FINS2624 PORTFOLIO MANAGEMENT

Table of Contents

FINS2624 PORTFOLIO MANAGEMENT .....................................................................................................0

FORMULAS .............................................................................................................................................2

I. Bond Fundamentals .............................................................................................................................3 Pricing Bonds ............................................................................................................................................................................... 3 Arbitrage Pricing ......................................................................................................................................................................... 3 Returns on Bonds ........................................................................................................................................................................ 3

II. Term Structure of Interest rates...........................................................................................................5 Term structure of interest rates ................................................................................................................................................. 5 Methods of inferring term structure: ......................................................................................................................................... 5 Arbitrage and Term Structure ..................................................................................................................................................... 5 Forward Rates ............................................................................................................................................................................. 6 Theories of Term Structure ......................................................................................................................................................... 6

Expectations Hypothesis ........................................................................................................................................................ 6 Liquidity Preference Hypothesis (Preferred Habitat) ............................................................................................................ 6 Hybrid of both is likely the correct theory ............................................................................................................................. 6

III. Duration ............................................................................................................................................8 Interest rate risk .......................................................................................................................................................................... 8 Duration ...................................................................................................................................................................................... 8 Convexity ..................................................................................................................................................................................... 9

Duration and Bond Parameters ............................................................................................................................................. 9 Price Yield Curve & Duration .................................................................................................................................................. 9 Approximation Error .............................................................................................................................................................. 9 Convexity and Price Changes ............................................................................................................................................... 10

Interest Risk Management ........................................................................................................................................................ 10 Portfolio duration ................................................................................................................................................................. 10 Asset liability matching ........................................................................................................................................................ 10

Immunisation ............................................................................................................................................................................ 10

IV. Markowitz’ Portfolio Theory ............................................................................................................ 11 Risk and Risk Aversion............................................................................................................................................................... 11 Preferences & Utility ................................................................................................................................................................. 11 Expected Return & Risk ............................................................................................................................................................. 12 Diversification ........................................................................................................................................................................... 12 Optimal Risky Portfolios ............................................................................................................................................................ 12

V. Optimal Portfolios ............................................................................................................................ 13 Optimal portfolio without risk free assets ................................................................................................................................ 13 Complete portfolio with risk free assets................................................................................................................................... 13 Separation Theorem ................................................................................................................................................................. 14

VI. CAPM .............................................................................................................................................. 16 CAPM Derivation ....................................................................................................................................................................... 16 Security Market Line ................................................................................................................................................................. 16 Systematic vs Idiosyncratic Risks .............................................................................................................................................. 17

VII. SIM and Factor Models ................................................................................................................... 18 Mispricing .................................................................................................................................................................................. 18 Exploiting Mispricing ................................................................................................................................................................. 18 Factor Models ........................................................................................................................................................................... 20

VIII. EMH and Behavioural Finance........................................................................................................ 21 Efficient Markets Hypothesis .................................................................................................................................................... 21 Behavioural Biases .................................................................................................................................................................... 22

Jess Williams (z5163715) FINS2624

2

FORMULAS Topic Formula 1 Bond

Fundamentals Price of a coupon bond 𝑃𝑟𝑖𝑐𝑒 = ∑

𝐶𝑡

(1 + 𝑌𝑇𝑀𝑡)𝑡𝑡

𝑃𝑟𝑖𝑐𝑒 =𝑐1

(1 + 𝑦1)1 +𝑐2

(1 + 𝑦2)2 + ⋯ +𝑐𝑡

(1 + 𝑦𝑡)𝑡 + ⋯

+𝑐𝑇 + 𝐹𝑉

(1 + 𝑦𝑇)𝑇

𝑃𝑟𝑖𝑐𝑒 =𝐶𝑟

(1 − (1

1 + 𝑟)𝑛

) +𝐹𝑉

(1 + 𝑟)𝑛

𝑃𝑟𝑖𝑐𝑒 = 𝐶𝑜𝑢𝑝𝑜𝑛 × 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 𝑓𝑎𝑐𝑡𝑜𝑟(𝑟, 𝑇) + 𝐹𝑉(𝑃𝑉 𝑓𝑎𝑐𝑡𝑜𝑟(𝑟, 𝑡)

Realised Compound yield 𝑅𝐶𝑌 = (

𝑉𝑁

𝑃 )1𝑁

− 1 = (1 + 𝑅)1𝑇

Holding Period Return 𝐻𝑃𝑅 =

𝑃1 + 𝐶1

𝑃0− 1

2 Term Structure

of Interest Rates YTM Given a sequence of forward Rates

(1 + 𝑦𝑛)𝑛 = (1 + 𝑟1)(1 + 𝑓1

2) + ⋯ + (1 + 𝑓𝑛−1

𝑛)

Forward rate of interest 𝑓𝑠

𝑡 = ((1 + 𝑦𝑡)𝑡

(1 + 𝑦𝑠)𝑠)

1𝑡−𝑠

− 1

Liquidity premium 𝑙𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 = 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 − 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑠ℎ𝑜𝑟𝑡 𝑟𝑎𝑡𝑒

Expectations Hypothesis 𝑓𝑠

𝑡 = 𝐸( 𝑦𝑠

𝑡)

Hybrid Liquidity and EH 𝑓𝑠

𝑡 = 𝐸( 𝑦𝑠

𝑡) + 𝐿

3 Duration Macaulay’s duration 𝜕𝑃

𝜕𝑦= − ∑

𝐶𝐹𝑡/(1 + 𝑦)𝑡

𝑃𝑟𝑖𝑐𝑒(𝑏𝑜𝑛𝑑)∙ (𝑡)

𝑇

𝑡=1

𝑤𝑡 =𝐶𝐹𝑡/(1 + 𝑦)𝑡

𝑃𝑟𝑖𝑐𝑒(𝑏𝑜𝑛𝑑)

𝐷 = − ∑ 𝑡 × 𝑤𝑡

𝑇

𝑡−=1

Proportional Change in Price Given a change in YTM(y)

∆𝑃𝑃

= −𝐷 × [∆(1 + 𝑦)

1 + 𝑦 ]

Modified Duration

∆𝑃𝑃

= −𝐷∗ × ∆𝑦

Duration Calculation Including Convexity

∆𝑃𝑃

= −𝐷∗∆𝑦 +12

× 𝐶𝑜𝑛𝑣𝑒𝑥𝑖𝑡𝑦 × (∆𝑦)2

Portfolio Duration 𝐷𝑝𝑜𝑟𝑡 = ∑ 𝐷𝑖𝑤𝑖

𝑁

𝑖−1

𝐷𝑝𝑜𝑟𝑡 = ∑ 𝐷𝑖 (𝑃𝑖

∑ 𝑃𝑖𝑁𝑖−1

) =𝑃𝑖

𝑃𝑝𝑜𝑟𝑡

𝑁

𝑖−1

Jess Williams (z5163715) FINS2624

11

o Also, cannot very long term securities do not exist Issues with Conventional Immunisation

• Uses the duration formula implying a flat- yield curve • Duration matching will immunise portfolios only for parallel shifts in the yield curve • Immunisation can be seen as inappropriate in an inflationary environment

IV. Markowitz’ Portfolio Theory Risk and Risk Aversion

• Risk averse investors prefer certain outcomes to random (stochastic) outcomes must give risk averse investors to take on risks, we need to give them an incentive in the form of a risk premium

• For a higher risk, want a higher return to compensate → Although don’t know the exact outcome, have an idea based on mean, volatility and correlation

Preferences & Utility Preferences Interested in satisfying as many preferences as possible use a utility function to mathematically express preferences by

assigning a value to each outcome so the preferred outcomes receive higher values Utility Concept

Utility refers to the total satisfaction from consuming a good or service → economic utility is essential to understand because it will directly influence the demand and therefore the price of a good or service.

• Cannot directly measure the benefit, satisfaction or happiness, therefore generally use wealth in finance to determine the amount of goods or services we will consume → different investment outcomes mean different payoffs and different resulting levels of wealth

o The more money an individual has, the more outcomes are achievable ( higher utility) o Wealth level depends on investment returns and hence is uncertain – many possible wealth levels with

specific probabilities ▪ Only know what specified investment outcome, therefore utility level will end up with unless

put money in risk free investment • Total Utility = Amount of satisfaction a person can receive from consumption of all units of a product or service • Marginal Utility = additional utility gained from consumption of an additional unit

Utility functions based on wealth

• Satisfaction depends on the resulting wealth level, as this is not precise, generally based on two common features:

o (1) The more the better o (2) The surer the better

• i.e. where U(W)=F(W) (F = functional form) o (1) 𝑈(𝑊2) > 𝑈(𝑊1) 𝑖𝑓 𝑊2 > 𝑊1 o (2) 𝑈(�̅�) > 𝑈(𝑊̅̅ ̅̅ ̅̅ ), 𝑤ℎ𝑒𝑟𝑒 �̅�means the expected/

average value of a random variable W ▪ i.e. average utility is lower than average

wealth of two • Conditions equivalent to calculus terms

o (1) Calculus: 𝜕𝑈𝜕𝑊

> 0→ Upward sloping (U increases with W)

o (2) Calculus: 𝜕2𝑈𝜕𝑊2 < 0 → Concave (U increases with W at decreasing rate)

Utility of an investment with uncertain future payoffs

• Rank different assets/ investments with uncertain future payoffs ( uncertain levels of end-of-period wealth) • U is expected value of a random variable r~ related to: expected value (1st moment), variance (2nd Moment) and

higher order moments Mean Variance Criterion

• Investors only evaluate their investments based on mean and variance of returns • Only capture 2 aspects:

o The more the better- E(rA) ≥ E(rB) o The surer the better - A ≤ B

• U(W)=ln(W) (Concave function; Surer the wealth level, the higher the utility function)

𝑈 = 𝐸(𝑟) −12 𝐴𝜎2

(must be a decimal! NOT Percentage) Note: you can also interpret the utility as the certainty equivalent rate of return → i.e. the rate a RF investment would need to offer to provide the same utility as the risky portfolio

!

! 30!

XI.$Option$Valuation$Binomial$Model$

•! Option!Priced!using!no!arbitrage!argument!!!i.e.$the$value$of$the$derivative$written$on$the$stock$is$the$value$$of$the$derivative$at$t0$=$f0!

•! The!derivative!is!has!payoff!functions!!" = $(&"); !) = $(&));**!•! Process!of!solving!One!Period!binominal!model!!

o! Form!a!portfolio!that!combines!the!position!in!the!option!&!underlying!asset!to!replicate!RF!asset!!and!find!the!price!of!the!derivative!!

!Method$1:$Replicating$the$Risk$free$Asset$$

1.! Form!a!risk!free!portfolio!by!combining!the!underlying!asset!and!the!derivative!!a.! e.g.!buy!∆!stocks!and!short!one!derivative.!Based!on!your!risk!free!portfolio,!draw!

one!period!binomial!tree!!i.! portfolio!value!at!+,: ., = ∆&, − !,!ii.! Portfolio!value!at!maturity:!." = ∆&,1 − !"*23*.) = ∆&,4 − !)!

!!

2.! Since!we!are!replicating!risk!free!asset,!can!calculate!∆!by!setting*." = .)*(7. 8. ∆&,1 − !" = ∆&,4 − !))*

*∆=

!" − !)&,1 − &,4

=$(&") − $(&))&,1 − &,4

!

!3.! Calculate!the!portfolio!value!at!maturity!by!substituting!∆!back!to!."*9:*.) !

!4.! Value!of!the!RF!portfolio!must!earn!the!risk!free!rate.!Can!calculate!the!portfolio!value!

at!t0!(i.e.!P0)!by!discounting!the!portfolio!value!at!maturity!by!the!risk!free!value!by!discounting!the!portfolio!value!at!maturity!by!the!RF!rate.!Then,!f0#could!be!calculated!by!using!., = ∆&, − !, !

!!Method$2:$Risk$Neutral$Valuation$of$Options$Time!zero!price!of!the!option:!

•! !•! Given!that!∆= ;<=;>

?@"=?@)!

Pricing!equation:!

!As!A = BCD=)

"=)!*1 − A = "=BCD

"=) !!

"!can!directly!apply!the!pricing!formula!derived!from!the!binomial!model!to!check!whether!the!answer!is!correct!!

FG =[IFJ + (L − I)FM]

OPQ =R(FQ)OPQ $

Where!A = BCD=)"=)

!(the!risk!neutral!probability)!The#expected#value#of#the#option#at#time#T#is#discounted#by#the#risk#free#rate#

The#expected#value#of#the#option#is#calculated#using#a#set#of#hypothetical#probabilities#of#option#payoffs###

Q! NOTE:!The!option!price!does!not!equal!to!the!actual!expected!value!to!T,!discounted!by!the!RF!rate!!!Because!the!payoff!the!option!is!not!risk!free!!

Q! But!this!process,!defines!a!probability!for!the!option’s!future!payoffs,!and!make!the!option!price!equal!to!the!expected!value!at!time!T!calculated!on!the!basis!of!that!probability,!discounted!by!the!RF!rate!!

!!

Interpreting!∆:!Q! Change!in!option!price!as!a!result!

of!a!$1!change!in!the!stock!price!Q! Measures! the! sensitivity! of! the!

option! value! to! the! underlying!asset!price!!

Q! Change!in!the!value!of!the!shorting!option! will! offset! the! value! of! ∆!shares! of! stock,! therefore! the!portfolio! value! will! not! change!(therefore!risk!free)!!

Q! Also!the!hedge!ratio!of!the!option,!as!the!value!change!in!one!option!position! (whether! long! or! short)!will!be!hedged!by!holding!∆!shares!of! stock! in! the! opposition!direction!!

!