19th letter to friends of Brunei [Portfolio Management (Part 5)]

The 3rd commonly used risk-adjusted portfolio return measure is the Jenson measure. This measure, denoted by α (alpha), compares the actual return of a portfolio against its expected return. (As we have learnt earlier, the expected return of a portfolio can be found using the CAPM model)

α = Actual Portfolio Return – Expected Portfolio Return

= Actual Portfolio Return - [Rf + β (Rm – Rf)]

Let's do an example: A-fund has a beta of 1.5 in STI. In 2008, A-fund recorded a 18% return, and STI recorded a 10% return. Calculate its Jenson measure. (assuming risk free return = 4%)

α = Actual Portfolio Return (18%) – Expected Portfolio Return [4% + 1.5 ( 10% - 4%)]
= 18% - [4% + 9%]
= +5%

From this α (of +5%), A-fund manager is said to have outperformed the market.

(A positive α means a portfolio manager outperforms the market, a negative α means the portfolio manager underperforms the market)

The hedge fund managers always use α to denote how good (or bad) they are (compared to each other).

18th letter to friends of Brunei [Portfolio Management (Part 4)]

Another commonly used risk-adjusted portfolio return measurement is the Treynor Ratio.

Treynor ratio calculates the excess returns per unit of systematic risk.

(whilst Sharpe ratio calculates the excess returns per unit of total portfolio risk; i.e. unsystematic AND systematic risk of a portfolio)

Consider a portfolio A,

Treynor A = [Rp - R f] / β p

Rp = Portfolio return
Rf = Risk free asset return
βp = Portfolio beta, which represents the systematic risk.

Let's do an example:

Portfolio A: [Return 12%] ; [Beta 1.2]
Portfolio B: [Return 14%] ; [Beta 1.1]
Portfolio C: [Return 16%] ; [Beta 1.4]

Assuming T-bill rate is 7%.

Calculate and rank the portfolios using Treynor ratio.

Treynor A = R A - R T-bill / Beta A = 12% - 7% / 1.2 = 4.17


Treynor B = 14% - 7% / 1.1 = 6.36


Treynor C = 16% - 7% / 1.4 = 6.43


Ranking = Portfolio C (rank = 1) , Portfolio B (rank = 2), Portfolio A (rank = 3)

i.e. Portfolio C produces the most return per unit of systematic risk.

17th letter to friends of Brunei [Portfolio Management (Part 3)]

Having assembled a porfolio of assets, we need to have a system of evaluating the performance. We can use:

A. Non-Risk Adjusted Measurements

1. Net Asset Value(NAV): a quick measurement on the net asset value(assets - liabilities) per unit of the portfolio;

2. Total return: for a quick measuring of a return in dollar amount after an investment period;

3. Arithmetic Mean: measures the average return of the portfolio;

4. Geometric Mean: measures the annual compounded return of the portfolio; &

5. More complex measurements like Dollar-Weighted Rate of Return (DWRR) and Time-Weighted Rate of Return (TWRR).

B. Risk-Adjusted Portfolio Returns

I will illustrate more on these type of portfolio returns, which is commonly cited in major business papers on investment value tables.

1. Sharpe Ratio

The Sharpe Ratio measures the excess return (or risk premium: Portfolio Return[Rp] - Risk free Return [R f]) per unit of Portfolio Total Risk (or portfolio's standard deviation: σp).

Sharpe Ratio = (Rp - Rf )/σp

On 1st glance, the following funds' ranking look like this:

Rank 1: China Growth Fund(return: 15%, risk: 16%)

Rank 2: India Growth Fund(return:13%, risk: 18%)

Rank 3: Emerging Market Growth Fund (return 12%, risk: 11%)

The ranking would be different if you use Sharpe Ratio (assuming risk-free return is 7%), the ranking is:

Rank 1: China Growth Fund: Sharpe Ratio: (15% - 7%) /16% = 0.5

Rank 2: Emerging Market Growth Fund: Sharpe Ratio: (12% - 7%) /11% = 0.45

Rank 3: India Growth Fund: Sharpe Ratio: (13% - 7%) /18% = 0.33

Above example shows that by taking risk into consideration, the decision on which fund to invest can be different.

16th letter to friends of Brunei [Portfolio Management (Part 3)]

Our study of portfolio management will lead us to one common sense question: What if I can invest in a risk-free asset, how much more (risk premium) should I get if I were to invest in a risky (of various degree) assets. In US, the risk-free asset is usually the T-bill issued by US government. In Singapore, it can refer to FD rate, CPF rate or Singapore government 10 year bond.

This question leads to a development of an important equation (known as CAPM: Captial Asset Pricing Model):

Expected Return (of a stock) = Risk free Return (Rf) + Risk Premium

where Risk premium = β (Rm – Rf)

[β(beta) = sensitivity of a stock to a market index; Rm = Expected Market Return]

Let's do an example: Mr. Tan is considering investing in a stock of DBS. Tan expects market return for bank's stock to be 14%. DBS's beta is 1.4 (on STI), a risk free Singapore government 10 years bond return is 3%. Calculate the expected return on DBS stock.

E (DBS) = R(f) + β (Rm – Rf )

E(DBS) = 3% + 1.4 (14% - 3%) = 18.4%

Another example: The return of DBS is 16%, the market return for bank's stock is 14.2%, risk-free S'pore 10 year bond is 3%. Determine the beta for DBS.

a. 0.65

b. 0.91

c. 1.42

d. 1.16

* Working: E (DBS) = R (f) +β (Rm – R f)

16% = 3% + β (14.2% - 3%)

(16% -3%) / (11.2%) = β

β = 1.16

I.e. When STI rises (or drops) by 1 %, DBS stock will rise (or drop) by 16% over and above STI.

15th letter to friends of Brunei [Portfolio Management(Part 2)]

Calculating the risk (or portfolio standard deviation) is not as straight forward as return (read my immediate previous blog entry). This is because we have to consider on important factor in the equation: the expected correlation of each asset with each of the other assets.

Correlation measures the extent to which the returns on 2 assets move together. If 2 assets move up and down together, we say they are positively correlated. If they move in opposite directions, we say they are negatvely correlated. If there are no particular relationship between the 2 assets, we say they are uncorrelated. Therefore, correlation factor between 2 assets lies between -1 to +1

Let me illustrate using a 2 assets portfolio example.

Stock X

Weightage(W(x)): 20%

Standard Deviation(risk R(x)): 15%

Stock Y

Weightage(W(y)): 80%

Standard Deviation (risk R(y)): 30%

Correlation factor (Corr(x,y)) between X & Y = +0.5

The portfolio risk [P(r)] can be calculated as follows:

P(r) = √ [W(x)2 R(x)2 + W(y)2 R(y)2 + 2 W(x)W(y)R(x)R(y)(corr(x,y))]

= √ [(0.2)2 (15)2 + (0.8)2 (30)2 + 2 (0.2)(0.8)(15)(30)(+0.5)]

= √ [(0.04)(225) + (0.64)(900) + 72]

=√ [9 + 576 + 72]

=√ [657]

=25.63%

(In real life, this is done through computer spreadsheets.)

14th letter to friends of Brunei [Portfolio Management (Part 1)]

When you have 2 or more assets in an investment plan, this is known as a "portfolio", and the calculation of its returns and risks has some variations as compared to single asset.

1. Calculating portfolio return

The return on a portfolio of assets is simply the weighted average return.

For example: Mary has a portfolio of 4 common stocks with the following market values and returns.

Stock [Market Value] (Stock Return)

A [$10k] (10 %)

B [$ 20k] (14 %)

C [$ 30k] (16 %)

D [$ 40k] (15 %)

Total $100k

Mary's portfolio return is:

[10k/100k](10%) + [20k/100k](14%) + [30k/100k](16%) + [40k/100k](15%)

= [0.1](10%) + [0.2](14%) + [0.3](16%) + [0.4](15%) = 14.6%

Let's try another example: The assets X,Y,Z have the following return statistics:

Assets (Returns)

X (18% )

Y (22%)

Z (26%)

What's the expected return of a portftolio containing 25% of X, 50% of Y, and 25% of Z?

a. 17.25%

b. 22.00%

c. 16.34%

d. 19.72%

You should have no problem in calculating portfolio returns after going through above two examples.

Cheers.

*Working for 2nd example:

Expected Return = (0.25)(18) + (0.5)(22) + (0.25)(26) = 4.5+11+6.5 = 22%

Therefore, the answer is b.