Portfolio Theory EZ - Portfolio Theory Econ 422 Investment...

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1 Portfolio Theory Econ 422: Investment, Capital & Finance University of Washington Spring 2010 1 E. Zivot 2006 R.W.Parks/L.F. Davis 2004 May 17, 2010 Forming Combinations of Assets or Portfolios Portfolio Theory dates back to the late 1950s and the seminal work of Harry Markowitz and is still heavily relied upon today by Portfolio Managers We want to understand the characteristics of portfolios formed from combining assets Given our understanding of portfolio characteristics, how does an individual investor form optimal portfolios, i.e., consistent within the economic models presented to date? 2 E. Zivot 2006 R.W.Parks/L.F. Davis 2004 What useful generalities or properties can we derive? How does this theory apply to the economy or capital markets (investors in the aggregate)? Is this theory consistent with behavior we observe in financial markets?
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2 Preliminaries: Portfolio Weights • Portfolio weights indicate the fraction of the portfolio’s total value held in each asset, i.e. x i = (value held in the i th asset)/(total portfolio value) • Portfolio composition can be described by its portfolio weights: x = {x 1 ,x 2 ,…,x n } and the set of assets {A 1 , A 2 , ….A n } • By definition, portfolio weights must sum to one: x 1 +x 2 +…+x n = 1 3 E. Zivot 2006 R.W.Parks/L.F. Davis 2004 • Initially we will assume the weights are non-negative ( x i > 0), but later we will relax this assumption. Negative portfolio weights allow us to deal with borrowing and short selling assets. Data Needed for Portfolio Calculations E(r i ) Expected returns for all assets i V( )S D ()Vi t dd d i t i f t f V(r i ) or SD(r i ) Variances or standard deviations of return for all assets i Cov(r i ,r j ) Covariances of returns for all pairs of assets i and j Where do we obtain this data ? 4 E. Zivot 2006 R.W.Parks/L.F. Davis 2004 • Estimate them from historical sample data using statistical techniques ( sample statistics ). This is the most common approach.
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3 Portfolio Inputs in Greek μ=E[R μ = E[R] σ 2 = var(R) σ = SD(R) σ ij = Cov(R i , R j ) C( R R 5 E. Zivot 2006 R.W.Parks/L.F. Davis 2004 ρ ij = Cor(R i , R j ) Note: σ ij = ρ ij * σ i * σ j A Portfolio of Two Risky Assets Real world relevance : 1. Client looking to diversify single concentrated holding in one particular asset. 2. Portfolio Manager looking to add an additional asset to a pre-existing portfolio. E(r) σ . (1) . (2) 6 E. Zivot 2006 R.W.Parks/L.F. Davis 2004 Points (1) and (2) show the expected return and standard deviation characteristics for each of the risky assets. What are the characteristics of a portfolio that is composed of these two assets with portfolio weights x 1 and x 2 of asset 1 and 2, respectively?
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4 Portfolio Characteristics n = 2 Case r x r x r = + 1 2 2 As you hold x 1 of asset 1 and x 2 of asset 2, you will receive x 1 of the return of asset 1 plus x 2 of the return of asset 2: Er xEr Vr xVr xxC o vrr p p p =+ =++ 1 1 11 2 2 1 2 12 2 21 2 2 Find expected return and variance of return.
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Portfolio Theory EZ - Portfolio Theory Econ 422 Investment...

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