final review

Arbitrage Opportunities in the APTWeeks 6-10Final Exam

Review

Juan Sotes-Paladino

FNCE30001Investments

Semester 2 2015

University of Melbourne FBE Review 1


Outline

A loose end from the 1st part

A quick review of weeks 6-10

More info on the final exam



A little more on APT



Example of Multifactor Arbitrage Opportunity in APT

Consider the following factor expected returns

Factor portfolio 1: E(rF1) = 10%

Factor portfolio 2: E(rF2) = 12%

Risk-free rate: 4%

In addition, a portfolio C is offered:

E(rC ) = 15%, βC ,F1 = 0.50, βC ,F2 = 0.75

Q: Is there an arbitrage opportunity?



Example of Multifactor Arbitrage Opportunity (cont.)

By definition, factor portfolios have

a beta of 1 w.r.t. themselves: βF1,F1 = βF2,F2 = 1

a beta of 0 w.r.t. any other factor: βF1,F2 = βF2,F1 = 0

⇒ A well-diversified portfolio A with weights wA,1,wA,2, (1− wA,1 − wA,2) in factors1, 2, and the risk-free asset will have betas of:

βA,F1 = wA,1βF1,F1 + wA,2βF2,F1 + (1− wA,1 − wA,2)× 0 = wA,1

βA,F2 = wA,1βF1,F2 + wA,2βF2,F2 + (1− wA,1 − wA,2)× 0 = wA,2



Example of Multifactor Arbitrage Opportunity (cont.)

Back to our question: Is there an arbitrage opportunity?

Create a well-diversified portfolio A such that βA,F1 = 0.50, βA,F2 = 0.75

� Easy: invest 50% of A in F1 (wA,1 = .5), 75% in F2 (wA,2 = .75) and remaining-25% (wA,f = 1− 0.50− 0.75 = −.25) in the risk-free asset:

E(rA) = (1− 0.50− 0.75)× rf + .5× E(rF1) + .75× E(rF2) = 13%

Arbitrage opportunity alert!

Arbitrage strategy: long C , short A

For every $1 invested in C , sell (short) $1 of A

Selling $1 of A implies, in turn, investing $0.25 at the risk-free rate for every $0.50 and$0.75 sold (short) of the factor portfolios 1 and 2

0 Net investment!

Arbitrage profit (%) = rC − rA = E(rC )− E(rA) = 2%



Weeks 6-10



Recap of Week 6

Portfolio performance evaluation comprises measurement (what is the ex postreturn/risk?), appraisal (is this better than a benchmark?) and attribution (whodeserves the credit?)

Performance measurement requires choices in return/risk measurement(dollar-weighted, time-weighted, simple, compound and standard deviation orbeta)

Performance appraisal utilises different measures (Sharpe, Treynor, M2, alpha, IR)for different portfolio scenarios

Performance attribution allocates excess return to strategic/tactical assetallocation and asset/stock selection choices



Sharpe (S) vs. Treynor (T ) vs. Information Ratio (IR)

Some guidance on using S vs. T vs. IR:

If the alternative portfolios represent the entire investment universe, the one withthe highest S (M2) is the one preferred

When the alternative portfolios are competing as one of a number of subportfolios,the one with the highest T is the preferred one

Finally, when alternative active portfolios can be mixed with the index portfolio theIR is the right measure to look at, since it penalizes deviations from the index



Recap of Week 7

Fixed, floating, and capital-indexed coupons characterize the core fixed incomesecurities

Convertible and callable bonds provide options to investors/borrowers

Pricing zero-coupon bonds is based on compound interest (zero rates)

Pricing money market securities is based on simple interest

The Zero Rate Curve is established from zero-coupon securities and isfundamental to pricing all (other) fixed income securities

Holding period returns are easily derived from the return on zeros



Recap of Week 8

Pricing a fixed-coupon bond occurs by pricing off the zero curve – mispricingwould attract arbitrage

Yield-to-maturity (ytm) is a bond’s internal rate of return, and is typically used bytraders as a “price” measure

Practical bond pricing requires fine tuning for calendar/trading days and coupondates

Distinguish the zero curve from the yield curve

Practitioners use basic bootstrapping of sequences of coupon bonds to find the(unobserved) zero rate curve

Holding period returns highlight reinvestment rate problems

Bond prices tend to converge to par value as the bond maturity approaches



Semi-annual and Annual Yields

Yields are compounded based on the coupon period

� So for a:

A semi-annual coupon bond,

with a reported yield of y pa and

2N periods (= N years) remaining to maturity, we have

P(y) =C

(1 + y/2)1+

C

(1 + y/2)2+ . . .+

C + Par

(1 + y/2)2N

For a bond that pays coupons annually with N periods to maturity (= N years)and yield y pa, we have:

P(y) =C

(1 + y)1+

C

(1 + y)2+ . . . +

C + Par

(1 + y)N



Key differences between yields and zero rates

A zero rate applies to every cash flow to be paid at that specific time

A yield applies to that specific bond

But not to other bonds, even from the same issuer

Zero rates can be used to compute discount factors

⇒ Zero rates translate directly to prices



Recap of Week 9

The term structure of interest rates (TSIR) reflects interest rates for variousterms to maturity embodied in the prices of default-free zero-coupon bonds

Forward rates are “breakeven” future interest rates that equate the total returnfrom a short-term rollover strategy to that of an equivalent maturity long-terminvestment strategy

The Expectations Hypothesis implies that forward rates are unbiased estimates ofexpected future interest rates. This then indicates that successive zero ratesreveal market expectations about future rates

The Liquidity Premium Hypothesis may bias longer term rates upwards due to aninvestor required excess rate of return to compensate for long-term investmentrisk. This makes inference of future interest rate expectations complicated



Interpreting the Slope of the TSIR

Today’s TSIR Pure expecta+ons:

All forward rates = 10% pa All expected rates = 10% pa

10%

Lt,t+1

Rising liquidity premium: All forward rates = 10% pa All expected rates = 10% – Lt, t+1

Term (years ahead)

Interest rate (pa)

INTERPRETING THE TSIR WITH A RISING LIQUIDITY PREMIUM University of Melbourne FBE Review 15


Recap of Week 10

All bonds are subject to interest rate risk with longer-term bonds, low-couponrate bonds, and low-yield bonds typically more sensitive

The average “life” of a standard coupon bond is measured with Macaulay’sduration, and is also an indicator of the sensitivity of a bond’s price to a change inits yield (interest rates)

Convexity (curvature of the price/yield curve) is desirable

Immunization requires both duration match and present value match

Immunization also requires rebalancing adjustments



Final Exam Info



Final Exam Information

2 parts (A and B)

A is multiple choice (15 questions)

B is mix of short essay and numerical questions

Roughly distributed across all Lectures

Formula sheet - may be detached

Advice:

Reading Time - Read and Rank!

Don’t write more than fits in the space provided

Attempt ALL questions!!



Final Exam Information (contd)

Examinable material:

Everything treated in the lectures and workshops

Exception: news articles, optional readings on LMS and lecture Appendices

All chapters of the textbook as indicated in the subject guide

Exceptions: Section 6.6, Yield to call (p.230-1), CDS (p240-1), Section 10.5

Review material

Lecture Notes

Tutorial Exercises

Textbook Concept Check Questions (solutions provided)

Practice Problems and Sample Final (see LMS)



More Information

Extra consultation hours

see schedule on LMS

Online Tutor

Finally: please fill out the SES questionnaire,

https://subjecteval.unimelb.edu.au


https://subjecteval.unimelb.edu.au


THANK YOU!


final review

Documents

f2 wa

f1 wa

arbitrage opportunities

arbitrage strategy

arbitrage profit

weights wa

arbitrage opportunity

portfolio c