.jpg)
Portfolio and Leverage
Market Optimal Portfolio
High leverage !
“ Leverage ratio ≑20 ”“ Expected Annual Return ≑ 40% ”
![Page 61: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/61.jpg)
Corporate Bond
預期報酬: 7.28%
波動度: 1.58%
Treasure Bond
預期報酬率: 5.75%
波動度: 1.90 %
相關係數: 0.9654
無⾵風險資產報酬: 5.36%
Input Data (⾼高度相關的兩個資產)
December 1997買
賣
![Page 62: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/62.jpg)
Output Data
Output Data(Position for 1 equity)
Corporate Bond: $19.66
Treasure Bond: -$15.60
Risk Free Asset: -$3.06
![Page 63: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/63.jpg)
Figure 6 Table 2
Optimal Portfolio Return
初始資產
⽉月報酬率
⽉月波動度
年報酬率
年波動度
Risk
$1 $4,700m
每⽇日波動度
3.1% $145m
8.1% $382m
37.0% $1,740m
28.1% $1,322m
1.8% $83m
![Page 64: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/64.jpg)
Ratio of equity to SD
= Equity Coverage(權益覆蓋率)
= Safety Factor(安全因⼦子)
i.e: Sum of Bond is $1 = $19.66 - $15.60 - 3.06Monthly volatility is 0.081.
Thus, safety factor value means $1/$0.081=12.3
![Page 65: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/65.jpg)
What’s the
Problem?
![Page 66: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/66.jpg)
Portfolio’s Problem
![Page 67: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/67.jpg)
𝞀 =0.9654
Does it make sense?
“”
Always?1993~1997
![Page 68: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/68.jpg)
兩年的移動平均 v.s 五年的移動平均
1998 𝞀=0.8
1992 𝞀=0.75
![Page 69: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/69.jpg)
1998
8.1%
13.65%
0.965
1997
19.24%
資產相關程度與⾵風險程度
![Page 70: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/70.jpg)
Ratio of equity to SD
= Equity Coverage (權益覆蓋率)
= Safety Factor (安全因⼦子)
1. December 1997(𝞀=0.965 S.D=8.1%)The safety factor = 1/0.081 = 12.34
2. In 1998(𝞀=0.8, S.D=19.24%)The safety factor = 5.20
![Page 71: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/71.jpg)
Rho S.D Safety Factor
Monthly Probability of Ruin Normal Student-6 Student-4
0.970 7.58% 13.19 0% 0.00059% 0.000954%
.
.
.
.
.
.
.
.
.
.
.
.
0.800 19.24% 5.20 0% 0.1009% 0.32637%
.
.
.
.
.
.
.
.
.
.
.
.
Table 3
171倍 342倍
![Page 72: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/72.jpg)
Ju & Pearson(1999)- (True VAR / Estimated VAR)
Model
Observation(T)
Number of Assets (N)
10 20 50 100
Moving Average
50 123% 164%
100 110% 124% 201%
200 105% 111% 133% 199%
1000 101% 102% 105% 111%
Exponential 135% 185% 485% 2174%
![Page 73: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/73.jpg)
LTCM’s Risk Profile
![Page 74: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/74.jpg)
1
2
⾵風險集中度⾼高
假設錯誤 ➜ 低估⾵風險
LTCM 的隕落...
![Page 75: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/75.jpg)
違約⾵風險
1 ⾵風險集中度⾼高
{ }市場⾵風險
流動性⾵風險
![Page 76: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/76.jpg)
Portfolio to Risk Factors
TradeLoss if risk factor increase
Long interest swap
Short equity options
Long off-the-run/ Short on-the-run Treasures
Long MBS securities(hedge)
Long sovereign debt
volatility default illiquidity
✔
✔
✔
✔
✔
✔
✔
✔
✔
✔
✔
Table 5
![Page 77: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/77.jpg)
2 假設錯誤 ➜ 低估⾵風險
常態分配
獲利&損失呈現對稱波動度不變
![Page 78: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/78.jpg)
Oct 245million�
Target 45million�
低估波動的的潛在⾵風險
![Page 79: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/79.jpg)
1998 年五⽉月 俄羅斯⾦金融危機爆發
債券價差的改變與預期相反
![Page 80: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/80.jpg)
short call
報酬分配的假設錯誤
![Page 81: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/81.jpg)
波動度常數假設錯誤
![Page 82: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/82.jpg)
LTCM vs. 投資銀⾏行
⾵風險因⼦子
⻑⾧長期資本管理公司 ⼀一般投資銀⾏行
較少 投資組合較多樣化
籌資速度 較慢 較快
勝!
勝!
波動度增加 損失 受益勝!
![Page 83: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/83.jpg)
Conclusion
![Page 84: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/84.jpg)
LTCM失敗的關鍵槓桿+部位 ➜ ⽅方向錯誤容易瞬間⽡瓦解
太
⼤大
⾵風險 ➜ 集中過
度
假設不合理{ 模型常態分配
變異數常數
資產相關係數固定
1
2
(流動性⾵風險⇡)
3
4 VAR 能將市場⾵風險數據化但沒有考慮到流動性⾵風險等其他⾵風險
![Page 85: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/85.jpg)
⿊黑天鵝效應
![Page 86: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/86.jpg)
沒有⼀一個模型可以完全預測市場“ ”
![Page 87: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/87.jpg)
Q & A ?
![Page 88: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/88.jpg)
Thanks foryour attention
![Page 89: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/89.jpg)
附錄
![Page 90: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/90.jpg)
Part 1
![Page 91: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/91.jpg)
Robert C. Merton
哥倫⽐比亞⼯工程數學學⼠士
加州理⼯工理學碩⼠士
⿇麻省理⼯工經濟學博⼠士,師承Paul Samuelson
研究投資組合、選擇權定價理論
將⽤用於定價選擇權的數學分析⽅方法更廣泛地應⽤用在⾦金融領域上,
此後財務⼯工程學興起,進⽽而促進全球衍⽣生性⾦金融商品的繁榮
![Page 92: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/92.jpg)
Dream TeamJohn Meriwether Robert C. Merton Myron Scholes
David Mullins Lawrence Hilibrand
Eric Rosenfeld Gregory Hawkins
William Krasker
Richard Leahy
![Page 93: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/93.jpg)
資產(Asset)� 負債(Liability)
現金 12.5億 REPO� 20億
同業拆放 20億 賣出公債 20億
買進公債 20億 資本(Equity) 12.5億
52.5億 52.5億
槓桿比率(Leverage ratio)=52.5/12.5=4.2
當期國債與⾮非當期國債交易 (On the run vs. Off the run bond)
![Page 94: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/94.jpg)
1995
核⼼心資本
50億
將30億資本抵押借⼊入
1250億
透過1250億擔保,操作表外衍⽣生性⾦金融商品
1.25兆
x25
x10
x250
![Page 95: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/95.jpg)
x25x28
x31
x55
![Page 96: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/96.jpg)
Part 2
![Page 97: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/97.jpg)
u Measure
1. Horizon
- Related to liquidity of the assets.
- The normal reaction time should run into difficulties.
2. Confidence level
- High enough that the probability of exceeding VAR is very low.
Value At Risk
![Page 98: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/98.jpg)
Value At Risk
LTCM
By the end of August, 1998
Equity capital 2.3 b
Excess liquidity 1 b
Reduce risk1
Raise fresh capital2
The size of its positions is
too enormous to cut its
risk exposure fast.
LTCM unable to attract
new investors.
Raise fresh capital2
![Page 99: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/99.jpg)
Three major approaches of measurement:
JPM Risk Metrics (or variance/covariance approach)
Historic or Back Simulation
Monte Carlo Simulation
Value At Risk
![Page 100: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/100.jpg)
Value At Risk-RiskMetrics Model
Daily Earning at Risk (DEAR)
= Dollars market value of position x price volatility
DEAR⬇p = √ DEAR⬇A ⬆2 + DEAR⬇B⬆2 + 2ρ⬇AB x DEAR⬇A DEAR⬇ B
Value at Risk = DEAR x √days
![Page 101: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/101.jpg)
Revalue portfolio based on actual prices (returns) on the
assets that the day before
calculate 5% worst-case outcomes
Value At Risk-Historic Simulation
![Page 102: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/102.jpg)
Value At Risk
Future
Face value 100m
Margin 10mPayoff 5m
u VAR resolves the paradox of how to
calculate rate of return that require
no upfront investment.
u Just consider the amount of equity
capital to cover most of the potential
losses at the predetermined
confidence level.
![Page 103: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/103.jpg)
Part 3
![Page 104: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/104.jpg)
Notation of Portfolio Optimization
µ = µ1 µ2 ... µn⎡⎣
⎤⎦t
ω = ω1 ω 2 ... ω n⎡⎣
⎤⎦t
∑ =Var(µ) = σ ij⎡⎣ ⎤⎦n×n =
σ 12 σ 12 ... σ 1n
σ 21 σ 22 ... σ 2n
! ! ! !σ n1 σ n2 ... σ n
2
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
µp =ω1µ1 +ω 2µ2 + ...+ω nµn = ω iµii=1
n
∑
![Page 105: Long-Term Capital Management Intro](https://reader030.vdocuments.net/reader030/viewer/2022020123/55ceee9fbb61ebdb7f8b46de/html5/thumbnails/105.jpg)
λ
𝛔 𝛔 𝛔
預期報酬 預期報酬 預期報酬λ >0 λ < 0
⾵風險趨避 ⾵風險中⽴立 ⾵風險愛好