2. Mean-variance optimization with unlimited borrowing and lending at a risk-free rate
Sharpe ratio of H < Sharpe ratio of M
The combination of risk-free asset and M dominates the combination
Debt 8% 12%
Equity 13% 20%
When ρDE = -1,
wE
D D
E
1 wD
When ρDE = 0,
Bodie et al. (2014), Table 7.3, p. 211
Dr Ekaterina Svetlova
1. Brief revision: Lecture 2
Source: Bodie et al. 2014: p. 220
Dr Ekaterina Svetlova
1. Brief revision: Lecture 2
Diversifiable (non systematic) risk vs undiversifiable (systematic) risk
We construct risky portfolios varying xD and xE
to provide the lowest possible risk for any given level of expected return
E(rp) = wD E(rD) + wEE(rE)
2 p
of risk-free asset and H
Source: Perold 2004
Dr Ekaterina Svetlova
2. Mean-variance optimization with unlimited borrowing and lending at a risk-free rate
Harold J. Bowen III
Dr Ekaterina Svetlova
2. Mean-variance optimization with unlimited borrowing and lending at a risk-free rate
Dr Ekaterina Svetlova
2. Mean-variance optimization with unlimited borrowing and lending at a risk-free rate
- Standard deviation of the return: σ = 0
Dr Ekaterina Svetlova
2. Mean-variance optimization with unlimited borrowing and lending at a risk-free rate
If you invest in asset H and riskless asset: xH and xf = 1 - xH
Trustees
the Tampa firefighters and police officers pension fund
The Investment Management Firm
Investment consultants
As for being diversified, which is the mantra of nearly all institutional money managers and consultants, [the Tampa fund] isn’t. … [T]he fund’s assets are concentrated in a relatively small number of stocks and fixed-income investments. In short, the Tampa pension fund pretty much breaks all the conventional rules of fund management.
Tobin separation theorem:
Portfolio choice problem can be separated in two tasks: 1. Identify the optimal risky portfolio 2. Identify the capital allocation between
zero
Sharpe ratio: 0.305 (higher than 0.25 for M and 0.175 for H) All investors will hold assets M and H in proportions 74/26 Source: Perold 2004
Dr Ekaterina Svetlova
ErH - Rf
σH
Sharpe ratio
Dr Ekaterina Svetlova
Source: Perold 2004
(ErH - Rf)
Risk premium
2. Mean-variance optimization with unlimited borrowing and lending at a risk-free rate
Unlimited borrowing and lending at a risk-free rate:
- Riskless asset is an asset with a certain return for the given time horizon.
- For example: US Treasury bonds that automatically adjust for inflation (TIPS: Treasury inflation protected securities) or short term US Treasury bills (US T-bills)
Foundations of Financial Analysis and
Investments
Lecture 3: Capital Asset Pricing Model (CAPM)
Dr Ekaterina Svetlova
Today‘s lecture
1. Brief revision: Lecture 2 2. Mean-variance optimization with unlimited
Sharpe ratio of asset H:
Sharpe ratio of asset M:
(12% - 5%)/ 40% = 0.175
(10% - 5%)/ 20% = 0.25
Important: all combinations of asset H with risk-free borrowing and
2. Mean-variance optimization with unlimited borrowing and lending at a risk-free rate
Combining equations for portfolio return and risk, we obtain :
Erp = Rf
How much of each risky asset should one hold in the portfolio?
New efficiency line when risk-free lending/borrowing is allowed
Correlation between M and H assumed to be
1. Brief revision: Lecture 2
Expected return E(r) Standard deviation
Bodie et al. (2014), Table 7.1, p. 208
Debt 8% 12%
Equity 13% 20%
B A
Bodie et al. (2014), Table 7.3, p. 211
3. Mean-variance optimization with unlimited borrowing and lending at a risk-free rate
In case of many risky assets:
Risk aversion
Risk seeking
Source: Perold 2004
Dr Ekaterina Svetlova
1. Brief revision: Lecture 2
The portfolio consists of two risky assets D (debt) and E (equity)
Their weights in the portfolio are xD and xE (xD + xE = 1; xD ≥ 0, xE ≥ 0)
wD2
2 D
wE2
2 E
2wD wE Cov
rD , rE
Cov(rD,rE) = DEDE
Success of diversification depends on the correlation coefficient
Dr Ekaterina Svetlova
Bodie et al. 2014, Ch. 7
Erp = (1 - xH) Rf + xH RH = Rf + xH(ErH - Rf) σp = (1 - xH)2 σf + xH2 σH2 + 2xH (1 - xH) ρfH σf σH As σf = 0, we obtain: