FINS3616 Lecture Notes - Lecture 9: Sharpe Ratio, Market Portfolio, Risk Aversion
8 – International Capital Market Equilibrium
Risk and return of international investments
• Two risks of investing abroad
Returns of the international asset in its local currency
Variatios i the alue of the foreig urre relatie to iestors urre
Return of investment = return of asset + return of currency
(a) 1 + r (t+1, $) = S (t+1) / S (t) * [1 + r (r+1, £)]
(b) r (t+1, $) = [1 + s (t+1)] * [1 + r (t+1, £)] – 1
(c) r (t+1, $) = r (t+1, £) + s (t+1) + r (t+1, £) * s (t+1)
• Volatility of international investments
Volatility of currency and equity returns
Volatility is not additive
(a) Var [r (t+1, FC) + s (t+1)] = Var [r (t+1, FC)] + Var [s (t+1)] + 2Cov [r (t+1, FC), s
(t+1)]
• CovarianceAB = Correlation * VolA * VolB
Correlation = p
If p < 1, there is a diversification benefit
• Sharpe ratios
Measured as the average excess return relative to the volatility of the return
Sharpe Ratio = (E[r] – rf) / Vol [r]
Benefits of international diversification
• Risk reduction through international diversification
Nonsystematic variance (~60-75% of ariae i firs retur
Idiosyncratic variance changes over time
(a) Some say it has increased – firms more focused (less diversified) and go
public earlier (i.e. average firm is younger)
(b) Some say no – little evidence
• What drives correlation of return
Trade
Geographic proximity
Industrial structure
Irrational investors (i.e. contagion)
• Asymmetric correlations?
International diversification benefits evaporate when you need them most (in
bear markets)
(a) Investors still better off if they diversify internationally though
• Effect of international diversification on Sharpe Ratios
When does international diversification improve the Sharpe ratio?
(a) P < 1; the lower the better
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2
Investment hurdle rates – lowest possible expected return that allows for an
improvement in the Sharpe ratio when they invest in that foreign market
Optimal portfolio allocation
• Optimal portfolio – maximises utility function of an investor
U = E[rp] – (A/2) * 2p
(a) Where A is your risk aversion (higher = more risk averse)
• One risky asset
Capital allocation line (CAL)
(a) rp = w*r + (1-w) *rf = rf + w*(r – rf)
(b)
2p = w2
2 and
p = w
i. Solving for w: w = p /
(c) E[rp] = rf + [(E[r] – rf) / ] p
(d) Optial portfolio depeds o the iestors appetite for risk i.e. here o
the horizontal axis the investor prefers to be)
(e) (E[r] – rf) /
is the slope of the CAL
(f) Lending region – left of along horizontal axis
(g) Borrowing region – right of along horizontal axis
• Mean standard deviation frontier
Once we add risky assets to the mix, it gets complicated
To siplif e eed to get rid of portfolios that aret effiiet usig a iiu
variance approach (i.e. how much risk we are willing to accept for a given level of
return)
(a) Minimise variance given the weights add up to one and the returns add to
our target return – efficient frontier
• Finding the mean-variance-efficient portfolio – where the CAL is a tangent to the
efficient frontier
CAL slope = Sharpe ratio
The Capital Asset Pricing Model (CAPM)
• E[rj] = rf + ßjm [E(rm) – rf]
All investors hold the same portfolio for risky assets – the market portfolio
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Document Summary
Risk and return of international investments: two risks of investing abroad. Returns of the international asset in its local currency. Variatio(cid:374)s i(cid:374) the (cid:448)alue of the foreig(cid:374) (cid:272)urre(cid:374)(cid:272)(cid:455) relati(cid:448)e to i(cid:374)(cid:448)estor(cid:859)s (cid:272)urre(cid:374)(cid:272)(cid:455) Volatility is not additive (a) var [r (t+1, fc) + s (t+1)] = var [r (t+1, fc)] + var [s (t+1)] + 2cov [r (t+1, fc), s (t+1): covarianceab = correlation * vola * volb. If p < 1, there is a diversification benefit: sharpe ratios. Measured as the average excess return relative to the volatility of the return. Sharpe ratio = (e[r] rf) / vol [r] Benefits of international diversification: risk reduction through international diversification. Nonsystematic variance (~60-75% of (cid:448)aria(cid:374)(cid:272)e i(cid:374) fir(cid:373)(cid:859)s retur(cid:374)(cid:895) International diversification benefits evaporate when you need them most (in bear markets) (a) investors still better off if they diversify internationally though: effect of international diversification on sharpe ratios.
