Finansiering 2
Introduktionsföreläsning
Course content in brief
Financial Instruments and Investment Companies (Descriptive)
- What? How? Who? Where?
Portfolio Theory and the Capital Asset Pricing Model (Analytical)
- Relationship between risk and return
- Optimal portfolio choice
- Capital Asset Pricing Model, Index Model
Market Efficiency and BehavioralFinance (Descriptive)
- Efficiency, Pricing anomalies, Investor behavior, Limits to arbitrage, Technical
analysis
Portfolio Performance Evaluation, International Diversification (Analytical)
Föreläsning 1 (Investment environment, asset classes and financial instruments)
How is wealth created?
- The wealth of society is determined by the goods and services its members can
produce.
- The productive capacity depends on its real assets.
How much we can produce depends on the real assets, which are: Knowledge, Machines,
Buildings and Land
We also have financial assets: Fixed Income, Equity, Derivatives. They Do not contribute to
the productive capacity of the economy. So why should we care about them?
The use of financial assets
Resource Allocation
- Stock markets help allocate money to firms with best prospects
Time Allocation Of Consumption
- Store wealth over time
Risk allocation
- Spread Risk of real investments to individuals with various risk appetite
Separation of ownership & management
- Manyowners, represented by management
Separation of Ownership & Control
Benefits
- Funding: Many Owners Facilitates Large Investments
- Stability: Stocks can be sold without affecting the business
Idea: The manager act as an agent for the many owners
Agency Problems
- Do managers really try to maximize firm value?
- Maybe they want an empire building
- Or are scared to get sacked and therefore take too little risk
- Executive Compensation Plans –too much risk
The investment process
- An investor’s portfolio is simply her collection of investment assets.
- Rebalancing the portfolio means to change the composition and/or size of the
portfolio.
- Asset allocation: Investors choose among broad asset classes, for example choosing
how much money to put in risk free or risky assets. Usually measured in %.
- Security selection: within an asset class-what securities to hold,
- Portfolio Construction Principles:
- Top-down
- Bottom-up
Top-down
- Examples of investors applying the top-down approach includes
- Pension funds
- Mutual funds
- Private investors
- Need to store a lot of money for a long time
- Start with the board members
Bottom-up
- Start with security selection and do it the other way around
- The investor that would use this strategy is
- Hedgefonder
- (Private investor)
Competitive Markets & the no-free-lunch idea
Question 1: How can you benefit if an apple costs 5kr at Östermalmstorg and 2 kr at
Hötorget?
Answer: Buy an apple for two and sell for five at Östermalmstorg and earn 3, if there is no
interest rate. You can get 3kr per apple without any risk, an arbitrage opportunity.
Question 2: What will happen to apple prices at Östermalmstorg and Hötorget in a
competitive market?
Answer: The prices at hötrget will get pushed up and the price at Östermalm will get pushed
down.
No-free-lunch: Efficient Markets
In fully efficient markets when prices quickly adjust to all relevant information, there should
be neither underpriced nor overpriced securities
- Passive Management (pension funds, etc)
- Diversified portfolio, Not seeking undervalued securities, not attempting to
time the market. Not chasing arbitrage.
- Active Management (hedge funds, etc)
- Finding mispriced securities, trying to time the market. Chasing arbitrage.
No-free-lunch: The Risk & Return trade off
- Investments have uncertain returns. The more uncertainty, the higher the risk.
- Investors prefer more return compared to less, all else equal.
- Investors are in general averse to risk: prefer less risk compared to more, all else
equal.
- The no-free-lunch: Imagine two investments of equal risk but different return. Which
one do you prefer? What happens to prices and returns?
- Risk-return tradeoff: you only get higher returns if accepting higher risk.
Market Players
- Firms-demanders of capital, for investment to expand etc.
- Households–Suppliers of capital
- Governments–Can be both borrowers or lenders
- Swedish Government Debt as of 2022-09-30:
073 000 000 000 SEK
Banking Activities
The Money Market
- Fixed Income Instruments
- Informally: “Money-like” instruments
- Usually these instruments are
- Short term
- Highly marketable
- Liquidly traded
- Low risk
- Large denominations
- Individuals access money market via Money market mutual funds
The Money Market Instruments
- Treasury bills: Short-term government debt (risk free, because its guaranteed by the
government)
- Certificates of deposit (CD): Time deposit with a bank
- Commercial paper: Short-term, unsecured debt of a company
-
Bankers’ Acceptances: Like a postdated check –the bank will pay a sum of money
on a future date to the holder. The bank helps secure the payment.
Eurodollars: Dollar-denominated time deposits in banks outside the U.S.
Repos and reverses: Short-term buy-back agreement.
Fed funds: Short term deposits at the FED
Libor: London Interbank Offered Rate -short term loans between banks in London
(key reference rate-now alternatives include SOFR(USA), SONIA(UK))
Bond market securities:
- Do you remember what a coupon bond is?
The Bond Market Securities
- Treasury Notes (<10 years) and Treasury Bonds (>10 years)
- Long dated, Face value of $1000, coupon payments,
- Inflation-Protected Treasury Bonds (TIPS)
- Real interest rate (payments linked to CPI)
- International Bonds
- Used to borrow abroad. Example Samuaribond
- Municipal Bonds
- Issued by municipals (kommun)
- Corporate Bonds
- Issued by firms, can have embedded options
- Mortgages and Mortgage-Backed Securities
- A pool of loans, secured by houses
Key Properties of Bonds
- The end date (maturity) is fixed
- The sequence of all future payments are known today, i.e. fixed income
- The face value at maturity is known
- All coupon payments are known
- The price the investor should pay when buying depends on factors including:
- Coupon of the Bond
- Discount rate
- Credit quality of issuer (probability of bankruptcy)
- Liquidity of the bond (get out of position)
- Investor cannot loose more than the initial investment
- Compared to many other assets, bond prices has moderate price fluctuations
Equity Securities
- Common stock represent ownership shares in a firm.
- In general each share of common stock entitles the holder to one vote at the firms
annual meeting
- Shareholders elect board of directors for the firm
- The board of directors select managers to run the firm on a day-to-day basis
- Common stock allow separation of ownership and control
- Common stock can be bought and sold at one or more stock exchanges
Limited liability and Residual claim
- Residual claim: In case of bankruptcy, all other claimholders (bondholders,
employees, suppliers) of the firm get payed first. The share holder is last in line
- Limited liability:
- While unincorporated firm owners have to use their own personal assets
(house car, etc) to repay other claimholders of the firm, a share holder does
not.
- A shareholders liability is limited to the initial investment, i.e. equity cannot
become negative
Key properties of Equity
- There is no fixed end date
- The company will live on unless the company goes bankrupt
- The sequence of future payments are unknown today,
- No known coupons but instead uncertain dividend payments
- No known face value, but instead uncertain residual claim in case of
bankruptcy
- Shareholder cannot loose more than the initial investment (limited liability)
- Compared to bonds, equity prices has large price fluctuations
Types of stocks
- Common stock
- Pay (variable) dividend
- Give holder voting power
- Residual claim but limited liability
- Preferred stock
- Ranks after bonds but before common stock in event of bankruptcy
- No voting power
- Promise to pay a fixed amount each year forever
Market Indexes -what and why
- Examples S&P500, OMXS30
- Fictive portfolio of assets
- Quick measure to read off the market development
- Used by mutual funds as a benchmark to evaluate investments against
- Used as an underlying portfolio for derivatives
- Can be traded by making use of derivatives or Exchange Traded Funds (ETF
Common Market Indexes
- Dow Jones Industrial Average (DJIA)
- Includes 30 large and stable corporations in the US
- Price-weighted average
- Standard & Poor’s 500 (S&P500)
- Broadly based index of 500 firms in the US
- Market-value-weighted index
- Non-US stock indexes:
- DAX, FTSE, Nikkei, OMXS30, …
- Non-stock indexes:
- Bloomberg Barclays US Aggregate Bond Index
- Reuters/Jefferies CRB Index
Market Indexes -Construction Principles
- Price Weighted
- Weight of each company in the index is proportional to its share price (holding
one stock of each)
- DJIA
- Market-value weighted
- Weight of each company in the index is proportional to company's market
value
- S&P500, OMXS30
- Equally weighted
- S&P Equal Weight
What is equally weighted?: 33,3 on all the stocks
Man får 13,28 genom att ta market value weighted gånger stock return på alla aktierna.
Här gör man samma som ovan men med equally weighted istället (som det egentligen ska
stå) siffrorna.
Derivatives
A financial derivative is an instrument that is defined in terms of othermore basic underlying
assets such as stocks, bonds, commodities or market indexes
Derivative Instruments
- Forward/futures
- Agreement to trade the underlying asset at a predetermined time at a
predetermined price
- Long position: agreement to buy
- Short position: agreement to sell
- Options
- Call option:Gives the holder the right to buy the underlying asset at a
predetermined time at a predetermined price.
- Put option:Gives the holder the right to sell the underlying asset at a
predetermined time at a predetermined price
Föreläsning 2 (Trading Securities, Mutual funds and other Investment companies)
New capital
● Firms can raise capital:
○ Borrowing money
○ Sell Shares
● Investment banks are hired to manage the sale of the new securities → Primary
Market
● Later these existing securities can be traded again→ Secondary Market
A. The primary market
● Privately held firms
● Publicity traded companies
Privately held firms
● Many times a young firm (but not always)
● Owned by a small number of people (many times family)
● Have fewer obligations to release financial statements and other information
● Raise funds via wealthy investors in a private placement
○ Must have limited number of shareholders
○ No formal secondary market
○ Low liquidity
○ Xtradiscount to attract investors
Publicly Traded Companies
● A firm can raise capital from a wide range of investors if it decides to go public
● The first issue of shares to the general public is called the firm's initial public offering
(IPO)
● IPOs are marketed by investment bankers as a role of underwriters
● Registration of the prospectus and the price must be filed with the Securities and
Exchange Commission (SEC)
● Later on, if firm needs additional capital, the sale of additional shares is called a
seasoned equity offering
It’s a good deal when the bank themselves have the shares to convince potential buyers.
Initial Public Offerings (IPO)
● Road shows
○ Create interest
○ Bookbuilding
● Book building: Determine demand for a new issue by aggregating potential investors
interest to provide valuable information about appropriate quantity and price
● Underwriter try to convince investors to buy shares:–Need to offer investors a
discount to incentives investors –IPOs are commonly (but not always) underpriced
compared to the first official trading day
B. The secondary market
● Types of markets
● Types of orders
● Bid-Ask Spread
Types of markets
1. Direct search markets, least organized where the buyer and seller need to find each
other. Ex blocket.
2. Brokered markets, when you have a broker in between the buyer and the seller. Ex
real estate
3. Dealer markets, when you don't connect the buyer and seller, ex car shops
4. Auction markets, buyer and seller gather at the same spot
Price quotes and orders (ways to buy a stock)
● Market orders, matcha order till aktiepris (Executed immediately, take price)
● Price contingent orders, matcha inte order till aktiepris (Executed later - depending on
price-level)
○ Limit order
○ Stop order
Market Order-a first example
● Call your broker (or look at your digital broker screen)
● Ask for best available price for Intel
○ Best bid: $ 20.77
○ Best ask: $ 20.78
● So if you are buying one share you pay $ 20.78 and if you are selling one share you
will receive $ 20.77 (unless the market moves before the order is executed).
● The bid-ask spread is in this example $0.01
Question
You pay 100,5 because it is the most expensive price. You will lose 5 öre if you sell it
immediately
Limit order
● Call your broker (or look at your digital broker screen)
● Ask for best available price for Intel
○ Best bid: $ 20.77
○ Best ask: $ 20.78
● “Tell your whish” I would like to buy 35725 stocks for $20.76 or less
○ Your order will be put in “que”
● Someone else might be willing to sell 31800 stocks for $20.79
○ such order is also put in “que”
● The que contains price and quantity wishes-it is known as the limit order book
Market order. a second example
Man kommer börja med att köpa alla aktier till 20,78 och 20,79 och 20,80 då summan av
mängden aktier i dessa priser blir antalet aktier du vill ha. Sedan tar du antalet aktier du
behöver gånger priset och delar på antalet aktierna du vill köpa. Då blir priset 20,79
Price contingent orders
Limit order är när man vill köpa för fixed price eller bättre, det som är bättre i exemplet om
lunch är att priset ska vara under 100 för att vara bättre. Man kan även göra sell limit order då
säljer man för det ficerade eller bättre i detta fall är bättre högre än 120.
När man sätter ett minimum för hur mycket man vill sälja aktien för. Detta genom att limit
ens loss. I detta fall har vi en aktie för 100 och 50 är vår threshold.
Samma när man sålt en aktie för 100 kr om man inte vill förlora pengar vid ett visst pris. I
detta fall 150 så kommer man köpa aktien när den är vid 150.
Answer: Limit buy order
C. Electronic Trading and Related Strategies
● Trading systems
● Trading Strategies in Electronic Markets
Trading Systems
● Dealer Markets (Many Bond Markets)
○ Dealers quote prices at which willing to trade
○ Brokers contact a dealer to execute a trade
● Electronic Communication Networks (ECN) (Avanza) (Most markets today)
○ Market and Limit orders are posted
○ Orders are matched and executed automatically without broker
■ Cheaper, faster, anonymous
■ Latency: time to accept, process, deliver order (0.0002 sek)
● Specialist Markets (Old fashioned)
○ A designated specialist keep the order book and make market
Electronic Trading Strategies
● Algorithmic trading
○ Use computers to chase price-anomalies or other discrepancies in the market
■ Cross-market discrepancies
■ Short-term (seconds) trends
■ Pairs-trading
■ Index/assets-trading
● High-Frequency trading
○ Xtremly rapid Algo Trading, när man handlar aktier väldigt snabbt ex när man
köpt en aktie billigt när folk är villiga att betala mer.
○ Increased interest to co-locate servers. När man co-locatear har man datorn vid
exchangen istället för hemma.
● Dark Pools
○ Large trades (blocks), när aktieägare men en stor mängd aktier vill sälja av de
bör de ske i mer invisible markets. Om allmänheten ser att personen vill sälja
så mycket kan det oroa dem och få dem att vilja sälja av sin aktier vilket
kommer ha en negativ påverkan på aktiepriset och därför ge säljaren en
orättvis deal.
○ Traded in less visible market (anonymity)
D. Trading cost and margin
● Trading costs
● Buying on Margin
● Shortselling
Trading Costs
● Broker Commission
○ Fee to broker for trading
● Bid-Ask Spread
○ Not one price but you buy high sell low, i.e. spread is indirect cost
● Price Impact
○ If trading large posts, (eating limit order book), exemplet med 890000 aktier
Buying on Margin (Broker’s call loan)
● Buy stocks
● Pay only 50%
● Borrow the rest (against stocks)
● The stock is the safety because the broker can take it if you don't pay
● The initial payment (50kr) is deposited to an account, called margin account
● If stock price drop below a specified amount (maintenance margin) → margin call:
deposit more money
● Buying on Margin = Levered position
Short selling
● A stock is trading at 100 kr.
● You believe price will increase to 120 kr in 1 month:
○ Buy the stock today, pay 100 kr
○ Wait 1 month
○ If correct you will sell the stock for 120 kr
○ Net: 20 kr profit and no stock !
● You instead believe price will decrease to 80 kr in 1 month
○ How benefit?
○ Short-sell: Borrow the stock from a friend
○ Sellstock in the market, get 100 kr
○ Wait 1 month
○ If correct buy back stock for 80 and return the stock to your friend
○ Net: 20 kr profit and no stock !
○ You need to post margin with the broker to cover potential losses (if you are
wrong and market goes up).
Question: Do you need to know friends who can lend you the sock?
Answer: No use lending institutes (broker), because they have long position stocks that will
be there for a long time
A. Introduction
● What is an investment company?
● What functions does it perform for investors?
● Investors wealth in the investment company
Investment companies
● Investment companies are financial intermediaries that collect funds from individual
investors and invest those funds in a wide range of assets.
● Investment companies enables small investors to team-up and pool their assets
Functions
1. Record keeping and administration
2. Diversification and divisibility
3. Cost-sharing
a. Professional management
b. Lower transaction cost
Investors wealth in relation to the investment company
● An individual investor buy shares in an investment company.
● The value of each share is called net asset value (NAV)
Net asset value = Market value - Liabilities / Shares outstanding
● Question: Mutual fund has a portfolio of securities worth $120 million. The fund
owes $20 million for rent, wages, and other costs. The fund has 5 million shares
outstanding. What is the NAV? Answer: 20 miljoner
B. Types of Investment companies
● Unmanaged Investment companies (Unit trusts )
● Managed Investment Companies
● Other Investment Organizations
Unmanaged investment companies - Unit trusts
● Pool money and invest in a fixed portfolio
○ static buy and hold portfolio (unmanaged)
● Often investments in uniform assets
○ Mortgage bonds only
○ BBB or better corporate bonds
● Investors that want to liquidate their holdings can sell shares back the trust
certificates. (open end)
● Not so common anymore
Managed Investment companies
Pool money and continuously buy and sell securities
● Open-end funds
○ When investors invest, fund issues new shares(no. of shares change)
○ Share are cashable when investors want get out(no. of shares change)
○ Shares are priced at NAV
● Closed-End funds
○ Do not issue or cash-in shares after the IPO (no. of shares constant)
○ Is traded on an exchange just like a stock.
○ Their price can differ from NAV
Other investment organizations
● Hedge funds
○ Similar of open-end fund, but often with
○ Lock-up periods
○ Minimum investment amounts, ex 1 million
○ Minimal regulation (derivatives allowed)
● Real Estate Investments Trusts (REITS)
○ Similar to closed-end funds
○ Invest in real estate (Equity trust) or loans (Mortgage trusts)
● Commingled funds
○ Partnership of investors
○ Similar to open-end funds (units instead of shares)
C. Mutual Funds
● Introduction
● Investment styles and policies
● Investment costs
● Performance
● Sources of information
Mutual funds: Introduction
● Common name for open-end investment companies
● Most common type of investment company
● Each mutual fund has a specified investment style
● The management company typically manage a collection of various funds
Investment styles
● Money market, fund will only invest in short
● Equity
● Sector
● Bond
● International
● Balanced funds (life-cycle). Several are funds of funds
● Asset allocation
● Index
Investment costs & mutual fund fees
1. Operating Expenses
- Costs for operating the fund incl administrative expenses and advisory fees to
the investment manager
- Usually 0.2%-2% of total assets under management
2. Front-end load
- Commission or sales charge paid when you buy the shares
3. Back-end load
- Exit fee
4. 12-b1
- Named after the SEC rule that permits them
- Use fund assets to pay for distribution costs such as advertising and
promotional literature
Performance
● The return of a fund before fees is given by:
● Example:
○ NAV equals 100 in the beginning of the year
○ NAV equals 110 at year end
○ No income and capital gain distributions
○ Return = (110-100)/100=10%
Performance & fees
Example: Operating expenses
● As before NAV0=100 NAV1=110
● If the operating expense is 1% then this amounts to 0.01 x 100=1.0
● NAV1 after operating expenses is 110-1.0=109.0
● The net return after expenses is Return= (109-100)/100=9%
● Fees eat up return!
Example: Front-end load
● If instead a front-end fee of 5% is charged then of $100 paid only 95 is actually
invested
● As before NAV0=100 NAV1=110 so return before fees are 10%
● Now you only get 10% on the 95 invested, i.e. the net return is only $ 9.5
● The net return after expenses is Return= 9.5/100=9.5%
● Again Fees eat up return!
Example: Back-end load
● If instead a back-end load fee that starts at 5% and is reduced by 1% for each year
(until year five) is charged then
● If a $100 is invested, the rate of return is 10% per year and the money is withdrawn
○ after one year, net return is 100(1+0.10)(1-0.05)=104.50
○ after two years, net return is 100(1+0.10)²(1-0.04)=116.16
● Again Fees eat up return!
Performance & fees
● In practice there can be a combination of the various fee structures.
● Which fee structures should I choose?
○ All else equal, small fees eat up less
○ If not all else equal, it depends on you investment horizon
○ If not all else equal, it depends on how much more value an actively managed
high fee fund can create
ETF (Exchange traded funds)
● “Fairly new” products: exist since 1993
○ Rapid growth
● In the beginning: passive management index tracking
○ Spider (S&P 500), Diamond (Dow jones), Webs
● Now: also actively managed
○ Levered (Om index går upp går den upp dubbelt går den ner får den dubbelt
ner) ETFs, Inverse ETF (Går i motsatt håll som index, bra för riskspridning)
● Hybrid of open-end fund: NAV calculated but traded whole day at the exchange like
stocks (not only once a day).
○ Price can depart from NAV
● Can be sold short or purchased on margin.
● Low cost in particular on passive ETFs
● Must be traded from a broker
Information
● Details of a funds investment strategies, advisors and fees can be found the prospectus
● Additional information is found in the funds annual report
● Information about available funds and guides on how to choose funds are available on
many places online including
○ Morningstar
○ Pensionsmyndigheten
Föreläsning 3 (Risk & Return in Financial Markets)
A. Interest rates & inflation
● Price of money
● Real and nominal interest rates
● Inflation
● Fisher Equation
Example
● You put 1000 SEK in the bank 1 year ago. Grown to 1100 SEK today.
● How much richer did you get?
○ It depends if you measure richness in nominal terms or in real terms
(purchasing power)!
● In nominal terms: increase is 100 SEK or: (1 100)/(1 000)−1=10%
● How much purchasing power did you get (in terms of carrots)
○ When you started a year ago, a carrot cost 1 SEK
○ Today it costs 1.06 SEK (i.e. inflation is 6%)
○ So you can afford to buy 1100/1.06=1038 carrots
○ In real terms you earned 3.8% (the real increase is only 3.8%)
● In this example nominal rate is 10%, the real rate is 3.8% and inflation is 6% and they
relate as
Real, nominal and inflation rates
More generally the real and nominal rates are linked by the inflation rate:
Can also stated
as:
If i is small; can be approximated with:
Remark: sometimes this is called the fisher equation, but more often as in the book.
The Fisher equation:
(used when looking forward)
Inflation measures
● Most common: Consumer price index (CPI) :basket of goods and services
● Variations of CPI excluding certain parts of economy
○ Riksbankens Inflation target (2%) is measured using KPIF
○ KPIF Fixed interest rate
● Updated numbers…
Interest: price of money
● What determines the level of the (real) interest rate?
○ Supply of money (households)
○ Demand for money (companies)
○ Governments net demand (central bank actions)
Tax-effects
● Taxes are paid on nominal interest rate earnings
● Thus you pay tax also on the part of the interest rate which is just an inflation
compensation (inflation penalty)
● Alternative tax-system is an inflation protected version where you only pay tax on the
real rates
B. Return and standard deviation
● Holding period return
● Effective Annual Rate (EAR)
● Annual Percentage rates (APR)
● Continuously compounded rates
● Expected returns & standard deviation
○ Scenarios, Arithmetic & Geometric
● Risk & return relations
Holding-period return
● Your realized return called the holding-period return from a (stock/bond) investment
over period [0,T] will depend on:
1. The price per share in the beginning and in the end of the investment period,
P0 and PT
2. The income payments (dividends/coupons)
● The holding-period return is calculated as:
Example
Question: What is the holding period return of investment A and B?
Hint: Recall that
Answer for A: (220-216+3)/216= 3.24%
Answer for B: (100-98+10)/98=12.24%
Question: Which investment is better?
Q: How compare investments with different horizon? We have to calculate APR!!!
Annual percentage rates (APR)
● For short investments horizons, up to 1 year we often use
● The annual percentage rate
● Return to our example:
● Question: What is the annual percentage rate for investment A and B?
● Answer: A) 3.24/0.25=12.96%
B) 12.24/0.5=24.49%
Effective Annual Rates (EAR)
● For long investments horizons, over 1 year we often use
● The effective annual rate is a compounded rate, calculated as the EAR that solves
● Example: Calculate the effective annual rate for
A) A 1 year investment with holding period return of 4.69%
B) A 25 year investment with holding period return of 329.18%
● Solution:
A) EAR=4.69%
B) EAR=(1+3.2918)(1/25)-1=6%
Continuous compounding frequency
● In practice no investments are continuously compounded
● However many textbook formulas (in particular for derivatives) are using
continuously compounded rates
● So what is it?
● An investment of 1 kr made over 1 year with a period interest rate of 5.8% will give
us 1.058 after 1 year. The 1 times compounded interest rate is 5.8%
● An investment of 1 kr made over 6 months with a period interest rate of 5.8/2=2.9%
will if we can repeat this investment 1 more time give us (1+0.029)(1+0.029) =
1.0588 after 1 year. The 2 times compounded interest rate is 5.88%
● …
● An investment of 1 kr over 3 months with a period interest rate of 5.8/4=1.45% will if
we can repeat this investment 3 more times give us(1+0.0145) (1+0.0145) (1+0.0145)
(1+0.0145)=1.0593 after 1 year. The 4 times compounded interest rate is 5.93%
● An investment of 1 kr made over 1 month with a period interest rate of
5.8/12=0.483% will if we can repeat this investment 11 more times give us
(1+0.00483)¹²=1.0596 after 1 year. The 12 times compounded interest rate is 5.96%
● An investment of 1 krover a “nanosecond” that can be repeated over and over again
until a year has passed will give us (1+0.058/m)m where m goes to
infinity=e0.058=1.05971 after 1 year. The continuously compounded interest rate is
5.97%
Expected return & variance
● Investors aim at forecast return and risk (standard deviation)
● Methods for forecasting:
○ Scenario analysis
○ Assume history repeat
● Recall from statistics (p(s) probability of a state and r(s) the return of that state)
Time Series of Past Rate of Returns
● In historical data we treat each observation as an equally likely scenario.
● If n observations, probability is 1/n for each observation
● Expected return is estimated as
● Similarly we want to estimate variance as
● When E[r] is estimated, an unbiased estimator of the variance is the sample-variance:
Another measure of average return
● Lets look at an example of past returns of investment over 2001-2005
● $1 invested in the beginning of 2001 gives $1.0275 but holding period return vary
every year.
● What would be the fixed annual holding period return compounded annually that
gives the same terminal value?
●
● Note the geometric average is smaller than the arithmetic average 2.1% (always true if
normaldist)
Relationship between risk and return
● The return on very safe investments, like Treasury bills are considered as virtually
risk free
● Risk averse investors want to get compensated for taking on additional risk
● To invest in a stock that has higher risk than T-bills, risk-averse investors want higher
return, they want positive excess return: Excess return = Return – Risk Free Rate
● Excess return is not known a priori. Its expected value is called the risk-premium:
Risk premium = Expected return – Risk Free Rate
Sharpe ratio
● How do we compare investment opportunities?
● What is a good investment?
○ Care about a return
○ Care about risk
○ Idea: calculate the reward-to-risk measure!
● The most well-known reward-to risk measure: the Sharpe ratio
Sharpe ratio-example
● Consider an investment with an expected return of 12% per annum and a standard
deviation of 20%. The risk free rate is 1%.
● Calculate the Sharpe ratio
● The Sharpe ratio is: (0.12-0.01) / 0.2 = 0.55
C. Normal distribution and return data
● Normal distribution
● Stock return
The Normal distribution
● Many empirical applications including finance is making use of the normal
distribution
○ Finance assumption: risky asset returns are normally distributed
○ In many cases it is a decent approximation
● The bell shaped density:
○ Illustrate probabilities (comp. histogram)
○ Symmetric around the mean
The Normal distribution properties
Normal distribution has many nice properties and is therefore easy and popular to work with:
● The whole distribution is completely characterized by only 2 measures, the mean and
the standard deviation
○ In a normal distribution, risk can be measured by standard deviation
● normal+normal=normal
● Multivariate normal-> marginal distribution is normal
○ Correlationcoefficient suffice to describe dependence
What about reality-are stock returns normally dist?
Standardized daily Volvo-returns 2018.05-2019.11
● Sort of bell-shaped!
What about reality-comparison to normaldist
Are stock returns normally distributed?
● Big picture: decent approximation
● Closer look: fatter tails, not symmetric
Fat tail (Kurtosis)
If real data have fatter tails than the normal distribution:
● Extreme events are more likely
● Larger risk of extreme loss (and profit) compared to what the normal distribution
suggest
Not symmetric (Skew)
If real data is negatively skewed
● Standard deviation is not sufficient risk measure (stdev will underestimate risk)
● Larger risk of extreme losses than what the normal distribution suggest
D. Other measures
● Lower partial standard deviation
● Sortino ratio
● Value at Risk
● Expected Shortfall
Lower partial standard deviation (LPSD)
● The variation of returns measured by variance and standard deviation comes from
both positive return (profits) and negative returns (loss)
● Investors care about loss but do not mind profits
● Idea: measure only the variation due to
negative return
● Lower Partial Standard Deviation:
○ Standard deviation only considering negative deviations (downside)
○ Typically calculating the deviations as deviations from the risk-free return
rather than from the mean
Sortino ratio
Sharpe ratio with lower partial standard deviation
Value at Risk (VaR)
● Simple risk measure:
○ easy to interpret,
○ easy to calculate (given strong assumptions),
○ too easy... complement.
● Standard measure used by banks and investment managers.
● Standard measure used by regulatory agencies that regulates banks and investment
funds. (Basel)
● Another name for the quantile of a distribution.
● A short name for Value-at-risk is VaR
Value at Risk (VaR)
If a portfolio's 1 day Value-at-Risk at 95% confidence level is 2 million kr, then the
probability that the portfolio will lose 2 million kr or more over the next day is 5% or less.
Value at Risk (VaR)
Two ways to calculate VaR
1. Empirical approach (assume history repeat)
Rough idea:
● Look at past returns (example 100 observations)
● Sort return with worst first (i.e. negative return)
● The 95% VaR is the 5th worst return (in SEK)
2. Model approach (assume specific model)
If normal distribution is assumed:
● 99%-VaR= meanreturn-2.33 * StDev
● 95%-VaR= meanreturn-1.645 * StDev
Question:
● If we use the model approach (assuming returns are normally distributed) when
calculating VaR
● If actual returns have fatter tails than the normal distribution
● Will we over or under estimate VaR? We will underestimate!
Expected shortfall (ES)
● Expected shortfall is the expected value of the loss, given that that the value-at-risk
will be exceeded
● Another name is conditional tail expectation
Föreläsning 4 (Capital allocation)
A. Investor preferences
● Introduction
● Risk-aversion
● Utility
● Indifference curve
Introduction
● Risk Averse Investor
Requires an expected return higher than the risk free rate to take on risk
● Risk Neutral Investor
Requires an expected return equal to the risk free rate to take on risk
● Risk Loving Investor
Requires an expected return lower than the risk free rate to take on risk
● Risk Premium
Expected return-risk free rate
Question
What expected rate of return do you require on H&M stock-more or less than on Treasury
bills? A majority of people would want a higher ror.
Speculation & gamble
● Gamble: take on risk for the fun of it
● Speculation: take on risk to make more money than the risk free rate
● Fair game: Risky investment with risk premium of zero (gamble)
● Risk-averse investors will reject gambles but not all speculative positions
Utility (nytta)
● Risk-averse investors considers only risk-free or speculative prospects with positive
risk premiums
● Risk-averse investors “penalizes” the expected return of a risky portfolio to account
for the risk involved.
● The greater the risk, the larger the penalty.
● The more risk-averse the investor is the larger is the penalty
● The mean-variance utility (Markowitz): U=E(r) - ½ Aơ²
○ maximize the utility!
Utility
● Utility is the satisfaction from wealth (the goods and services that can be bought for
the money)
● Risk-averse investors prefer more compared to less
● Risk-averse investors prefer more at a decreasing speed (decreasing marginal utility)
● In this course we use mean variance utility
Other include log-utility, exponential utility
Risk aversion
● Mean-variance utility: U=E(r) - ½ Aơ²
● A is the risk-aversion parameter
○ Risk-averse investors: A>0 (textbook examples include A=4)
○ Risk-neutral investors: A=0
○ Risk-lover investors: A<0
● How do we quantify the risk-aversion parameter A?
○ Difficult: quiz can guide you
○ Depends on:
■ How much investment risk can you afford?
■ How much risk can you tolerate-sleep good at night?
Example & Poll Question
A risk-averse investor with mean-variance utility and risk aversion coefficient of A=2 is
facing three investment opportunities. Which one will she choose?
U=E(r) - ½ Aơ²
L: U=0.07-0.5 x 2 x 0.05² = 0.0675
M: U=0.09-0.5 x 2 x 0.1²= 0.08
H: U=0.13-0.5 x 2 x 0.2²= 0.09 (this is the investment a high risk taker will choose (A=2))
Mean-variance criterion
● Portfolio A dominates portfolio B if
E(rA) > E(rB) and 𝜎A < 𝜎B
● Portfolio A dominates portfolio B if
E(rA) > E(rB) and 𝜎A < 𝜎B
Indifference curve
● An investor like Portfolio Q and Portfolio
P equally much.
● All points on the blue line are portfolios
with the same utility score
U=E(r) - ½ Aơ²
● The investor is equally happy with all
these portfolios (indifferent)
● The curve is called the indifference curve
Poll: Indifference curve
● Indifference curve of two investors with
blue and curves respectively.
● Both have utility function
U=E(r) - ½ Aơ²
● But they have different aversion to risk:
○ One is more risk averse and has
thus a higher A
● Question (Poll) : Who is more risk averse?
○ black or blue. Answer: blu
Indifference curve
● The more risk averse investor require a
greater increase in return to compensate
for more risk.
● The more risk averse, the steeper is the
indifference curve
B. Possible risk and return levels
● The risk free asset (ex government paper, treasury bill)
● The risky portfolio
● The complete portfolio
○ The return of the complete portfolio
○ The risk of the complete portfolio
○ The capital allocation line (CAL)
The risk-free asset
● In practice there actually is not a risk-free asset
● Treasure bills are regarded as essentially risk-free assets
● Money-market funds hold a lot of Treasury bills and can be used as a proxy for
risk-free assets
The risky portfolio
● The risky portfolio consists of all the risky investments
● At this point we assume that the investor has already decided on the composition of
the risky portfolio P
● The focus is instead on capital allocation between the risky portfolio P and the risk
free asset.
Capital allocation possibilities
● The capital allocation is the proportion (y) of the investment budget that is allocated
to the risky portfolio P.
● The remaining proportion (1-y) is invested in the risk-free asset.
● We call the portfolio of y risky assets and 1-y risk-free assets the complete portfolio
Return of the complete portfolio
The return on the complete portfolio C:
E[rC]
= y E[rP] + (1-y) rf
= y 0.15+(1-y)0.07
= y 0.15 +0.07 – y 0.07
= 0.07+y(0.15-0.07)
Expected return = risk free rate + riskpremium * proportion invested in P
Risk of the complete portfolio
The return on the complete portfolio C:
Var[rC]
= y² Var[rP] + (1-y)² Var(rf)
= y² 0.22²+(1-y)0
= 0.22² y²
ơC= 0.22 y
St.Dev(C )= St.Dev(P) * proportion invested in P
Risk and return relation of the complete portfolio = CAL
(p, rf uncorrelated)
Risk and return if borrowing rate is not the risk-free rate
C. Maximize investor utility
● Where on the CAL should you be?
● Optimal proportion of risky assets
● Graphical solution in the return-risk diagram
● The Normality assumption
● The Capital Market Line (CML)
Where on the CAL should you be?
● I.e. which combination of risk & return should you pick?
○ i.e. what proportions should be invested in risky and risk-free assets?
● Pick the combination that maximizes your utility!
○ Markowitz: maximize U= E[rC] -½ A ơC² = rf+y(E[rP]-rf) -½ Ay²ơ P²
○ Slide 20-21: E[rC]= rf+y(E[rP]-rf) and ơC=yơP
Optimal proportion of risky assets (y*)
Graphical solution…
● Recall the indifference curve:
○ Portfolio Q is equally preferred to the investor as portfolio P.
○ All points on the blue line are portfolios with the same utility score
○ The investor is equally happy with all this portfolios (indifferent)
○ The curve is called the indifference curve
…Graphical solution
● Now combine the investor preferences (indifference curve) with the capital allocation
possibilities (CAL)
Graphical solution with indifference curves
● The more risk averse, the steeper the indifference curve, the less the proportion of
risky assets (the optimal complete portfolio is more to the left on the CAL).
Normal distributed return
● The Markowitz optimal capital allocation proposed here is based on the assumption
that returns are normally distributed.
● Empirical evidence suggests that returns are not exactly normally distributed.
● Many times normal returns have fatter tails (extreme events more likely).
● Extensions of this framework involves other risk measures than standard deviations
including VaRand Expected Shortfall
Passive strategy and the CML
● To draw the Capital allocation line (CAL) we needed the risky portfolio P
○ So far we have assumed the investor have already decided about the
composition of the risky portfolio P (slide 18).
● Which assets to include into portfolio P may result from a passive or an
activestrategy.
● Natural candidate for a passive strategy is a well diversified portfolio such as a
marketindex of common stocks.
● When the strategy is passive the capital allocation line has a special name: the capital
market line (CML)
○ CML is created from a risk free asset (such as a money market fund) and a
fund of common stocks that mimics a broad market index.
Föreläsning 5 (Optimal Risky Portfolios & Capital Allocation)
A. The opportunity set of risky assets
● Introduction
● Two risky assets
● Many risky assets
Introduction
Question: What sources of risk would affect an investor holding H&M stocks?
Possible answers:
● Inflation
● Metal prices
● Technical innovations
● Business efficiency
● Management skills
● ....
Diversification
● If you invest in both Volvo and in H&M,
○ A price shock in metal prices does not affect Volvo and H&M the same way,
○ Thus the portfolio risk is reduced as compared to holding only Volvo stocks
● If you invest in more stocks you will reduce risks even further
● The effect of adding more and more stock will have a smaller
and smaller effect on risk reduction
● When common sources of risk affect all firms even extensive
diversification cannot eliminate all risk
○ The risk that remains is called market risk, or systematic
risk or non-diversifiable risk
○ The risk that can be eliminated by diversification is
called
unique risk, or firm specific risk or non systematic risk
or diversifiable risk
Diversification
● How can we diversify efficiently?
● We want to construct portfolios of risky assets that provide the lowest possible risk for
any level of expected return.
● We start by looking at a portfolio of 2 assets (equity & bond)
● Let us look at what portfolios are possible to form and what is the return-risk profile
(opportunity set) ?
Example: One portfolio of 2 risky assets
● Buy a proportion of WE=40% of the equity fund
○ Expected return E[rE]=13%, standard deviations σE=20%
● Buy WD=60% of the bond fund. Note (WE+WD=1) bond fund
○ Expected return [rD]=8%, standard deviations σD=12%
● What is the risk and return on your portfolio of the 2 risky assets?
○ Need to know covariation or correlation coefficient since normaldist.
cov(rE,rD)=σEσDρ
○ Suppose correlation coefficient is ρ=0.3
Question: Another portfolio of 2 risky assets
● Buy a proportion of WE=50% of the equity fund
○ Expected return E[rE]=13%, standard deviations σE=20%
● Buy WD=50% of the bond fund. Note (WE+WD=1) bond fund
○ Expected return [rD]=8%, standard deviations σD=12%
● Question: What is the risk and return on your portfolio of the 2 risky assets?
○ Suppose correlation coefficient is ρ=0.3
● Answer:
Expected return E[rp]=WEE[rE]+WDE[rD]=0.5x0.13+0.5x0.08=10.50%
Variance 𝜎𝑃²=𝑤𝐷²𝜎𝐷²+𝑤𝐸²𝜎𝐸²+2𝑤D𝑤𝐸𝜌𝜎𝐸𝜎𝐸=
0.5x(0.2)²+0.5²x(0.12)²+2x0.5x0.5x0.3x0.2x0.12
Standard deviation 𝜎𝑃=𝜎𝑃¹/²=13.11%
The portfolio opportunity set of 2 risky assets
● We have calculated 2 different combinations of portfolios so far:
● We can calculate all combinations of portfolio ER and StD that can be constructed
from the 2 available assets (by changing the weights WE and WD).
● It is called the portfolio opportunity set
The portfolio opportunity set of 2 risky assets
All possible combinations of expected return and standard deviation
The minimum variance
All possible combinations of expected return and standard deviation
The portfolio opportunity set of 2 risky assets
Question: What if correlation coefficient is not 0.3, is there still a diversification effect?
Answer: Yes whenever correlation coefficient is strictly
smaller than 1
Diversification effects
● If assets are perfectly correlated ρ=1 there is no diversification effect
● If assets are not perfectly correlated there is a diversification effect
● The smaller the correlation coefficient, the larger is the diversification effect.
● If assets are perfectly negatively correlated ρ =-1, there exist a perfect hedge with
minimum variance 0.
Question: What 2 risky assets have correlation coefficient equal to -1.000?
Answer: In practice very few assets
Many risky assets
● So far we have identified the opportunity set of risky assets when we have only 2
risky assets
● Now is time to generalize to the case of several risky assets
● As before form portfolios of the risky assets, but now we have more than 2 assets. The
weights sum up to 1.
● Identify the minimum variance frontier, which are the portfolios with the smallest
possible variance for any given level of portfolios expected return
The portfolio opportunity set for many risky assets
B) Optimal portfolio of risky assets
● Idea
● Graphical solution
● Mathematical solution
Optimal risky portfolios
● So far we have identified the opportunity set , i.e all possible combinations of
portfolio expected return and standard deviation that can be constructed from the 2
risky assets (by changing the weights WE and WD).
● Now given the opportunity set, which risky portfolio is optimal (P) ?
How can we pick the optimal risky portfolio (P) ?
● Idea: Suppose we could also invest into a riskfree asset:
○ Then we know from chapter 6 that by investing a proportion y into the risky
portfolio asset and 1-y into the riskfree asset we will create a capital allocation
line (CAL)
○ Recall the slope for the CAL is the Sharpe ratio (excess return per risk unit)
○ The optimal risky portfolio will be the one that gives us the steepest possible
CAL (highest Sharpe ratio)!
Finding the optimal portfolio of risky assets
● Look at the global minimum variance portfolio A.
● Is portfolio A the optimal risky portfolio?
● A is on the efficient frontier, good. Is it impossible to find a capital allocation line that
is steeper?
○ No it is not impossible since B has a steeper
CAL. Thus A is not the optimal risky
portfolio.
● Question (poll): Is portfolio B the optimal risky
portfolio?
● Answer: No, it is possible to make CAL steeper by
choosing a portfolio on the blue line that lies to
northeast of B.
The optimal portfolio of risky assets
● The optimal risky portfolio (P) has CAL as its
tangent. Den mest optimala är alltid en tangent vilket
innebär att CAL snuddar vid OSRA
● The optimal risky portfolio (P) has the highest
Sharpe ratio (slope) on the efficient frontier
The optimal portfolio of 2 risky assets in mathematical terms
The optimal portfolio of 2 risky assets in mathematical terms
The optimal portfolio of 2 risky assets in mathematical terms
C) The optimal complete portfolio
● Where on the CAL should you be?
● Optimal proportion of risky/risk free assets
● Graphical solution in the return-risk diagram
● The Normality assumption
● The Capital Market line (CML)
Where on the CAL should you be?
● I.e. which combination of risk & return should you pick?
○ i.e. what proportions should be invested in risky and risk-free assets?
● Pick the combination that maximize your utility!
○ Markowitz: maximize U=E(rc) -½ Aơ c²= rf+y(E[rP]-rf) -½ Ay²ơ²
● This is exactly what we looked at in chapter 6.
● The solution is the optimal complete portfolio
Optimal complete portfolio
● The optimal proportion of assets invested in to
the risky portfolio P is y*.
● Is determined graphically by picking the point C
at which the CAL(P) is tangent to the indifference
curve.
The 2 fund separation property
You can find the optimal complete portfolio by following 2
separate steps:
1. Find the optimal risky portfolio P
● The point on the efficient frontier that is tangent to CAL
● Remark: Does not depend on preferences
2. Find the complete portfolio C
● The point on the indifference curve that is tangent to CAL
● Remark: Dependson investor preferences
Föreläsning 6 (Asset Pricing Models)
A) Introduction and the single-index model
● The input list of Markowitz model
● The single factor model
● The single index model
Large input list
● When applying Markowitz model to portfolio optimization we specify an input-list
● If we want to consider a market of 30 stocks in the model, we need to feed the model
with the following input:
○ 30 expected values
○ 30 variances
○ (30 x 30 -30) /2 =435 covariance's
○ TOTAL: 495 inputs
● Nobody know the true values of these inputs so they need to be estimated!
● Clearly difficult to get an accurate model!
● The problem get worse:
○ With 500 stocks you need to estimate 125 750 inputs
○ With 3000 stocks you need to estimate approx. 4.5 million inputs
○ With n stocks you need to estimate (n*n -n)/2 + 2n
Markowitz model in practice
Two practical problems with Markowitz portfolio selection model:
1. Depends on a very large number of input parameters
2. May to introduce inconsistency errors in variances
● Easy to arrive at estimates which imply negative variances
● Such model is of course completely unreasonable
IDEA: Look for a simpler model
● Easier to apply in practice
● Fewer parameters
● Simpler risk structure
Single factor model
● Return data show: many assets are positively correlated. Common risk source?
● Idea: Assume uncertainty in stock return is due to;
1. Shocks in macroeconomic factors that affect all stock prices
2. Firm-specific uncertainty
● The variance:
From single to index model
● Idea: Use a market index as a proxy for common factor m. I.e. choose m=rm in the
single factor model
● Subtract risk free rate on both sides to get excess returns
● Remark: This is a regression equation (excess return of
asset i is regressed on the excess return on market index)
The index model
● The index model is a linear regression equation of an assets excess return on the
market excess return
● Taking expectations we obtain
● Interpretation: an assets risk premium consists of 2 parts;
○ the non market premium(alfa) and the systematic premium.
● The variance and covariance of 𝑟𝑖−𝑟𝑓 and 𝑟𝑗−𝑟𝑓 are:
●
● Remark 2: if high beta, market risk has large impact on stock variance.
● Remark 1: Fewer parameters to estimate!
Remark 1: Input list Markowitz vs Index model
● No of parameters (inputs)
○ Markowitz: (n*n -n)/2 + 2n
○ Index model: 3n+2
Estimation is much easier in the index model!
Remark 2: Conclusions on alpha and beta?
The index model in expectation:
● Beta is a sensitivity parameter-the higher the beta the higher is the stocks
risk-premium in relation to the market risk premium.
○ Makes sense since we saw high beta results in higher risk via
● Alpha is non-market risk premium
○ It will be large if you think stock is underpriced
What about the no-free lunch?
● If you estimate that a stocks alpha is positive
● It means you think the stock is underpriced(a good deal)
● What Will You Do?
○ Buy the stock
● If efficient markets and investors have the same information..
● All investors will find alpha positive and so all investors want to buy the stock…
● Question: What will happen to stock price? What will happen to alpha?
● Answer: price increase, alpha decrease
● Question: What if alpha is negative?
● Answer: price decrease, alpha increase
Equilibrium idea:
● It is reasonable to believe that in equilibrium, there is no free lunch and so alpha will
be (close to) zero. That is
● Remark:This is the very same equation as the well known CAPM equation that is
formally derived by Sharpe, Lintner and Mossinin a different way.
The index model & Diversification
● The risk of a portfolio of many assets
in the index model will decrease when
the number of assets increase.
● However some risk will stay: the
systematic risk
B) CAPM
● Assumptions
● Capital Allocation Line (CAL) & Capital Market Line (CML)
● Market Price of Risk
● CAPM equation
● Security Market Line (SML)
● Extensions
CAPM Assumptions
● Individuals
1. Rational mean-variance optimizers
2. Homogeneous expectations (use same input lists)
3. Same planning horizon
● Markets
1. All assets are publicly held
2. All information is available
3. No taxes
4. No transaction costs
Under theses assumptions…
● Every investor
○ optimize risk-return a la Markowitsand
○ is using the same input lists in their estimations
○ is facing the same investment opportunities
● Then all investors will arrive at
○ the same efficient frontier
○ and the same risk-free rate and
○ the same optimal risky portfolio (P)
● So all investors will face the same tangent Capital Allocation Line (CAL)
Capital Market Line
● Since every investor will face the same risky portfolio P
○ Every investor will choose identical weights for
each risky asset.
○ What more can be said about these weights?
● The market portfolio is the aggregation of all risky portfolios.
○ Since all risky portfolios have identical weights also
the weights in the aggregated market portfolio must
be the same weights
● So if all risky investors pick the same risky portfolio this is
the market portfolio, so
○ proportion of each security is its market value as a
percentage of total market value
○ Capital Asset Line = Capital Market Line
The mutual fund theorem
● Under the CAPM assumptions,
○ all investors know that everyone will face the same optimal portfolio of risky
assets; the market portfolio
○ No need to perform security analysis, just need to look at the market weights
and voila this is the optimal risky portfolio.
○ In fact investors would not mind if they could not invest in individual
securities but only could invest in a mutual fund providing the risk and return
of the market portfolio. This is the mutual fund theorem.
Remark 1: If everyone is just passive, not performing analysis, free lunches
may start to appear and it may payoff to do analysis.. More on efficiency later
Remark 2: In reality all assumptions do not hold, but market portfolio seems
like a reasonable first start
The risk premium on the market is proportional to its risk
● Recall from ch6 that the investor with risk aversion A will have the optimal
proportion of risky assets equal to
● In the simplest CAPM economy risk free investments involve borrowing and lending
among investors and thus across all investors borrowing and lending sum to zero, so
The reward to risk ratio in equilibrium…
Measuring risk as the covariance to the market portfolio:
● The market portfolios reward-to-risk ratio is:
○ Also known as the market price of risk
● For risky asset i the reward-to risk ratio is:
● In equilibrium the risk-reward ratios (slopes) should be equal…
... Capital Asset Pricing Model (CAPM)
In equilibrium the reward-to-risk ratio is equal:
Multiply both sides with 𝐶𝑜𝑣(𝑟𝑀,𝑟𝑖):
Identify market beta:
Equal reward to risk
● We assumed reward-to-risk ratio is equal in equilibrium.
● How can we see that this must be the case?
● Assume one asset has higher reward to risk ratio.
● This is attractive to investors so demand for this asset increase.
● That will push demand up for this asset
● So price will increase for this asset.
● In other words the return will decrease.
● So the reward-to-risk ratio will also decrease…
● ...until it is equal to the reward-to-risk ratio of the other assets!
CAPM Application
● Question: The risk-free rate is 6% and the expected market rate of return is 12%.
Calculate the expected rate of return according to the CAPM model of the security
XYZ if the security has a beta of 1.5.
● Answer: E(R) = 6 + 1.5(12 -6) = 15%.
A close up on beta in CAPM
● The beta of asset i in CAPM is
● Question: What is the beta of the market portfolio?
● Answer: β𝑀=1
● Stocks with
beta>1 aggressive (high sensitivitet to market swings)
beta<1 defensive (low sensitivitet to market swings)
The security market line
● The security market line (SML) is the relationship
between a securities return and beta.
● This is a linear function according to CAPM:
𝐸(𝑟𝑖)=𝑟𝑓+β(𝐸(𝑟𝑀)-𝑟𝑓)
● In equilibrium there is no free lunch and all assets
return-beta should be fair and lie on the SML
The security market line (SML)
● Off equilibrium a stocks return-beta can plot off the line.
● A stock is a good buy if it plots above the SML since it
will provide a return in excess of the fair return.
○ We say it has positive alpha. Recall the index model
Limited dependence structure
● The index model and CAPM benefit: simple models easy to estimate
● Do these models also have limitations?
● Yes, we model the depends structure between all securities only via their dependence
on the one common macroeconomic/systematic factor:
○ Two stocks in different industries may behave rather different than two stock
in the same industry: Ford vs Toyota as compared to Ford vs Apple
● Is one macroeconomic factor enough to describe economy?
○ Most people agree this is too simplistic
○ More realistic to have multiple factors (Two: inflation, real interstrate)
○ How many factors, 2, 3, 4, 5,...
○ In many cases 3 is enough (Mathematical fact)
Multifactor models in general
● The CAPM/index model are single factor models
● Rewrite to simplify notation𝑙𝑒𝑡 𝑅𝑖=𝑟𝑖−𝑟𝑓
A famous multifactor model
Fama & French three factor model:
Factors in addition to CAPM:
● SMB: Small-Big (return on portf of small stocks in excess of return on portf of large
stocks)
● HML: (return on portf of stocks with high book-to-market ratio in excess of return on
portf of stocks with low book-to-market ratios)
These two additional factors was chosen because of empirical observations suggest the
relationship: high factors high return. Factors are thought to be proxies of macroeconomic
variables.
Föreläsning 7 (Swedbank Robur, ESG)
The Alphabet soup of Sustainability (A practitioners observations from the handicraft hall)
Development of sustainability
2000
- UN Global Compact
- UN GRI
- CDP
2005
-
UN PRI
2015
-
Paris Agreement COP21
SDGs - 17 goals for 2030
SBT - GHG reduction
The two most highlighted ones are PRI & SDGs
Evolution of sustainability
Klipp in bild
My building blocks for fundamental portfolio management
Philosophy
- Long term returns are mainly derived from a company’s capabilities to create cash
flows and earnings
- “The stock market” is a complex adaptive system
- Think about sustainability as “new analytic tools”
Process - Long term and Thematic
Screening→Thematic approach →Fundamental analysis →Portfolio Construction
→Portfolio maintenance
Thematic approach
- Climate
- Demographics
- Health and wellness
- Digitalisation
Building blocks for fundamental analytics
- 33% Growth/SDG
- 33% Quality/ESG
- 33% Valuation
Föreläsning 8 (Efficient Market Hypothesis and Behavioral Finance)
Efficient Market Hypothesis
Example. Merck announces a new allergy drug to prevent hay-fever. How should Merck's
share price react to this news? There are 3 scenarios:
● Increase Immediately to a new equilibrium level, om hypotesen stämmer och
marknaden kan bearbeta informationen.
● Increase Gradually To The New Equilibrium Level
● Firstover-shoot and then settle back to new equilibrium level
What do you think?
Efficient Market Hypothesis
Efficient Market Hypothesis: Market prices of securities reflect all available information
about their value. This means that the market is efficient.
A more precise definition of EMH needs to answer two questions:
1. What Is``all available information''?
2. What Does It Mean To``reflect available information''?
Answer:
1. All Available Information Includes:
- Pastprices - Weak form (easy to get)
- Public Information (prices,news,...) - Semi-Strong Form (able to get, harder)
- All Information Including Inside Information- Strong Form (hard to get)
2. ``Prices reflect all available information'' means that financial transactions at market
prices, using the available information, are zero NPV activities
Implications Of Market Efficiency:
- No free lunch (no arbitrage) in financial markets
- Prices fully reflect all available information
- Prices follow random walks
- Trade-off between risk and expected return
- “Active” asset management does not add value
Empirical tests of EMH
1.Weak form of EMH is supported by the data.
- Technical trading rules are not consistently profitable.
-
Serial correlation in daily stock returns is close to zero
Example. Trading can be hazardous to your wealth
Example. Gender Issues in finance
Män är mer självsäkra än kvinnor och därför är sannolikheten större att de tar mer risker.
Turnover is the total sales made by a business in a certain period. It's sometimes referred to as
'gross revenue' or 'income'. This is different to profit, which is a measure of earnings
2. Semi-strong form of EMH is generally supported by the data.
- Prices react to news quickly (corporate actions, accounting changes...
Cumulative Abnormal returns (CAR) before and after Dividend announcements
Hur marknaden reagerar efter en dividend increase samt decrease. CAR är returnen man vill
räkna ut kopplat till nyheterna som kommit ut.
2 Empirical Tests of EMH
Här har priset höjts innan informationen kommit ut vilket indikerar att man förmodligen fick
veta om infon innan.
3. Strong-form of EMH has mixed evidence:
- Money managers cannot consistently outperform.
Mutual Fund Performance (Gross of Expense)
Performance of Average Equity Mutual Funds
Average presterar sämre än indexet
Inside-trading is not profitable – or is it?
Ambiguous evidence
2. Smooth dividends but volatile prices (Shiller)
Real S&P Index p versus Ex Post Rational Price p*(1871-1979)
Questions about EMH
1. How does information get into prices?
2. If prices reflect all available information,who has the incentive to collect costly
information?
3. How about anomalies, crashes, crises?
Practical Issues about EMH
4. Transactions costs
5. Regulatory restrictions
6. Missing risk factors
7. Liquidity
8. Taxes
9. Micro vs. macro efficiency…
Lessons from EMH
1. Trustmarketprices.
● Buying and selling assets are zero NPV activities.
● Market prices give best estimate of value for projects.
● Firms receive “fair” value for securities they issue.
2. Read into prices.
● If market price reflects all available information, we can extract information
from prices.
3. There are no financial illusions.
● If Market price reflects value only from an asset's payoff.
● It is not easy to trick the market.
4. Value comes from economic rents such as superior information, superior technology,
access to cheap resources..
EMH makes two important predictions
1. Security prices property reflect whatever information is available to investors
2. Active traders will find it difficult to outperform passive strategies such as holding
market indexes
Tests of market efficiency have focused on the performance of active trading
strategiesBehavioral finance assumes investors are not rational
Behavioral finance
The behavioral critique
Two broad categories of irrationalities
1. Investors do not always process information correctly and therefore infer incorrect
probability distributions of future returns
2. Even when given a probability distribution of returns, investors may make
inconsistent or suboptimal decisions
Information processing
Limited attention, underreaction, and overreaction
● Reliance on heuristics due to limited time/attention
Overconfidence
● People tent to overestimate the precision of their beliefs or forecasts, and they tend to
overestimate their abilities
Extrapolation and pattern recognition
● Representativeness bias
○ Individuals are adept at discerning patterns, even perceiving patterns that may
be illusory
○ Overly prone to believe these patterns are likely to persist
Behavioral Biases
Framing
● Decisions affected by how choices are described, such as whether uncertainty is posed
as potential gains from a low baseline levels, or as losses from a higher baseline value
Mental accounting
● Specific form of framing in which people segregate certain decisions
Regret avoidance
● Individuals who make decisions that turn out badly have more regret when that
decision was more unconventional
Affect and feelings
● Investors tend to choose stocks with high affect, driving up prices while
simultaneously driving down returns. Ex att man väljer H&M och Volvo när man har
tillgång till aktier över hela världen.
Behavioral Biases: Prospect Theory
Conventional view: Higher wealth provides higher utility, but at a diminishing rate.
Behavioral view: Utility depends on changes in wealth from current levels, not the level of
wealth
Risk vs Uncertainty
Urn A Contains 100 Balls:
● 50 Red, 50 Black
● Pick a color, then draw a ball
● If you draw your color,$10,000 prize
● What color would you prefer?
● How much would you pay to play this game?
Urn B Contains 100 Balls:
● Proportion unknown......
● Pick a color, then draw a ball
● If you draw your color,$10,000 prize
● What color would you prefer?
● How much would you pay to play this game?
Limits to arbitrage
Behavioral biases would not matter if rational arbitrageurs could fully exploit the mistakes of
behavioral investors
Fundamental risk
● “Markets can remain irrational longer than you can remain solvent” –Keynes
● Intrinsic value and market value may take too long to converge
The Dutch Book Theorem
Ac är ifall det inte sker.
Implementation Costs
● Transactions costs and restrictions on short-selling can limit arbitrage activity
Model Risk
● What if you have a bad model and the market value actually is correct?
Limits to arbitrage and the law of one price
“Siamese Twin” Companies
● Royal Dutch should sell for 1.5 times Shell
● Deviated from parity ratio for extended periods
● Example of fundamental risk
Equity Carve-Outs
● Examples: 3Com and Palm
● Arbitrage limited by availability of shares for shorting
Closed-End Funds
● May sell at premium or discount to NAV
● Can also be explained by rational return expectations
Bubbles and Behavioral Economics
Bubbles are easier to spot after they end
Dot-com bubble
● 6-year period beginning in 1995
● Overconfidence and representativeness biases
Housing bubble
● Set off worst financial crisis in 75 years
Technical Analysis and Behavioral Finance
Technical analysis attempts to exploit recurring and predictable patterns in stock prices to
generate superior investment performance
● Prices adjust gradually to a new equilibrium
● Market values and intrinsic values converge slowly
Disposition effect
● Demand for shares depends on price history
● Can lead to momentum in stock prices
Trends and Corrections
Momentum and moving averages
● The moving average is the average price over a given time interval, where the interval
updates as time passes
● Bullish signal signifies a shift from a falling trend to a rising trend
● Bearish signal signifies price series crossing from above the moving average to below
it, representative of the beginning of a downward trend in stock prices
Share Price and 50-Day Moving Average for INTC
Technical Analysis: Relative Strength
Relative strength
● Measures the extent to which a security has outperformed or underperformed either
the market as a whole or its particular industry
○ Calculated as the ratio of the price of the security to a price index for the
industry
● Strength of industry relative to the whole market
○ Computed by tracking the ratio of the industry price index to the market price
index
Technical Analysis: breadth
Breadth: Measure of the extent to which movement in a market index is reflected widely in
the price movements of all the stocks in the marke. Most common measure is the spread
between the number of stocks that advance and decline in price
Technical Analysis: Sentiment indicators
Trin statistic
● Ratios above 1.0 are bearish
● Rising volume in a rising market should not necessarily indicate a larger imbalance of
buyers versus sellers
Confidence index
● Ratio of the average yield on 10 top-rated corporate bonds divided by the average
yield on 10 intermediate-grade corporate bonds
● Ratio will always be below 1
● Higher values are bullish signals
Short interest
● Total number of shares of stock current sold short
● Increased short interest reflects negative sentiment and is a warning sign concerning
the stock’s prospects
Put/call ratio
● Ratio of outstanding put options to outstanding call options
● Rising ratio is taken as a sign of broad investor pessimism and a coming market
decline
Technical Analysis: A Warning
Actual and Simulated Levels for Stock Market Prices of 52 Weeks
Actual and Simulated Changes in Stock Prices for 52 Weeks
Föreläsning 8 (Portfolio performance evaluation & International diversification)
Overview
● If markets are efficient, investors must be able to measure performance of their asset
managers
● Discuss methods to evaluate investment performance
● Conventional approaches to risk adjustment
Time-Weighted Returns
Time-weighted average
● Geometric average is a time-weighted average
● Each period’s return has equal weight
Dollar-Weighted Returns (1)
Dollar-weighted rate of return is the internal rate of return on an investment
● Returns are weighted by the amount invested in each period
Dollar-Weighted Returns (2)
Consider a stock paying a dividend of $2 annually that currently sells for $50. You purchase
the stock today, collect the $2 dividend, and sell it for $53 at year-end.
Using DCF approach, the IRR is equal to:
The time-weighted (geometric average) return is 7.81%
Adjusting Returns for Risk
Simplest and most popular way to adjust for risk is to compare rates of return with those of
other investment funds with similar risk characteristics
● Comparison universe is the set of money managers employing similar investment
styles, used for assessing the relative performance of a portfolio manager
Universe Comparison
Risk-Adjusted Performance: Sharpe
Sharpe’s ratio divides average portfolio excess return over the sample period by the standard
deviation of returns over that period. Measures reward to (total) volatility trade-off
Risk-Adjusted Performance: Treynor
Treynor’s measure is a ratio of excess return to beta, like the Sharpe ratio, but it uses
systematic risk instead of total risk
Risk-Adjusted Performance: Jensen
Jensen’s alpha is the average return on the portfolio over and above that predicted by the
CAPM, given the portfolio’s beta and the average market return
Risk-Adjusted Performance: Information Ratio
Information ratio divides the alpha of the portfolio by the nonsystematic risk of the portfolio,
called “tracking error” in the industry
● Measures abnormal return per unit of risk that in principle could be diversified away
by holding a market index portfolio
M² Measure and the Shape Ratio
Focuses on total volatility as a measure of risk, but its risk adjustment leads to an
easy-to-interpret differential return relative to the benchmark index
M² of Portfolio P
Appropriate Performance Measure
The role of Alpha in Performance Measures
A positive alpha is necessary to outperform the passive market index
● Though necessary, it’s not enough to guarantee a portfolio will outperform the index
Most widely used performance measure
Performance statistics
Interpretation of Performance Statistics
If P or Q represents the entire investment, Q is better because of its higher Sharpe measure
and better M²
If P and Q are competing for a role as one of a number of subportfolios, Q also dominates
because its Treynor measure is higher
If we seek an active portfolio to mix with an index portfolio, P is better due to its higher
information ratio
Realized Returns versus Expected Returns
Must determine “significance level” of a performance measure to know whether it reliably
indicates ability
● To estimate the portfolio alpha from the SCL, regress portfolio excess returns on the
market index
● Then, to assess whether the alpha estimate reflects true skill and not just luck,
compute the t-statistic of the alpha estimate
Even moderate levels of statistical noise make performance evaluation extremely difficult
Survivorship Bias and Portfolio Evaluation
Regardless of the performance criterion, some funds will outperform their benchmarks in any
year, and some will underperform
Recall, performance in one period is not predictive of future performance
Limiting a sample of funds to those for which returns are available over an entire sample
period introduces survivorship bias
Style Analysis
Style analysis,a tool to systematically measure the exposures of managed portfolios, was
introduced by William Sharpe
● Idea is to regress fund returns on indexes representing a range of asset classes
○ Regression coefficient on each index would then measure the fund’s implicit
allocation to that “style”
○ R² of regression would measure percentage of return variability attributable to
style choice rather than security selection
○ Intercept measures average return from security selection of the fund portfolio
Style Analysis for Fidelity’s Magellan Fund
Fidelity Magellan Fund Cumulative Return Difference
Performance Measurement with Changing Portfolio Composition
Risk-adjustment techniques all assume that portfolio risk is constant over the relevant time
period, which isn’t necessarily true
Performance Manipulation and the MRAR
● Managers may try to game the system, given their compensation depends on
performance
● Only measure impossible to manipulate is MRAR
MRAR Scores with and without Manipulation
Market Timing
In its pure form, market timing involves shifting funds between a market-index portfolio and
a safe asset
Treynor and Mazuy:
Henriksson and Merton:
Characteristic Lines
Performance of Bills, Equities, and Perfect (Annual) Market Timers
Valuing Market Timing as a Call Option
Performance Attribution Procedures (1)
Performance attribution studies attempt to decompose overall performance into discrete
components that may be identified with a particular level of the portfolio selection process
A common attribution system decomposes performance into three components:
1. Broad asset allocation choices across equity, fixed-income, and money markets
2. Industry (sector) choice within each market
3. Security choice within each sector
Performance Attribution Procedures (2)
Bogey is designed to measure the returns the portfolio manager would earn if he or she were
to follow a completely passive strategy
● In this context, “passive” has two attributes
1. It means the allocation of funds across broad asset classes is set in accord with
a notion of “usual” allocation across sectors
2. It means that within each asset class, the portfolio manager holds an indexed
portfolio
Asset Allocation Decisions
Superior performance relative to the bogey is achieved by:
● Overweighting investments in markets that turn out to perform well
● Underweighting investments in poorly performing markets
Contribution of asset allocation to superior performance equals the sum over all markets of
the excess weight in each market times the return of the index for each market
Sector and Security Selection Decisions
Overview
U.S. equities represent about 40% of world equities and a far smaller fraction of total world
wealth
International investing:
● Similar to earlier treatment of portfolio selection, except a larger menu of assets
● Pose some problems not encountered in domestic markets
○ For example, exchange rate risk, restrictions on capital flows across national
boundaries, an added dimension of political risk and country-specific
regulations, and differing informational transparency in different countries
Global Markets for Equities
By 2018, more than 25 countries had stock markets with market capitalization above $100
billion. U.S. accounts for 40.6% of world stock market capitalization. Developed countries
account for 56% of world GDP and 76.8% of world market capitalization. Portfolio of
equities of just the six countries with the largest capitalization would make up over two-thirds
of the world portfolio.
Market Capitalization and GDP of Developed Countries (1)
Market Capitalization and GDP of Developed Countries (2)
Market Capitalization of Stock Exchanges in Emerging Markets
Market Capitalization and GDP
Development of equity markets will serve as a catalyst for enrichment of the population
● Developed code of business laws, institutions, and regulations that allow citizens to
legally own, capitalize, and trade capital assets is an important requirement for
economic advancement
Countries with larger capital markets also tend to have higher levels of per capita GDP
Per Capita GDP versus Market Cap as a Multiple of GDP
Home-Country Bias
Investors everywhere tend to overweight investments in their home countries (relative to
representation in the world portfolio) and underweight investments in foreign assets. This
patterns holds true for investors around the world, not just U.S. investors
Exchange Rate Risk and International Diversification
When a U.S. investor invests abroad, the dollar-denominated return depends on two factors:
1. Performance of the investment in the local currency
2. Exchange rate at which that investment can be brought back into dollars
Exchange Rate Risk (1)
The dollar-denominated return on the following investment in British bills is the following:
The pound-denominated return
The exchange rate “return”
Exchange Rate Risk (2)
Exchange rate risk arises from uncertainty in exchange rate fluctuations
● Currency volatility can be quite high, as evidenced by the data in Table 25.3
● However, exchange rate risk may be mostly diversifiable
● Investors can hedge exchange rate risk using a forward or futures contract in foreign
exchange
Hedging Exchange Rate Risk
Interest rate parity relationship (or covered interest arbitrage relationship)
rearranged:
Stock Market Returns
Exchange Rate Volatility
Investment Risk in International Markets
Recall that estimates of mean returns are extremely unreliable without very long data series
Estimates of volatility can be informed by return variation within the sample period
● Increasing frequency of observations can increase accuracy of risk estimates, which
means precise estimates are feasible even with relatively short sample periods
Stock Market Volatility
Index Model Regressions
International Diversification
Benefits from diversification depend on the correlation structure among securities
● International correlations have increased over time
● However, even naïve diversification provides considerable benefit
CAL supposed by the U.S. index has a Sharpe ratio of 0.217, which is the highest ratio of any
country or region in the sample
● Despite this, its Sharpe ratio is considerably less than the tangency portfolio
Efficient Frontier and CAL using Country and Regional Stock Indexes
Are Benefits from International Diversification Preserved in Bear Markets?
Some studies suggest that correlation in country portfolio returns increases during periods of
turbulence in capital markets
● If so, benefits from diversification would be lost exactly when they are needed the
most
○ Roll’s findings suggest a common factor underlying the movement of stocks
around the world
● Market behavior repeated itself in the crisis of 2008, vindicating Roll’s prediction
from his study of the October 1987 crash
Beta and SD of Portfolios
Political Risk
In principle, security analysis at the macroeconomic, industry, and firm-specific levels is
similar in all countries
In practice, getting information about foreign assets can be quite difficult
PRS Group (Political Risk Services) assesses political risk by country
● Provide country composite risk ratings on a scale of 0 (most risky) to 100 (least risky)
Composite Risk Ratings
Variables used in PRS’s Political Risk Score
Country Risk Ranking by Category
Political Risk Points by Component
Political Risk Points by Component
Performance attribution
1. Currency selection
2. Country selection
3. Stock selection
4. Cash/bond selection
Example of Performance Attribution: International (1)
Example of Performance Attribution: International (2)
Föreläsning 9 (Fixed Income)
Overview: Asset classes (Ch. 2)
Part 1: Bond Prices and Yields
● Basics & Bond Characteristics
● Bond Pricing
○ Prices and Yields
○ Prices over time
● Default Risk and Bond Pricing
Bond Characteristics
● A borrower issues (sells) a bond to the lender (creditor) for cash.
● Face value(par value) is the principal repaid at maturity.
● The coupon rate determines the interest payment (“coupon payments”) paid
semiannually
● Maturitydate is the last day of the bond contract
Example: Coupon Bond
● Face value $1000
● Annual coupons at a coupon rate of 5%
● Time to maturity: 3 years
What is a zero coupon bond?
We just saw an example of a 3 year long coupon bond. There is also something called zero
coupon bonds. What is that?
Example: Zero coupon Bond
● Face value $1000
● Annual coupons at a coupon rate of 0%
● Time to maturity: 3 years
Various Bonds
● Treasury bonds and bills
Government bonds issued by the treasury (RGK). Bills (SSVX) have maturity ≤ 1
year, bonds have maturity > 1 year.
● Corporate bonds
Issued by private companies. Typically graded according to the creditworthiness of
the issuing company.
● Other
Municipal bonds, foreign bonds, Euro bonds, Catastrophe bonds, Index-linked bond
Bond Price
● The value today of a future fixed income is simply the present value (PV) of the
future cash flow.
● The present value is calculated by discounting the future cash flow by the appropriate
discount rate
Zero Coupon Bond Price Example
Questions (zero coupon bond price)
Question 1: What is the present value (PV) of receiving $50 in 1 year if the annual interest
rate for a 1 year investment is y1=4%
Question 2: What is the present value (PV) of receiving $50 in 2 years if the annual interest
rate for a 2 year investment is y2=5%
Question 3 (Poll) : What is the present value (PV) of receiving $1050 in 3 years if the annual
interest rate for a 3 year investment is y3=6%
Solution: 𝟏𝟎𝟓𝟎/(𝟏+𝟎.𝟎𝟔)³=𝟖𝟖𝟏.𝟔𝟎
Question 4 (coupon bond price)
What is the value of the Coupon Bond if the annual interest rate for a 1 year investment is
y1=4%, the annual interest rate for a 2 year investment is y2=5%, and the annual interest rate
for a 3 year investment is y3=6%?
Bond facts: Face value $1000, Annual coupons of 5% , Time to maturity: 3 years
Question 4 cont’d
What is today's value of the 3 year coupon bond if y1=4%, y2=5% and y3=6%?
Answer: It is the sum of all present value of future cash flow: 48.08 + 45.35 + 881.60 =
975.03
Lesson: A coupon bond can be regarded as a portfolio of zero coupon bonds!
Bond Price Formula
Par value, Premium and Discount
● The value of a bond is in general different from its face value. So in general the bond
trades at a price different from the face value.
● For the special case when the bond trades at a price equal to the face value, the bond
is said to trade for par value. (Common when a bond is issued)
● Bonds that trade at a price larger than the face value, are called premium bonds.
● Bonds that trade at a price smaller than the face value, are called discount bonds.
Bond Yield-to-Maturity (YTM)
Bond Yield-to-Maturity (YTM) cont’d
Bond Yield-to-Maturity (YTM) cont’d
● Is there really no better way than guessing y?
● Yes and no: you start with a first guess and then you let the computer do the trial and
error for you.
● In Excel, solver is one possibility to find numerical solution
Remark: Inverse relationship between Bond Price and Yield
Remark: Bond Yield-to-Maturity and Realized Return
● The realized return over the life of a bond is not known today, but only at the maturity
date.
● The yield-to maturity can be calculated today.
● Clearly realized return and yield to maturity are not the same thing.
● But under the assumption that you:
1. Keep the bond to maturity date
2. Can reinvest the coupon to the yield to maturity
Then the realized return will be equal to the yield-to-maturity.
Bond Price over Time
● The price of a bond will change as time pass
● The closer time get to maturity date the bond price will approach the face value.
Bond Price and Default Risk
● Although bonds promise a fixed income to the holder, it is not risk free unless the
investor can be sure that the bond issuer will not default on the obligation.
● Government bonds are typically considered as (almost) risk free
● Company bonds and municipal bonds have a bond default risk, typically referred to as
credit risk.
● How much credit risk does a bond have?
Rating
Yield-to-maturity and Default risk
● The promised (stated) yield will be realized only if the bond issuer do not default.
● The promised yield is the maximum possible yield to maturity.
● Because of the risk of default, the investor will expect to receive less than the stated
yield.
● The default premium is a measure of what inventors require as compensation for the
possibility of default
● The default premium (credit spread)is: promised yield on a risky bond minus the yield
on a corresponding risk-free(government) bond
Yield Spreads (Figure 14.11)
Mörkblå- junk bonds
Part 2: The Term structure of interest rates
● The yield curve
Spot rates (zero rates)
Real world Example (SEB october 30, 2022, Bolåneräntor)
Remark: Spot rates change over time!
The yield curve (term structure of interest rates)
● The yield curve is a graph that displays the relationship between yield-to-maturity
versus time to maturity.
● When the yield curve is based on zero coupon yields (i.e. spot rates) it is referred to as
a pure yield curve or a zero curve.
● Many central banks publish a zero curve based on government bonds
Treasury yield curves (Fig 15.1)
The yield curve change over time
● The yield curve is derived from bond prices.
● Bond prices changes every day
● Yield curves change every day
Part 3: Bond Portfolio Management
● Interest rate risk
● Duration
Interest rate sensitivity (inverse & asymmetry)
Interest rate sensitivity (maturity)
Interest rate sensitivity (maturity)
Interest rate sensitivity (coupon)
Interest rate sensitivity: Malkiels bond-pricing relations
1. Bond prices and yields are inversely related.
2. An increase in a bond’s yield to maturity results in a smaller price change than a
decrease of equal magnitude.
3. Long-term bonds tend to be more price sensitive than short-term bonds.
4. As maturity increases, price sensitivity increases at a decreasing rate
5. Interest rate risk is inversely related to the bond’s coupon rate
“Effective maturity” and interest rate risk
● We just concluded that maturity has an impact on the interest rate sensitivity. The
longer the maturity, the larger is the sensitivity to yield changes.
● Compare 2 bonds: Is a zero coupon bond with the 3 years to maturity equally
sensitive to yield changes as a coupon bond with 15% annual coupons maturing in 3
years?
○ No, we just concluded that the smaller the coupon, the more sensitive is the
bond.
● Even though both bonds matures in 3 years, the coupon bond is paying some money
already in 1 and in 2 years time. So the “actual length” of this bond is less than 3
years, i.e. a coupon bond effectively behaves like a shorter bond than a zero coupon
bond with the same maturity which explains why it is not as sensitive.
● The vague concept of “actual length” seem to be important for interest rate risk. Let
us formalize it
Duration
● A formalized measure of the “actual length” of a bond
● Duration is a weighted average of the times until each payment is received, with the
weights being the fraction of the bond value that is being paid out at each point in
time.
● It is shorter than maturity for all coupon bonds, and is equal to maturity for zero
coupon bonds.
● Duration can also be used as a measure of interest sensitivity
Macaulay Duration
Duration is a weighted average of the times to each cash flow t, where the weights is
determined as that’s cash flows present value over total value, i.e.
Duration-example
Another way to think about duration (physics)
● In a rocking board the support must be closer to the heavy person
● Where support should be is calculated using force (kraftmoment)
● Idea with duration is the same
Duration-question
What is the duration of a bond with coupon rate of 8% (paid annually), face value 1000 and
time to maturity of 4 years if YTM is 8%?
Solution:
1. Calculate present values of each of 4 cash flows:
● 1 year: 80(1.08)¹=74.07
● 2 year: 80(1.08)²=68.58
● 3 year: 80(1.08)³=63.51
● 4 years: 1080(1.08)⁴=793.83
2. Calculate weights:
● 74.07/1000=0.074
● 68.58/1000=0.069
● 63.51/1000=0.064
● 793.83/1000=0.794
3. Calculate Duration:
● 0.074×1 +0.069×2+0.064×3+0.794×4=3.58 years
Duration as a price sensitivity measure
● It turns out that the bonds price change is proportional to duration
● Alternative formulation
Revisit example: modified duration
● We calculated duration of the 4y ear coupon bond to D=3.58 and since
● the bond yield was y=8%,
● the modified duration is D* = D/(1+y)= 3.58/1.08= 3.31
● Interpretation: If the bond yield increase by 1%, the bond price will decrease by
approximately 3.31%