FNCE30001 Course Summary Notes AWS
FNCE30001 Course Summary Notes
Table of Contents:
Returns
Page 2
Portfolio Theory
Page 2
Portfolio (Investment Fund) Performance Evaluation
Page 14
Fixed Income Valuation
Page 18
Interest Rate Swaps
Page 29
1
Returns
r1 is the return from t=0 to t=1, which is: 𝑟1 =
𝑃1 + 𝐷1 −𝑃0 𝑃0
o
𝑟̃ – unknown future return (tilde indicated variable with probability distribution) 𝑟̃ (s) – return in state ‘s’, which has probability of occurring: p(s) o E(𝑟̃ ) – expected future return = ∑ 𝑝(𝑠) 𝑟̃ (s) o r – realised (actual) return Variance (σ2) = ∑ 𝑝(𝑠) [𝑟̃ (𝑠) − 𝐸(𝑟̃ (𝑠))]2 = ∑ 𝐸[𝑟̃ − 𝐸(𝑟̃ )]2
Distribution of returns
Returns can be modelled to the ‘normal distribution’ o
𝑧̃ =
𝑟̃𝑡 −𝜇 , 𝜎
where 𝑟̃𝑡 ~𝑁(𝜇, 𝜎)
Can then calculate probabilities using a standard normal table or Excel Can approximate returns in a normal distribution by the ‘68.2, 95.4, 99.7 rule’ o NB: natural logarithm returns are closer to being accurate, but have too “fat tails” Skewness – ‘which way it leans’ o
Skew =
𝐸(𝑟̃ −𝜇)3 𝜎3
o
Positive skew – less negative tail Stock market is this (as cannot go below $0 value), meaning a normal distribution overestimates downside risk o Negative skew – less positive tail o NB: skewness decreases when considering data over time – has been tiny since 1950’s Kurtosis (K) – ‘how fat the tails are’ o Normal distribution = 3 o Platykurtic: K3 (thinner in middle, fatter tails) o
Excess kurtosis (any kurtosis over 3) =
𝐸[(𝑟̃ −𝜇)4 ] − 𝜎4
3
Average excess kurtosis approx. 0.4 since 1950’s
Portfolio Theory Capital Allocation Line (CAL) The balance between risky (‘y’ proportion) and risk-free (‘1 – y’ proportion) investments
CAL considers all the available risk-return combinations 𝐸(𝑟̃ ) = 𝑟𝑓 (1 − 𝑦) + 𝑦𝐸(𝑟̃𝑝 ) o ‘y’ can be >1 (i.e. borrowing at risk-free rate and investing in risky portfolio) 𝜎𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 = 𝑦𝜎𝑝 (only includes risky, as risk-free has σ=0) CAL plots 𝐸(𝑟̃ ) against σ, for given values of ‘y’ o
𝐸(𝑟̃ ) = 𝑟𝑓 +
𝐸(𝑟̃𝑝 )−𝑟𝑓 𝜎𝑝
𝜎𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑
Capital Market Line (CML) o A special case of the CAL, when P=M (the “market portfolio”)
Sharpe ratio (a.k.a. ‘reward-to-volatility’ ratio)
The rate at which return increases when risk (σ) increases – the gradient of the CML 2
o
𝑆𝑝 =
So all the points on the CAL have the same Sharpe ratio 𝐸(𝑟̃𝑝 )−𝑟𝑓 𝜎𝑝
o
𝐸(𝑟̃𝑝 ) = return on risky assets only
Higher Sharpe ratio indicates better performance
Utility function
Represents an investor’s attitude to risk o Determine where along the CAL the investor will fall (a retrospective measurement, not practically useful) o Aim to maximise utility (i.e. make U as big as possible)
2 𝑈𝑖 = 𝐸(𝑟̃) 𝑖 − 𝐴𝜎𝑖
1 2
2 𝐸(𝑟̃) 𝑖 = expected return of combined (risky and risk-free) portfolio; 𝜎𝑖 = variance on combined portfolio Risk averse: A>0 Risk neutral: A = 0 (irrelevant) Risk seeking: A < 0 (irrelevant)
o o o o
Qn1. Given the following portfolio options, with a risk-free rate of 4%, which would an investor with A = 2 choose?
1 2
Ans: 𝑈𝑖 = 𝐸(𝑟̃ ) − 2𝜎𝑖 2 = 𝐸(𝑟̃ ) − 𝜎𝑖 2 Low risk: 𝑈𝑖 = 0.04 + 0.04 − 0.062 = 0.0764 Medium risk: 𝑈𝑖 = 0.06 + 0.04 − 0.122 = 0.0856 High risk: 𝑈𝑖 = 0.10 + 0.04 − 0.192 = 0.1039 Utility is maximised with the high risk portfolio, therefore the investor would invest in that one
Selecting ‘y’ so as to maximise utility 1 2
Substituting in 𝐸(𝑟̃ ) = 𝑟𝑓 + 𝑦[𝐸(𝑟̃𝑝 ) − 𝑟𝑓 ] to 𝑈𝑖 = 𝐸(𝑟̃ ) − 𝐴𝜎𝑖 2 gives:
o
1
𝑈𝑖 = 𝑟𝑓 + 𝑦[𝐸(𝑟̃𝑝 ) − 𝑟𝑓 ] − 𝐴𝜎𝑖 2
To get the maximum utility, we differentiate and solve = 0, which gives:
2
𝑦∗ =
𝐸(𝑟̃𝑝 )−𝑟𝑓 𝐴𝜎𝑝 2
=
𝑆𝑝 𝐴𝜎𝑝
Qn2. Given 𝑟𝑓 = 4%, 𝐸(𝑟̃𝑝 ) = 18% and 𝜎𝑝 = 25%, what proportion would an investor with A = 5 put into the risky portfolio? Ans: 𝑦 ∗ =
𝐸(𝑟̃𝑝 )−𝑟𝑓 𝐴𝜎𝑝 2
=
0.18−0.04 5 × 0.252
= 0.384
So they would put 38.4% of their funds in the risky portfolio, and the remaining 61.2% in the risk-free asset 𝐸(𝑟̃𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 ) = 𝑟𝑓 + 𝑦[𝐸(𝑟̃𝑝 ) − 𝑟𝑓 ] = 0.04 + 0.384[0.18 − 0.04] = 10.61% 𝜎𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 = 𝑦𝜎𝑝 = 0.384 × 0.25 = 9.6% 𝑆𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 =
0.1061−0.06 0.096
= 0.48 (which = SP, proving all the points on the CAL have the same Sharpe ratio)
Optimal portfolio choice Diversification (which risky asset, and how much in each)
Sources of risk o Market (systematic) risk 3
Table of Contents:
Returns
Page 2
Portfolio Theory
Page 2
Portfolio (Investment Fund) Performance Evaluation
Page 14
Fixed Income Valuation
Page 18
Interest Rate Swaps
Page 29
1
Returns
r1 is the return from t=0 to t=1, which is: 𝑟1 =
𝑃1 + 𝐷1 −𝑃0 𝑃0
o
𝑟̃ – unknown future return (tilde indicated variable with probability distribution) 𝑟̃ (s) – return in state ‘s’, which has probability of occurring: p(s) o E(𝑟̃ ) – expected future return = ∑ 𝑝(𝑠) 𝑟̃ (s) o r – realised (actual) return Variance (σ2) = ∑ 𝑝(𝑠) [𝑟̃ (𝑠) − 𝐸(𝑟̃ (𝑠))]2 = ∑ 𝐸[𝑟̃ − 𝐸(𝑟̃ )]2
Distribution of returns
Returns can be modelled to the ‘normal distribution’ o
𝑧̃ =
𝑟̃𝑡 −𝜇 , 𝜎
where 𝑟̃𝑡 ~𝑁(𝜇, 𝜎)
Can then calculate probabilities using a standard normal table or Excel Can approximate returns in a normal distribution by the ‘68.2, 95.4, 99.7 rule’ o NB: natural logarithm returns are closer to being accurate, but have too “fat tails” Skewness – ‘which way it leans’ o
Skew =
𝐸(𝑟̃ −𝜇)3 𝜎3
o
Positive skew – less negative tail Stock market is this (as cannot go below $0 value), meaning a normal distribution overestimates downside risk o Negative skew – less positive tail o NB: skewness decreases when considering data over time – has been tiny since 1950’s Kurtosis (K) – ‘how fat the tails are’ o Normal distribution = 3 o Platykurtic: K3 (thinner in middle, fatter tails) o
Excess kurtosis (any kurtosis over 3) =
𝐸[(𝑟̃ −𝜇)4 ] − 𝜎4
3
Average excess kurtosis approx. 0.4 since 1950’s
Portfolio Theory Capital Allocation Line (CAL) The balance between risky (‘y’ proportion) and risk-free (‘1 – y’ proportion) investments
CAL considers all the available risk-return combinations 𝐸(𝑟̃ ) = 𝑟𝑓 (1 − 𝑦) + 𝑦𝐸(𝑟̃𝑝 ) o ‘y’ can be >1 (i.e. borrowing at risk-free rate and investing in risky portfolio) 𝜎𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 = 𝑦𝜎𝑝 (only includes risky, as risk-free has σ=0) CAL plots 𝐸(𝑟̃ ) against σ, for given values of ‘y’ o
𝐸(𝑟̃ ) = 𝑟𝑓 +
𝐸(𝑟̃𝑝 )−𝑟𝑓 𝜎𝑝
𝜎𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑
Capital Market Line (CML) o A special case of the CAL, when P=M (the “market portfolio”)
Sharpe ratio (a.k.a. ‘reward-to-volatility’ ratio)
The rate at which return increases when risk (σ) increases – the gradient of the CML 2
o
𝑆𝑝 =
So all the points on the CAL have the same Sharpe ratio 𝐸(𝑟̃𝑝 )−𝑟𝑓 𝜎𝑝
o
𝐸(𝑟̃𝑝 ) = return on risky assets only
Higher Sharpe ratio indicates better performance
Utility function
Represents an investor’s attitude to risk o Determine where along the CAL the investor will fall (a retrospective measurement, not practically useful) o Aim to maximise utility (i.e. make U as big as possible)
2 𝑈𝑖 = 𝐸(𝑟̃) 𝑖 − 𝐴𝜎𝑖
1 2
2 𝐸(𝑟̃) 𝑖 = expected return of combined (risky and risk-free) portfolio; 𝜎𝑖 = variance on combined portfolio Risk averse: A>0 Risk neutral: A = 0 (irrelevant) Risk seeking: A < 0 (irrelevant)
o o o o
Qn1. Given the following portfolio options, with a risk-free rate of 4%, which would an investor with A = 2 choose?
1 2
Ans: 𝑈𝑖 = 𝐸(𝑟̃ ) − 2𝜎𝑖 2 = 𝐸(𝑟̃ ) − 𝜎𝑖 2 Low risk: 𝑈𝑖 = 0.04 + 0.04 − 0.062 = 0.0764 Medium risk: 𝑈𝑖 = 0.06 + 0.04 − 0.122 = 0.0856 High risk: 𝑈𝑖 = 0.10 + 0.04 − 0.192 = 0.1039 Utility is maximised with the high risk portfolio, therefore the investor would invest in that one
Selecting ‘y’ so as to maximise utility 1 2
Substituting in 𝐸(𝑟̃ ) = 𝑟𝑓 + 𝑦[𝐸(𝑟̃𝑝 ) − 𝑟𝑓 ] to 𝑈𝑖 = 𝐸(𝑟̃ ) − 𝐴𝜎𝑖 2 gives:
o
1
𝑈𝑖 = 𝑟𝑓 + 𝑦[𝐸(𝑟̃𝑝 ) − 𝑟𝑓 ] − 𝐴𝜎𝑖 2
To get the maximum utility, we differentiate and solve = 0, which gives:
2
𝑦∗ =
𝐸(𝑟̃𝑝 )−𝑟𝑓 𝐴𝜎𝑝 2
=
𝑆𝑝 𝐴𝜎𝑝
Qn2. Given 𝑟𝑓 = 4%, 𝐸(𝑟̃𝑝 ) = 18% and 𝜎𝑝 = 25%, what proportion would an investor with A = 5 put into the risky portfolio? Ans: 𝑦 ∗ =
𝐸(𝑟̃𝑝 )−𝑟𝑓 𝐴𝜎𝑝 2
=
0.18−0.04 5 × 0.252
= 0.384
So they would put 38.4% of their funds in the risky portfolio, and the remaining 61.2% in the risk-free asset 𝐸(𝑟̃𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 ) = 𝑟𝑓 + 𝑦[𝐸(𝑟̃𝑝 ) − 𝑟𝑓 ] = 0.04 + 0.384[0.18 − 0.04] = 10.61% 𝜎𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 = 𝑦𝜎𝑝 = 0.384 × 0.25 = 9.6% 𝑆𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 =
0.1061−0.06 0.096
= 0.48 (which = SP, proving all the points on the CAL have the same Sharpe ratio)
Optimal portfolio choice Diversification (which risky asset, and how much in each)
Sources of risk o Market (systematic) risk 3
