Today’s Agenda: Wed Sep 17, 2020
• Return and Risk Concepts
Investment Analysis &
Portfolio Management
Session 2
Two Components of Return
• Risk-free rate
– compensation for time: delayed consumption, opportunity cost
& expected inflation, estimated as risk-free rate, rf
• Risk premium
– compensation for risk, defined as risk-premium ri - rf
3
Comparing Security Returns
• Assume that investor X traded securities A & B
– X bought A for Rs 200 and sold it for Rs 240 in 1 year; A also provided a
dividend of Rs 10.
– X bought B for Rs 1000 and sold it for Rs 1100 in 3 months; B provided no
dividend.
• To compare profitability on investing in A & B
– we need to add dividend income to capital gain/loss
– we need to consider difference in amounts invested
– we need to consider difference in holding periods
• We can compare profitability by estimating holding period return
(HPR) 4
Holding Period Return
(𝑃1 − 𝑃0 +𝐷)
• Holding period return, 𝐻𝑃𝑅 =
𝑃0
– HPR on A = (240-200+10)/200 = 25% in 1 year
– HPR on B = (1100-1000+0)/1000 =10% in 3 mth, or 1/4th of a year
• Since the periods are unequal, we further need to annualise the
returns
1
• Annualised HPR = 1 + 𝐻𝑃𝑅 𝑡 −1
– on A = (1 + 0.25)1 −1 = 0.250 = 25.00%
– on B = (1 + 0.10)4 −1 = 0.4641 = 46.41%
=> B provides higher annualised holding period returns than A
6
Average Returns over Multiple Periods
• Suppose security A provides following returns (HPRs including dividends)
over 5 years:
r1 = 10% r2 = 2% r3 = 0% r1 = 8% r1 = 5%
• Average return on A:
𝑟 (10+2+0+8+5)
𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑚𝑒𝑎𝑛 𝑟𝑒𝑡𝑢𝑟𝑛, 𝑟 = = = 5%
𝑛 5 1
𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑚𝑒𝑎𝑛 𝑟𝑒𝑡𝑢𝑟𝑛 = [ 1 + 𝑟1 × 1 + 𝑟2 ×. . 1 + 𝑟5 ] −1 5
1
= 1.10 × 1.02 × 1.00 × 1.08 × 1.03 5 − 1= 0.0453 = 4.53%
• When do we use arithmetic mean & geometric mean to estimate average
returns ?
7
Arithmetic vs Geometric Mean Returns
• Arithmetic return provides a statistically unbiased estimate of
expected return for the next period based on sample of past
annual returns
− However, it provides upward biased forecasts of future returns when
compounded over multiple periods
• Geometric mean return provides a better estimate of expected
return over multiple investment periods over long horizon
– geometric mean return is always lower or equal to arithmetic mean
return (GM = AM only when returns are equal across all periods)
9
Realised (ex post ) and Expected (ex ante) Returns
• Returns may also be categorised as past or realised return (ex
post) and expected or forecasted return (ex ante).
• The ex ante return estimate may be
– unconditional: if we do not expect it to vary with economic scenarios
– conditional: if it varies with economic scenarios
• Conditional forecasts may be stated in the form of estimates
under multiple scenarios with associated probabilities.
– an expected return can be calculated as the mean estimate by using
probabilities as weights.
10
Expected Returns based on Probabilities
• An investor has the following return expectations from stock A
20 % return with probability of 50%
50% return with probability of 30%
0% return with probability of 20%
• Expected return, 𝐸 𝑟 = 𝑝𝑖 𝑟𝑖
= 0.5 × 20% + 0.3 × 50% + 0.2 × 0% = 25%
11
Nominal versus Real Returns
• Real return = (1 + nominal return)/(1 + inflation) – 1
• Real return ≈ nominal return – inflation
• Should we work with nominal or real returns?
- it is convenient to work with nominal returns as data including security
prices & financial statements is available in nominal terms
- however, since real returns should matter to the investor, the final
return estimate can be adjusted for inflation if required
- comparison of returns within a market may be done in either nominal or
real terms, as long as the same basis is consistently used
- comparison of returns across markets are more meaningful in real terms
Volatility and Risk
• Compare annual returns on 2 securities A & B over 5 years
10% 10%
8% 8%
6% 6% 6%
5% A
4% 4% 4%
B
2% 2%
0% 0%
1 2 3 4 5
• Both provide same mean return of 5% over 5 years
• However, A is riskier because its returns vary more than B & hence less
predictable
– we would be less confident that the next year’s return will be close to 5% in case
of A than in case of B
• Volatility or dispersion of returns can be measured using variance and
standard deviation, for securities whose returns are symmetrically
distributed 13
Estimating Variance & Standard Deviation
Year r of A (%) 𝒓−𝒓 (𝒓 − 𝒓)𝟐 Year r of B (%) 𝒓−𝒓 (𝒓 − 𝒓)𝟐
1 10 5 25 1 6 1 1
2 2 -3 9 2 4 -1 1
3 0 -5 25 3 4 -1 1
4 8 3 9 4 6 1 1
5 5 0 0 5 5 0 0
𝐫 of A 5 Total 68 𝐫 of B 5 Total 4
• Security A
(𝑟−𝑟)2 68
– v𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝜎 2
= = = 0.0017…note variance is in (%)2 => divide by 10,000
(𝑛−1) 4
– 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛, 𝜎 = 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 0.0017 = 4.12%
• Security B
(𝑟−𝑟)2 4
– 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝜎2 = = = 0.0001
(𝑛−1) 4
– 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛, 𝜎 = 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 0.0001 = 1.00% 14
Standard Deviation of Probabilistic Expected Returns
Scenario 𝒑𝒊 𝒓𝒊 𝒑𝒊 𝒓𝒊 𝒓𝒊 - E(𝒓𝒊 ) 𝒑𝒊 (𝒓𝒊 − 𝑬(𝒓𝒊 ))𝟐
1 0.50 20% 10 -5 12.5
2 0.30 50% 15 25 187.5
3 0.20 0% 0 -25 125
Mean, E(𝒓𝒊 ) 25 Variance, 𝜎 2 325
• 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝜎 2 = 𝒑𝒊 (𝒓𝒊 − 𝑬(𝒓𝒊 ))𝟐 = 0.0325
• 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝜎 = 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 18.03%
15
Annualising Standard Deviation
• If returns are serially independent (uncorrelated from one month
to another),
– to annualise monthly variance, multiply by 12
– to annualise monthly standard deviation, multiply by √12
• In general, if σ is given for t years (for ex. 1 month = 1/12 years)
– annualised σ2 = σ2/t, ex. σ2/(1/12) = σ2 x 12
– annualised σ = σ/√t, ex. σ/√(1/12) = σ x √12
16
Risk-Return Trade-off
• On a standalone basis
• Compare 3 securities: - security A appears to be dominated by
– A: E(rA) = 5%, σA = 4.1% both C (in terms of returns) and B (in
– B: E(rB) = 5%, σB = 1% terms of risk)
- it appears that an investor can choose
– C: E(rC) = 7%, σC = 4.1%
between B & C based on risk tolerance
versus targetted return
• Which security is better?
• ..but individual securities do not
8% constitute the complete
6%
C
opportunity set of investments
B A
• The investor can also invest in
E(r)
4%
2%
portfolios
- portfolios containing A, B & C in
0% varying proportions offer more choices
0% 1% 2% 3% 4% 5% - some portfolios may offer better
σ reward-to-risk tradeoff than the
17
individual securities
