投资与风险

债券债券

时间价值

货币是有时间价值的

金融工具分类与时间价值简易贷款年金附息债券贴现债券

现值和终值简易贷款年金附息债券

贴现债券

Tr

FVP

1

r

c

r

c

r

cP

211

到期收益率简易贷款年金附息债券贴现债券

TT

Y

Fc

Y

c

Y

cP

111 2

21

利率折算惯例

比例法复利法

名义利率与实际利率差别在于是否考虑了通货膨胀的影响

即期利率与远期利率利率水平的决定

可贷资金模型流动性偏好模型

利率的结构预期假说市场分割假说偏好停留假说

面值（ Face or par value ）息票率（ Coupon rate ）

零息票债券利息支付方式债券契约

债券特征

各类债券国债企业债地方政府债券海外债创新债券

指数化债券浮动和反向债券

Money Markets

US Treasury Bills (T-Bills) Certificates of Deposit (CD) Commercial paper (CP) Bankers’ acceptances Eurodollars Repos and Reverses Federal Funds

US Treasury Bills

Initial maturities are91-182 days, offered weekly52 weeks, offered monthly

Competitive and noncompetitive (10-20%) bids.

The investor buys the instrument at discountbid-ask (spread) represents the profit for the

dealerquotes use the bank discount yield.Exempt of state and local taxes.

Bank Discount Yield

$10,000 par T-bill at $9,600 with 182 DTM.$400(360/182) = $791.21

thus the bank discount yield is 7.91% rBD=(10,000-P)/10,000 ·360/n

effective annual yield is:(1+400/9600)2-1=8.51%

bond equivalent yield is:rBEY=(10,000-P)/P ·365/n

Certificates of Deposit

Time deposits with commercial banks.It may not be withdrawn upon demand.Large CDs can be sold prior to maturity.Insured by FDIC up to $100,000(Federal Depository Insurance Corporation)

Commercial Paper

Unsecured short term debt (corporations).Maturity is up to 270 days.CP is issued in multiples of $100,000.Small investors buy it through mutual

funds.Most issues have credit rating.Treated for tax purposes as regular debt.LC backed (letter of credit) optional.

Bankers’ acceptances

Orders to a bank by a customer to pay a given sum at a given date.

Backed by bank. Traded in secondary markets.Widely used in international commerce,

because the creditworthiness is supplied by a bank.

Eurodollars

Dollar denominated time deposits in foreign banks.

Most are for large amounts and with maturity of less than 6 months.

Repos and Reverses

Repurchase agreements (RPs) used by dealers in government securities.

Term repo has a maturity of 30 days or more.

Reverse repo is the result of a dealer finding an investor buying government securities with an agreement to sell them at a specified price at a specified future date.

Federal Funds

Commercial banks that are members of the Federal Reserve System (Fed) are required to maintain a minimum reserve balance with Fed.

Banks with excess reserves lend (usually overnight) to banks with insufficient reserves.

Brokers’ Calls

Brokers borrow funds to loan to investors who wish to buy stock on margin.

The broker agrees to repay the loan upon the call of the bank.

The rate is higher because of the credit risk component.

LIBOR

London Interbank Offer Rate (LIBOR) is the rate at which the large London banks lend among themselves.

This rate serves often as an anchor for floating rate agreements which for example can be set at LIBOR + 3%

Yields on Money Market Instruments

In general, money market instruments are quite safe.However, T-bills are the safest of the money

instruments.As a result the other instruments provide a

slightly higher yield.

Fixed-Income Capital Markets

T-Notes - initial maturity of 10 years (or less).

T-Bonds - initial maturities of 10-30 years.Par (also called face or principal) $1,000.Interest (coupons) paid semiannualy.

Rate Mo/Yr Bid Asked Chg. Ask Yld83/4 Aug 00n 105:16 105:18 +8 7.55

Rate coupon payment 83/4% of $1,000;paid semiannually; $43.75 per bond each 6 mo.

Maturity = August 2000 n = note

Bid =105:16 means 10516/32=105.5at the price $1055 buyer is willing to buy.

Ask=105:18 means 10518/32=105.5625 at the price $1055.625 seller is willing to sell.

Municipal Bonds (Munis)

Issued by state and local governments and agencies. Interest (not capital gains!) is exempt from federal taxes.

General Obligations are backed by the taxing power of the issuer.

Revenue bonds are backed only by revenues from specific projects.

Industrial Development bond is issued to finance a private projects.

Interest from Munis

Is not subject to federal income tax.Hence the yields are lower ： r (1- t) = rm

r - before tax return on taxable bond rm - return on municipal bond

t - marginal tax rateAttractive to wealthy investors.

Corporate Bonds

Used to generate long-term funds.The primary difference is the default risk.Backed by specific assets (like mortgages).By the financial strength of the firm only

(debentures).Callable at a call price (firm).Convertible, may be exchanged to a stock

(investor).

信用 赎回条款转换条款回售条款浮动利率

债券条款

违约风险和评级评级公司

Moody’s Investor ServiceStandard & Poor’sFitch (Duff and Phelps)

两个大类投资类投机类

偿债能力（ Coverage ratios ）杠杆比率（ Leverage ratios ）流动性比率（ Liquidity ratios ）盈利能力（ Profitability ratios ）现金流（ Cash flow to debt ）

评级机构使用的指标

偿债基金未来债务红利限制抵押

违约风险保护

债券定价（ Bond Pricing ）

)1()1(1 rParValue

rCP T

T

T

tt

tB

PB = 债券价格Ct = 利息T = 付息次数R = 要求收益率

10 年期，面值 1000, 8% 息票率，半年付息一次

PB = $1,148.77

Ct = 40P = 1000T = 20 periodsr = 3%

)03.01(1000

)03.01(40

20

20

1

ttBP

要求收益率高则债券价格低

要求收益率为零则债券价格为未来现金流之和

债券价格与要求收益率之间的关系

Price

Yield

价格和要求收益率

)1(1000

)1(35950

20

1 rrT

tt

则，收益率 r = 3.8635%

10 年期，面值 1000 ，息票率 = 7% ，当前价格 = $950

收益率折算

折算为年收益率7.72% = 3.86% x 2

实际年收益率(1.0386)2 - 1 = 7.88%

当期收益率$70 / $950 = 7.37 %

实现的收益率和到期收益率再投资假设持有期收益

利率变化利息的再投资价格变化

持有期收益

I = 利息P1 = 卖出价格P0 = 买入价格

0

0

P

PPIHPR I

Example

息票率 = 8% 要求收益率 = 8% 期限 =10 年 P0 = $1000

由于要求收益率降到 7%P1 = $1068.55

HPR = [40 + ( 1068.55 - 1000)] / 1000

HPR = 10.85% ( 半年 )

积极策略预测利率走势寻找市场的非有效性

消极策略控制风险平衡风险与收益

债券投资的基本策略

债券定价基本性质价格和收益率的反向关系收益增加比收益减少引起的成比例的价格变化较小长期债券的价格比短期债券的价格对利率的敏感性

更强随着到期日的增加，价格敏感性的增加呈下降趋势利率敏感性与息票率呈反向关系当债券以一较低的到期收益率出售时，债券价格对

收益变化更敏感

久 期A measure of the effective maturity of a

bondThe weighted average of the times until

each payment is received, with the weights proportional to the present value of the payment

Duration is shorter than maturity for all bonds except zero coupon bonds

Duration is equal to maturity for zero coupon bonds

icePryCFwt

t t )1(

twtDT

t

1

tperiodforFlowCashCFt

久期的计算

8%Bond

Timeyears

Payment PV of CF(10%)

Weight C1 XC4

.5 40 38.095 .0395 .0198

1 40 36.281 .0376 .0376

1.5

2.0

40

1040

sum

34.553

855.611

964.540

.0358

.8871

1.000

.0537

1.7742

1.8853

一个例子

久期与价格之间的关系

= 连续复利

= 年复利

修正久期 D* = D / (1+y)

r

rD

P

P

1

rDP

P

r

r

Rules for Duration

Rule 1 The duration of a zero-coupon bond equals its time to maturity

Rule 2 Holding maturity constant, a bond’s duration is higher when the coupon rate is lower

Rule 3 Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity

Rule 4 Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower

Rules for Duration (cont’d)

Rules 5 The duration of a level perpetuity is equal to:

Rule 6 The duration of a level annuity is equal to:

Rule 7 The duration for a corporate bond is equal to:

1)1(

1

Ty

T

y

y

y

y)1(

yyc

ycTy

y

yT

]1)1[(

)()1(1

被动管理Bond-Index FundsImmunization of interest rate risk

Net worth immunizationDuration of assets = Duration of liabilities

Target date immunizationHolding Period matches Duration

Cash flow matching and dedication

Yield

Price

Duration

Pricing Error from convexity

久期和凸性

凸性修正

n

tt

t tty

CF

yPConvexity

1

22

)()1()1(

1

Correction for Convexity:

])([21 2yConveixityyD

P

P

Active Bond Management: Swapping Strategies

Substitution swapIntermarket swapRate anticipation swapPure yield pickupTax swap

Maturity

Yield to Maturity %

3 mon 6 mon 9 mon

1.5 1.25 .75

Yield Curve Ride

Contingent Immunization

Combination of active and passive management

Strategy involves active management with a floor rate of return

As long as the rate earned exceeds the floor, the portfolio is actively managed

Once the floor rate or trigger rate is reached, the portfolio is immunized

投资与风险

股票股票

基本面分析基本面分析

全球经济国内经济行业分析公司分析

从上到下的方法

全球经济国家和地区之间的巨大差异政治风险汇率风险

SalesProfitsStock returns

关键经济变量Gross domestic productUnemployment ratesInterest rates & inflationConsumer sentiment

政府政策财政政策

直接的效果缓慢的实施过程

货币政策 Open market operationsDiscount rateReserve requirements

冲击需求

税收政府支出

供给价格变化劳动力教育水平科技进步

经济周期经济周期

波峰波谷

行业与经济周期敏感 不敏感

指标领先

资本品的订单数消费者信心指数股价 ~~~~~

同步工业产量制造品与贸易销售额

滞后消费品价格指数失业平均期限

行业分析对经济周期的敏感度影响敏感度的因素

产品销售对经济周期的敏感程度 经营杠杆比率 （ DOL=净利润变化 / 销售额变化）财务杠杆比率

行业生命周期

Stage Sales Growth

Start-up Rapid & IncreasingConsolidation StableMaturity SlowingRelative Decline Minimal or Negative

行业生命周期

行业结构进入威胁现有企业之间的竞争来自替代品厂商的压力购买者的谈判能力供给厂商的谈判能力

2010 年中国统计数据初步核算， 2010 年中国 GDP总量为 397983亿元，按国际汇率计算，超过日本 4000多么亿美元，正式超越日本成为世界第二大经济体。按可比价格计算， GDP同比增长 10.3% ，在世界主要经济体中位居前列。

2010 年中国统计数据从 1970 年以来，中国经济总量大部分时间位居世界前十，只有在上世纪 80 年代末和 90 年代初由于人民币大贬值，有几年掉出过前十之列。

进入新世纪以来，中国厚积薄发，开始在 10 年之内接连超越意、法、英、德、日等国，跃居世界第二。 2001 年超越意大利、 2005 年超过法国， 2006 年超过英国， 2007 年超过德国， 2010 年超过日本。

在 1990 年，美国 GDP 是中国的 15倍左右，而到了 2010 年，中国已经上升到美国的 40.1% 了，假以时日，超过美国也不是不可能的。

当然，人均方面，中国目前还比较低，还有很长的路要走。

2010 年中国统计数据基础工业数据：

1 、 粗钢产量： 6.27亿吨， 同比增长 9.3% ， 占世界总产量的 44.3% ，超过第 2- 第 20 名的总和；

2 、 钢材产量： 7.98亿吨， 同比增长 14.9%3 、 水泥产量： 18.68亿吨， 同比增长 15.53% ，占世界总产量的 60%；

4 、 电解铝产量： 1565万吨， 同比增长 21.4% ， 超过世界总产量的 65%；

5 、 精炼铜产量； 457万吨， 同比增长 10.6% ， 占世界总产量的 24%；进口 429万吨，消费当量达880万吨，超过世界总产量的 50%；

6 、 煤炭产量： 32.4亿吨， 同比增长 8.9% ， 占世界总产量的 45%；

2010 年中国统计数据基础工业数据：

7 、 原油产量： 2.02亿吨， 同比增长 6.9%； 进口2.39亿吨，同比增长 17.4%；表观消费量达 4.39亿吨；

8 、 乙烯产量： 1418.9万吨，同比增长 31.7% ， 世界第二（美国第一）；当量消费 2400万吨，自给率约为59%；

9 、 化肥产量： 6740.6万吨，同比增长 5.6% ， 占世界总产量的 35%；

10、塑料产量： 5550万吨， 同比增长 20.9% ， 占世界总产量的 20%；

11、化纤产量： 3090万吨 ， 同比增长 12.44% ，占世界总产量的 42.6%；

12、玻璃产量： 6.3亿重量箱，同比增长 10.9% ，超过世界总产量的 50%

2010 年中国统计数据基础设施数据：

1 、电力：新增装机容量 9118万千瓦，总装机容量达到 9.62亿千瓦（美国为 10亿千瓦），同比增长10 ． 07％；

2 、发电量： 42065亿度，同比增长 13.2% ，占世界总发电量的 22% ，首次超越美国（美国 2010年约为 41100亿度），跃居世界第一；美国在这个位置上已经坐了 100多年了，发电量超过美国是

个里程碑的标志；

2010 年中国统计数据基础设施数据：

3 、公路：新增公路通车里程 10.5万公里（含高速），公路网总里程达到 398.4万公里；

新增高速公路 8258 公里，总里程达到 7.41万公里（美国 9 万公里）；

4 、铁路：新线 4986 公里，其中客运专线 1554 公里；投产复线 3747 公里；营业总里程达 9.1万公里（仅次于美国），其中高铁 8358 公里。

5 、能源：一次能源消费量为 32.5亿吨标准煤，同比增长 6% ，首次超越美国，跃居世界第一，这也是个里程碑的标志。

2010 年中国统计数据工业产品数据：

1 、 汽车产量： 1826.47万辆，同比增长 32.44% ，占世界总产量的 25%；销量 1806.19万辆，同比增长 32.37% ，超越了美国创造

的新车销售 1750万辆的历史最高纪录。2 、 船舶产量：造船完工量 6560万载重吨，同比

增长 54.6% ，占世界总量的为 41.9%；新接订单量 7523万载重吨，同比增长 290% ，占世界总量

的 48.5%；手持订单量 19291.5万载重吨，占世界总量的 40.8%；其中造船完工量和手持订单量所占世界份额均比 2009 年有

较大提升；出口额首次突破 400亿美元，同比增长 42.07%。

2010 年中国统计数据工业产品数据：

3 、 工程机械产量： 590亿美元，同比增长 20% ，占世界总产量的 43% （工程机械：挖掘机、装载机、推土机、起重机、混凝土泵 、叉车、压路机等）

4 、计算机产量： 2.46亿台，同比增长 35% ， 占世界总产量的 68% ，份额比 2009 年提升 8 个点；

5 、 彩电产量： 1.18亿台，同比增长 19.5% ，占世界总产量的 50% ，份额比 2009 年提升 2 个点；

2010 年中国统计数据工业产品数据：

6 、 冰箱产量： 7300万台，同比增长 23% ， 占世界总产量的 65% ，份额比 2009 年提升 5 个点；

7 、 空调产量： 1.09亿台，同比增长 35% ， 占世界总产量的 80% ，份额比 2009 年提升 10 个点；

8 、 手机产量： 9.98亿部，同比增长 61% ， 超过世界总产量的 70%；份额比 2009 年暴增 20 个点；

9 、洗衣机产量： 6100万台，同比增长 21% ， 占世界总产量的 44% ，份额比 2009 年提升 4 个点；

2010 年中国统计数据工业产品数据：

10、微波炉产量： 6800万台，同比增长 12% ， 占世界总产量的的 70%；

11、数码相机产量； 8200万台， 占世界总产量的65%；

12、数字电视机顶盒： 1.5亿台，占世界总产量的73%；

2010 年中国统计数据轻工产品：

1 、纱产量： 2717万吨，同比增长 17.5% ，占世界份额的 46%；

2 、布产量： 800亿米，同比增长 6.2%其他：

黄金产量： 340.876吨，同比增长 8.57% ，世界第一；

2009 年中国统计数据农业数据（中国的膳食比例应该是世界上最合理

的）粮食产量 5.31亿吨，占世界份额的 24%；肉类产量 7642万吨，占世界份额的 28%；禽蛋产量 2741万吨，占世界份额的 45%；牛奶产量 3518万吨，仅占世界份额的 5%；水产品产量 5120万吨，占世界份额的 40%；蔬菜产量 5.7亿吨， 占世界份额的 50%；水果产量 1.95亿吨，占世界份额的 18%；油料产量 3100万吨，占世界份额的 7.5%

（中国是世界上最大的大豆进口国）；白糖产量 1200万吨， 占世界份额的 7%

2009 年中国统计数据发展潜力

现在国内人均钢材消费量 400多公斤峰值：美国， 711 公斤；日本， 802 公斤

现在国内人均铜消费量 6 公斤峰值：日本， 12 公斤

国内水泥消费人均： 1300 公斤峰值：日本， 1000 公斤；美国， 1000 公斤

人均石油消耗量

人均天然气消耗量

人均能源消耗量 (换算成石油 )

资本估价模型基本方法

资产负债表估价法红利贴现法市盈率方法

评估增长率和增长机会

资产负债表估价法清算价值（净资产）重置成本托宾 Q

托宾 Q= 市值 / 重置成本

内在价值和市场价格内在价值市场价格交易信号

IV > MP BuyIV < MP Sell or Short SellIV = MP Hold or Fairly Priced爆仓

VD

ko

t

tt

( )11

VD

ko

t

tt

( )11

V0 = Value of StockDt = Dividendk = required return

红利贴现法的基本原理

VD

ko

Stocks that have earnings and dividends that are expected to remain constant

Preferred Stock

无增长模型

E1 = D1 = $5.00

k = .15V0 = $5.00 / .15 = $33.33

VD

ko

无增长模型的举例

VoD g

k g

o

( )1Vo

D g

k g

o

( )1

g = constant perpetual growth rate

稳定增长模型

VoD g

k g

o

( )1Vo

D g

k g

o

( )1

E1 = $5.00 b = 40% k = 15%

(1-b) = 60% D1 = $3.00 g = 8%

V0 = 3.00 / (.15 - .08) = $42.86

稳定增长模型的举例

g ROE b g ROE b

g = growth rate in dividendsROE = Return on Equity for the firmb = plowback or retention percentage rate (1- dividend payout percentage rate)

估计红利增长率

)1()1()1(...2

21

10

kPD

kD

kDV N

NN

PN = the expected sales price for the stock at time N

N = the specified number of years the stock is expected to be held

特定持有期模型

VE

kPVGO

PVGOD g

k g

E

k

o

o

1

11( )

( )

VE

kPVGO

PVGOD g

k g

E

k

o

o

1

11( )

( )

PVGO = Present Value of Growth Opportunities

E1 = Earnings Per Share for period 1

两分定价 :增长和无增长成分

ROE = 20% d = 60% b = 40%

E1 = $5.00 D1 = $3.00 k = 15%

g = .20 x .40 = .08 or 8%

两分定价举例

V

NGV

PVGO

o

o

3

15 0886

5

1533

86 33 52

(. . )$42.

.$33.

$42. $33. $9.

V

NGV

PVGO

o

o

3

15 0886

5

1533

86 33 52

(. . )$42.

.$33.

$42. $33. $9.

Vo = value with growth

NGVo = no growth component value

PVGO = Present Value of Growth Opportunities

两分定价举例

市盈率决定市盈率的两个因素

要求收益率红利预期增长

应用相对定价行业分析中的广泛应用

PE

kP

E k

01

0

1

1

PE

kP

E k

01

0

1

1

E1 - expected earnings for next yearE1 is equal to D1 under no growth

k - required rate of return

市盈率：无预期增长

PD

k g

E b

k b ROE

P

E

b

k b ROE

01 1

0

1

1

1

( )

( )

( )

PD

k g

E b

k b ROE

P

E

b

k b ROE

01 1

0

1

1

1

( )

( )

( )

b = retention ratio ROE = Return on Equity

市盈率：稳定增长

市盈率：无增长例子

E0 = $2.50 g = 0 k = 12.5%

P0 = D/k = $2.50/.125 = $20.00

PE = 1/k = 1/.125 = 8

市盈率：有增长例子

b = 60% ROE = 15% (1-b) = 40%E1 = $2.50 (1 + (.6)(.15)) = $2.73D1 = $2.73 (1-.6) = $1.09k = 12.5% g = 9%P0 = 1.09/(.125-.09) = $31.14PE = 31.14/2.73 = 11.4PE = (1 - .60) / (.125 - .09) = 11.4

市盈率分析中的误区使用会计数据收益随经济周期波动

通货膨胀影响

历史成本低估了经济成本实证研究表明高通货膨胀通常带来低的实际收益

可能的原因Shocks cause expectation of lower earnings

by market participantsReturns are viewed as being riskier with

higher rates of inflationReal dividends are lower because of taxes

投资与风险

资产组合资产组合

风险与风险厌恶风险与风险厌恶

单一前景的风险风险、投机与赌博风险厌恶与效用

2

2

1)( ArEU

2005.0)( ArEU

均值 2

1

0 2

1

0 标准方差

风险与风险厌恶无差异曲线特征

斜率为正下凸同一投资者有无限多条不能相交

资产组合风险资产风险与资产组合风险资产组合中的数学

一个例子无风险收益 5%

正常年份 异常年份牛市 熊市

概率 0.5 0.3 0.2

公司 A 收益率 25% 10% -25%

公司 B 收益率 1% -5% 35%

风险与风险厌恶概率分布的描述

一阶矩二阶矩高阶矩

正态分布和对数正态分布风险厌恶与预期效应

风险与无风险资产的配置将风险资产看作一个整体无风险资产——短期国债一种风险资产与一种无风险资产

资产配置线酬报与波动性比率

风险忍让与资产配置消极策略——资本市场线

最优风险资产组合分散化与资产组合风险两种风险资产的资产组合资产在风险与无风险之间的配置Markowitz 资产组合选择模型具有无风险资产限制的最优资产组合

资本资产定价模型股票需求与价格均衡

积极投资基金对股票的需求被动投资（指数）基金对股票的需求价格均衡

A 股票 B 股票每股价格 39元 39元流通股数 500万股 400万股

市值每年每股红利预期 6.4元 3.8元

要求收益率 16% 10%

年末每股价格预期资本收益率

红利率年度总预期收益率

收益率标准差 40% 20%

相关系数 0.2

无风险收益率 5%

A 股票 B 股票每股价格 39元 39元流通股数 500万股 400万股

市值 195百万元 156百万元每年每股红利预期 6.4元 3.8元

要求收益率 16% 10%

年末每股价格预期 40 38

资本收益率 2.56% -2.56%

红利率 16.41% 9.74%

年度总预期收益率 18.87% 7.18%

收益率标准差 40% 20%

相关系数 0.2

无风险收益率 5%

资本资产定价模型为什么所有投资者都持有市场资产组合？消极策略有效吗？市场资产组合的风险溢价单个证券的期望收益证券市场线

Capital Asset Pricing Model (CAPM)

Equilibrium model that underlies all modern financial theory

Derived using principles of diversification with simplified assumptions

Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development

Individual investors are price takersSingle-period investment horizonInvestments are limited to traded financial

assetsNo taxes, and transaction costsInformation is costless and available to all

investorsInvestors are rational mean-variance

optimizersHomogeneous expectations

Assumptions

All investors will hold the same portfolio for risky assets – market portfolio

Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value

Risk premium on the market depends on the average risk aversion of all market participants

Risk premium on an individual security is a function of its covariance with the market

Resulting Equilibrium Conditions

Capital Market Line

E(r)

E(rM)

rf

MCML

m

M = Market portfoliorf = Risk free rate

E(rM) - rf = Market risk premium

E(rM) - rf = Market price of risk

= Slope of the CAPMM

Slope and Market Risk Premium

2

The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio

Individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio

Expected Return and Risk on Individual Securities

Security Market Line

E(r)

E(rM)

rf

SML

M

ßß = 1.0

= [COV(ri,rm)] / m2

Slope SML = E(rm) - rf

= market risk premium SML = rf + [E(rm) - rf]

Betai = [Cov (ri,rm)] / m2

Betam = m2 / m

2 = 1

SML Relationships

E(rm) - rf = .08 rf = .03

x = 1.25

E(rx) = .03 + 1.25(.08) = .13 or 13%

y = .6

E(ry) = .03 + .6(.08) = .078 or 7.8%

Sample Calculations for SML

Graph of Sample CalculationsE(r)

Rx=13%

SML

m

ß

ß1.0

Rm=11%

Ry=7.8%

3%

xß

1.25

yß.6

.08

Disequilibrium Example

E(r)

15%

SML

ß1.0

Rm=11%

rf=3%

1.25

Suppose a security with a of 1.25 is offering expected return of 15%

According to SML, it should be 13%Underpriced: offering too high of a rate of

return for its level of risk

Disequilibrium Example

CAPM 模型的扩展形式零贝塔模型生命周期与 CAPM 模型CAPM 模型与流动性

Black’s Zero Beta Model

Absence of a risk-free assetCombinations of portfolios on the

efficient frontier are efficientAll frontier portfolios have companion

portfolios that are uncorrelatedReturns on individual assets can be

expressed as linear combinations of efficient portfolios

),(

),(),()()()()(

2QPP

QPPiQPQi rrCov

rrCovrrCovrErErErE

Efficient Portfolios and Zero Companions

Q

P

Z(Q)Z(P)

E[rz (Q)]

E[rz (P)]

E(r)

Zero Beta Market Model

2)()(

),()()()()(

M

MiMZMMZi

rrCovrErErErE

CAPM with E(rz (m)) replacing rf

CAPM & Liquidity

LiquidityIlliquidity PremiumResearch supports a premium for illiquidity

Amihud and Mendelson

CAPM with a Liquidity Premium

)()()( ifiifi cfrrErrE

f (ci) = liquidity premium for security i

f (ci) increases at a decreasing rate

Illiquidity and Average Returns

Average monthly return(%)

Bid-ask spread (%)

单指数证券市场系统风险与公司特有风险指数模型的估计指数模型与分散化

Reduces the number of inputs for diversificationEasier for security analysts to specialize

ri = E(Ri) + ßiF + eßi = index of a securities’ particular return to the

factorF= some macro factor; in this case F is

unanticipated movement; F is commonly related to security returns

Assumption: a broad market index like the S&P500 is the common factor

Single Index Model

(ri - rf) = i + ßi(rm - rf) + ei

Risk Prem Market Risk Prem or Index Risk Prem

i= the stock’s expected return if the market’s excess return is zero

ßi(rm - rf) = the component of return due to

movements in the market index

(rm - rf) = 0

ei = firm specific component, not due to market

movements

Single Index Model

Let: Ri = (ri - rf)

Rm = (rm - rf)

Risk premiumformat

Ri = i + ßi(Rm) + ei

Risk Premium Format

Security Characteristic LineExcess Returns (i)

SCL

....

........

.. ..

.. ...... ..

.... ..

.. ....

......

.. ..

.. ....

.. ....

.. ..

.. ....

.. ....

.. ..

..

.. ...... .... .... ..

Excess returnson market index

Ri = i + ßiRm + ei

Jan.Feb...DecMeanStd Dev

5.41-3.44

.

.2.43-.604.97

7.24.93

.

.3.901.753.32

ExcessMkt. Ret.

ExcessGM Ret.

Using the Text Example

Estimated coefficientStd error of estimateVariance of residuals = 12.601Std dev of residuals = 3.550R-SQR = 0.575

ßß

-2.590(1.547)

1.1357(0.309)

rGM - rf = + ß(rm - rf)

Regression Results

Components of Risk

Market or systematic risk: risk related to the macro economic factor or market index

Unsystematic or firm specific risk: risk not related to the macro factor or market index

Total risk = Systematic + Unsystematic

Measuring Components of Risk

i2 = i

2 m2 + 2(ei)

where;

i2 = total variance

i2 m

2 = systematic variance

2(ei) = unsystematic variance

Examining Percentage of Variance

Total Risk = Systematic Risk + Unsystematic Risk

Systematic Risk/Total Risk = 2

ßi2

m2 / 2 = 2

i2 m

2 / i2 m

2 + 2(ei) = 2

Index Model and Diversification

)(

1

1

1

222

1

1

1

PMPP

N

iPP

N

iPP

N

iPP

PPPP

e

eNe

N

N

eR

Risk Reduction with Diversification

Number of Securities

St. Deviation

Market Risk

Unique Risk

2(eP)=2(e) / n

P2M

2

CAPM 模型与指数模型实际收益与期望收益指数模型与已实现收益指数模型与期望收益的贝塔关系指数模型的行业版本

Industry Prediction of Beta

Merrill Lynch ExampleUse returns not risk premiumshas a different interpretation = + rf (1-)

Forecasting beta as a function of past betaForecasting beta as a function of firm size,

growth, leverage etc.

多因素模型经验基础理论基础经验模型与 ICAPM

Multifactor Models

Use factors in addition to market returnExamples include industrial production,

expected inflation etc.Estimate a beta for each factor using

multiple regressionFama and French

Returns a function of size and book-to-market value as well as market returns

套利定价理论套利机会与利润充分分散的投资组合证券市场线单个资产与套利定价理论套利定价理论与 CAPM 模型多因素套利定价理论

Arbitrage Pricing Theory

Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit

Since no investment is required, an investor can create large positions to secure large levels of profit

In efficient markets, profitable arbitrage opportunities will quickly disappear

Current Expected StandardStock Price$ Return% Dev.% A 10 25.0 29.58 B 10 20.0 33.91 C 10 32.5 48.15 D 10 22.5 8.58

Arbitrage Example from Text

Mean S.D. Correlation

PortfolioA,B,C 25.83 6.40 0.94

D 22.25 8.58

Arbitrage Portfolio

Arbitrage Action and ReturnsE. Ret.

St.Dev.

* P* D

Short 3 shares of D and buy 1 of A, B & C to form PYou earn a higher rate on the investment than you pay on the short sale

APT & Well-Diversified Portfolios

rP = E (rP) + PF + eP

F = some factorFor a well-diversified portfolio

eP approaches zero

Similar to CAPM

F

E(r)%

Portfolio

F

E(r)%

Individual Security

Portfolio &Individual Security Comparison

E(r)%

Beta for F

10

7

6

Risk Free 4

AD

C

.5 1.0

Disequilibrium Example

Disequilibrium Example

Short Portfolio CUse funds to construct an equivalent risk

higher return Portfolio DD is comprised of A & Risk-Free Asset

Arbitrage profit of 1%

E(r)%

Beta (Market Index)

Risk Free

M

1.0

[E(rM) - rf]

Market Risk Premium

APT with Market Index Portfolio

APT applies to well diversified portfolios and not necessarily to individual stocks

With APT it is possible for some individual stocks to be mispriced - not lie on the SML

APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio

APT can be extended to multifactor models

APT and CAPM Compared

投资与风险——期权

Option Terminology

Buy - Long Sell - ShortCallPut Key Elements

Exercise or Strike PricePremium or PriceMaturity or Expiration

Market and Exercise Price Relationships

In the Money - exercise of the option would be profitable

Call: market price>exercise price Put: exercise price>market priceOut of the Money - exercise of the option

would not be profitable Call: market price>exercise price Put: exercise price>market priceAt the Money - exercise price and asset

price are equal

American - the option can be exercised at any time before expiration or maturity

European - the option can only be exercised on the expiration or maturity date

American vs. European Options

Stock OptionsIndex OptionsFutures OptionsForeign Currency OptionsInterest Rate Options

Different Types of Options

Notation Stock Price = ST Exercise Price = XPayoff to Call Holder

(ST - X) if ST >X 0 if ST < X

Profit to Call HolderPayoff - Purchase Price

Payoffs and Profits on Options at Expiration - Calls

Payoff to Call Writer

- (ST - X) if ST >X 0 if ST < X

Profit to Call WriterPayoff + Premium

Payoffs and Profits on Options at Expiration - Calls

Payoff Profiles for Calls

Payoff

Stock Price

0

Call Writer

Call Holder

Payoffs to Put Holder0 if ST > X

(X - ST) if ST < X

Profit to Put Holder Payoff - Premium

Payoffs and Profits at Expiration - Puts

Payoffs to Put Writer0 if ST > X

-(X - ST) if ST < X

Profits to Put WriterPayoff + Premium

Payoffs and Profits at Expiration - Puts

Payoff Profiles for Puts

0

Payoffs

Stock Price

Put Writer

Put Holder

Investment Strategy Investment

Equity only Buy stock @ 100 100 shares $10,000

Options only Buy calls @ 10 1000 options $10,000

Leveraged Buy calls @ 10 100 options $1,000equity Buy T-bills @ 2% $9,000

Yield

Equity, Options & Leveraged Equity

IBM Stock Price

$95 $105 $115

All Stock $9,500 $10,500 $11,500

All Options $0 $5,000 $15,000

Lev Equity $9,270 $9,770 $10,770

Equity, Options & Leveraged Equity - Payoffs

IBM Stock Price

$95 $105 $115

All Stock -5.0% 5.0% 15%

All Options -100% -50% 50%

Lev Equity -7.3% -2.3% 7.7%

Equity, Options & Leveraged Equity

Protective Put

Use - limit lossPosition - long the stock and long the putPayoff ST < X ST > X

Stock ST ST

Put X - ST 0

Protective Put Profit

ST

Profit

-P

Stock

Protective Put Portfolio

Covered Call

Use - Some downside protection at the expense of giving up gain potential

Position - Own the stock and write a callPayoff ST < X ST > X

Stock ST ST

Call 0 - ( ST - X)

Covered Call Profit

ST

Profit

-P

Stock

Covered Call Portfolio

Option Strategies

Straddle (Same Exercise Price)Long Call and Long Put

Spreads - A combination of two or more call options or put options on the same asset with differing exercise prices or times to expiration

Vertical or money spreadSame maturityDifferent exercise price

Horizontal or time spreadDifferent maturity dates

ST < X ST > X

Payoff for

Call Owned 0 ST - X

Payoff for

Put Written-( X -ST) 0

Total Payoff ST - X ST - X

Put-Call Parity Relationship

Long Call

Short Put

Payoff

Stock Price

Combined =Leveraged Equity

Payoff of Long Call & Short Put

Arbitrage & Put Call Parity

Since the payoff on a combination of a long call and a short put are equivalent to leveraged equity, the prices must be equal.

C - P = S0 - X / (1 + rf)TIf the prices are not equal arbitrage will be

possible

Stock Price = 110 Call Price = 17Put Price = 5 Risk Free = 10.25%Maturity = .5 yr X = 105

C - P > S0 - X / (1 + rf)T

17- 5 > 110 - (105/1.05) 12 > 10

Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative

Put Call Parity - Disequilibrium Example

Put-Call Parity Arbitrage

Immediate Cashflow in Six MonthsPosition Cashflow ST<105 ST> 105

Buy Stock -110 ST ST

BorrowX/(1+r)T = 100 +100 -105 -105

Sell Call +17 0 -(ST-105)

Buy Put -5 105-ST 0

Total 2 0 0

Optionlike Securities

Callable BondsConvertible SecuritiesWarrantsCollateralized Loans

Exotic Options

Asian OptionsBarrier OptionsLookback OptionsCurrency Translated OptionsBinary Options

Option Values

Intrinsic value - profit that could be made if the option was immediately exercisedCall: stock price - exercise pricePut: exercise price - stock price

Time value - the difference between the option price and the intrinsic value

Time Value of Options: Call

Option value

XStock Price

Value of Call Intrinsic Value

Time value

Factor Effect on valueStock price increasesExercise price decreasesVolatility of stock price increasesTime to expiration increasesInterest rate increasesDividend Rate decreases

Factors Influencing Option Values: Calls

Restrictions on Option Value: Call

Value cannot be negativeValue cannot exceed the stock valueValue of the call must be greater than the

value of levered equityC > S0 - ( X + D ) / ( 1 + Rf )T

C > S0 - PV ( X ) - PV ( D )

Allowable Range for Call

Call Value

S0

PV (X) + PV (D)

Upper

bou

nd =

S 0

Lower Bound

= S0 - PV (X) - PV (D)

100

200

50

Stock Price

C

75

0

Call Option Value X = 125

Binomial Option Pricing:Text Example

Alternative Portfolio

Buy 1 share of stock at $100

Borrow $46.30 (8% Rate)

Net outlay $53.70

Payoff

Value of Stock 50 200

Repay loan - 50 -50

Net Payoff 0 150

53.70

150

0

Payoff Structureis exactly 2 timesthe Call

Binomial Option Pricing:Text Example

53.70

150

0

C

75

0

2C = $53.70C = $26.85

Binomial Option Pricing:Text Example

Alternative Portfolio - one share of stock and 2 calls written (X = 125)

Portfolio is perfectly hedgedStock Value 50 200Call Obligation 0 -150Net payoff 50 50

Hence 100 - 2C = 46.30 or C = 26.85

Another View of Replication of Payoffs and Option Values

Generalizing the Two-State Approach

Assume that we can break the year into two six-month segments

In each six-month segment the stock could increase by 10% or decrease by 5%

Assume the stock is initially selling at 100Possible outcomes

Increase by 10% twiceDecrease by 5% twiceIncrease once and decrease once (2 paths)

Generalizing the Two-State Approach

100

110

121

9590.25

104.50

Expanding to Consider Three Intervals

Assume that we can break the year into three intervals

For each interval the stock could increase by 5% or decrease by 3%

Assume the stock is initially selling at 100

S

S +

S + +

S -S - -

S + -

S + + +

S + + -

S + - -

S - - -

Expanding to Consider Three Intervals

Possible Outcomes with Three Intervals

Event Probability Stock Price

3 up 1/8 100 (1.05)3 =115.76

2 up 1 down 3/8 100 (1.05)2 (.97) =106.94

1 up 2 down 3/8 100 (1.05) (.97)2 = 98.79

3 down 1/8 100 (.97)3 = 91.27

Co = SoN(d1) - Xe-rTN(d2)

d1 = [ln(So/X) + (r + 2/2)T] / (T1/2)

d2 = d1 + (T1/2)

where

Co = Current call option value.

So = Current stock price

N(d) = probability that a random draw from a normal dist. will be less than d.

Black-Scholes Option Valuation

X = Exercise price.e = 2.71828, the base of the nat. log.r = Risk-free interest rate (annualizes

continuously compounded with the same maturity as the option)

T = time to maturity of the option in yearsln = Natural log functionStandard deviation of annualized

cont. compounded rate of return on the stock

Black-Scholes Option Valuation

So = 100 X = 95

r = .10 T = .25 (quarter)= .50d1 = [ln(100/95) + (.10+(5 2/2))] / (5.251/2)

= .43 d2 = .43 + ((5.251/2)

= .18

Call Option Example

N (.43) = .6664Table 17.2

d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700

Probabilities from Normal Dist

N (.18) = .5714Table 17.2

d N(d) .16 .5636 .18 .5714 .20 .5793

Probabilities from Normal Dist.

Co = SoN(d1) - Xe-rTN(d2)

Co = 100 X .6664 - 95 e- .10 X .25 X .5714

Co = 13.70

Implied VolatilityUsing Black-Scholes and the actual price of

the option, solve for volatility.Is the implied volatility consistent with the

stock?

Call Option Value

P = C + PV (X) - So

= C + Xe-rT - So

Using the example dataC = 13.70 X = 95 S = 100r = .10 T = .25P = 13.70 + 95 e -.10 X .25 - 100P = 6.35

Put Option Valuation: Using Put-Call Parity

Adjusting the Black-Scholes Model for Dividends

The call option formula applies to stocks that pay dividends

One approach is to replace the stock price with a dividend adjusted stock priceReplace S0 with S0 - PV (Dividends)

Hedging: Hedge ratio or delta The number of stocks required to hedge

against the price risk of holding one optionCall = N (d1)

Put = N (d1) - 1

Option ElasticityPercentage change in the option’s value given a 1% change in the value of the underlying stock

Using the Black-Scholes Formula

Portfolio Insurance - Protecting Against Declines in Stock Value

Buying Puts - results in downside protection with unlimited upside potential

Limitations Tracking errors if indexes are used for the

putsMaturity of puts may be too shortHedge ratios or deltas change as stock

values change