Chapter XVII (Word format).doc

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CHAPTER XVII

ASSET ALLOCATION AND PORTFOLIO MANAGEMENT

I. An Introduction to Asset Allocation

1.A Passive approach

• Passive fund manager reproduces a market index of all securities.

Passive manager = Lazy manager.Manager assumes zero forecast ability. Popular Indexes to follow: MSCI World/EAFE Index.

S&P 500.

• Q: Is this a good managerial approach?

TABLE XVII.1% Actively Managed Int. Funds Outperforming MSCI

Indexes (1987-1997)

Funds Investing in

3-year 5-year 10-year

Japanese Securities

62 % 37% 25%

European Securities

10% 26% 0%

• Foreign markets tend to be independent of each other: Different international asset allocations yield

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different returns.

• Passive International Managers face three decisions:

i. Country weights.ii. Hedging strategies.iii. International market index.

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1.B Active approach

• Managers try to time markets and switch currencies switching.

Active managers “have” forecasting ability.

• An active strategy can be broken down into three parts:

(1) Asset allocation (bonds, stocks, real state, etc.).(2) Security selection (specific bonds or stocks.)(3) Market timing (when to buy and sell).

• Two approaches to active international management:

i. Top-down approach1. Choice of markets and currencies

1.a Selection among asset classes (stocks, bonds, or cash).

2. Then, selection of the best securities available.

ii. Bottom-up approach1. Choice of individual stocks, regardless of country of

origin.2. Choice of markets and currencies is implicitly made

in (1).

Empirical Evidence:

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i. Securities within a single market tend to move together, but national markets and currencies do not.

ii. The performance of international money managers is attributable to asset allocation, not security selection.

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II. Evaluation of the Asset Allocation Process

2.B International Performance Analysis (IPA)

• IPA measures the return on a portfolio or a portfolio segment.

• IPA calculations are usually done on a monthly or quarterly basis.

• Tournament evaluation: good managers beat the competition.

Popular measures to beat: 1. MSCI World or EAFE Index.2. U.S. S&P 500 3. Mean return of managed portfolios.

● Issues:

1) Calculation of returns2) Risk considerations

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2.B.1 Calculating a rate of return (rT)

Three methods are very common: 1. Time-weighted rate of return (TWR)2. Internal rate of return (IRR)3. Money-weighted rate of return (MWR)

Notation:Vt: value of portfolio at time t.Ct+k: cash outflow at time t+k.

When no cash flows occur during period T all methods give the same rT.

rT = (Vt+T - Vt) / Vt. (Ct=0)

When cash flows occur the three common methods might diverge.

Consider the following example:

a. Vt = 100 (initial value)b. Ct+k = 50, k=30 days c. Vt+T = 60, T=365 days (end of period value).

Change in value = Vt+T + Ct+K - Vt = 10.

Now, methods differ on how to calculate the rate of return, rT.

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• A common approach is the MWR, which has the following formula:

MWR = (Vt+T + Ct+k - Vt)/(Vt - .5Ct+k)= (60+50-100)/(100-25)= 13.33%

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MWR does not take into account the exact timing of the cash flows: it assumes that they take place in the middle of the period.

To avoid this problem, we have modified MWR*:

MWR* = (Vt+T + Ct+k - Vt)/(Vt - ((365-t)/365)xCt+k)= (60+50-100)/(100-(335/365)x50)= 18.48%

If several cash flows take place, each is weighted accordingly.

• The IRR is the discount rate that equals Vt to the sum of the discounted cash flows including the Vt+T. In our example,

Vt = [Ct+k / (1+r)t+k/365] + [Vt+T / (1+r)].

In this example, the IRR = 18.90%

• TWR is calculated independently of cash flows.

TWR measures the performance that would have been realized had the same capital been under management over the period.

TWR calculations require knowledge of Vt before a cash flow occur.

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The first period rate of return, before the first cash withdrawal at t+k, rt+k-1 is given by:

1 + rt+k-1 = (Vt+k-1/Vt).

The rate of return for the second period, that is, from t+k to t+T, is given by:

1 + rt+T-k = (Vt+T/(Vt+k-1 - Ct+k).

The total rate of return, rt+T, is:

1+rt+T = (1+rt+k-1)(1+rt+T-k).

In the above example, assume Vt-k-1 = 95, then

1 + rt+k-1 = 95/100 = 0.95 (rt+k-1 = -5.00%)

1 + rt+T-k = 60/45 = 1.33 (rt+T-k = 33.33%)

1+rt+T = 1.27 (rt+T = 26.66%).

The various methods yield different results: from 13.33% to 26.66%

If the required information is available, TWR should be used.

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Example XVII.5: Situation:A U.S. manager manages USD 1,000,000.Cap on Japanese investments: 10%.USD Investment in the Nikkei Japanese Index: USD 100,000.

After two weeks the Nikkei rises 30%.

Japanese segment is above the cap transfer out USD 30,000.

Over the next two weeks the Nikkei loses 30%.

The following table summarizes the performance of the manager:

t=1 t=15 t=30 TWRMWR

Index 100 130 91 -9% -9%Portfolio 100

100 70 -9% 0%

Calculations for the portfolio TWR and MWR:

1 + TWR = (130/100).(70/100) = 0.91 (TWR = -9%)

MWR = (70+30-100)/(100-.5x(30)) = 0. ¶

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2.B.2 Designing an IPA

• Idea:

An active manager to be considered good has to beat a passive manager.

Active managers have to show: Better asset selectionBetter security selection

Q: Can the active manager beat the passive allocation of the MSCI?

Q: Can the active manager select local stocks or bonds and beat the local MSCI index?

• Basic steps:

A portfolio is broken down into various segments:stocksbondscash....currency

Each homogeneous segment is valued separately in its local currency.

Each homogeneous segment is compared to a passive standard.

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• IPA

The base currency rate of return is easily derived by translating all prices into the base currency D at exchange rate Sj:

rjD = (PjtSj

t + DjtSj

t - Pjt-1Sj

t-1) /(Pjt-1Sj

t-1),

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Let pj, dj and sj be the rate of change (percent) of Pj, Dj and Sj.

After some algebra:

rjD = pj + dj + sj (1+pj+dj) = pj + dj + cj.

Over period t, the total return on account r is computed in the base currency as follows:

r = Σj ωj rjD = Σj ωj (pj + dj + cj) =

= Σj ωj pj + Σj ωj dj + Σj ωj cj.

Capital gain Yield Currency = component + component + component.

Example XVII.6: We break down the total return of an account:

Total Return12.95

Capital gains (losses)11.33Fixed income 0.84Equity and gold 10.49

Market return 9.24Indiv. stocks selection 1.25

Currency movements 0.95Fixed income 0.23

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Equity and gold 0.72

Yield 0.67Fixed income 0.41Equity and gold 0.27

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The relative performance of a manager may be measured by making two comparisons:

1.- Security selection ability: determined by isolating the local market return of his/her account.

Let Ij be the return in local currency of the market index of segment j.

rjD = Ij + (pj - Ij) + dj + cj.

The total portfolio return may be written as:

r = Σjωj Ij + Σjωj (pj-Ij) + Σjωj dj + Σjωj cj =

Market Security Yield Currency= return + selection + component +component

component component

Example XVII.7: Example XVII.6 breaks down the performance of an equity investment.

Capital gain = Market return + security selection

10.49% = 9.24% + 1.25%. ¶

2.- A manager's performance is evaluated relative to a standard (I*).

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The objective is to measure (r - I*),

where I* is the return on an international index (MSCI World Index).

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Let IjD be the return on market index j, translated into base currency D. IjD = Ij + Ej

Ej: currency component of the index return in base currency. That is,

Ej = sj (1+Ij).

Let ωj* be the weight of market j in the international index chosen as a standard (these weights are known).

In base currency, the return on this international index equals

I* = Σ ωj* IjD.

Now, we can write r as:

r = Σjωj*IjD + Σj(ωj-ωj*)Ij + Σj(ωjcj-ωj*Ej) + Σjωj dj + Σjωj (pj-Ij)

Now, we can estimate the contribution to total performance of any deviation from the standard asset allocation (ωj - ωj*).

• Relative performance of a manager, r-I*, is the result of two factors:

i. An asset allocation different from that of the index: ωj

ωj*.

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ii. Superior Security selection (pj - Ij).

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2.C Risk

Final step: analyze the risk borne by the manager.

Traditional measure of total risk of an account: SD of its rate of return.

GRAPH XVII.1

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2.C.1 Risk Adjusted Performance

It seems attractive to use a single number that takes into account both performance and risk. Popular measures:

Reward to variability (Sharpe ratio): RVAR = Excess Return/SD.

Reward to volatility (Treynor ratio): RVOL = Excess Return/Beta.

Risk-adjusted ROC (BT): RAROC = Return/Capital-at-risk.

Jensen measure: Based on the Mean-Variance Efficiency of the Market Portfolio (recall the CAPM).

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2.C.1.i RVAR and RVOL

Measures:RVARi = (ri - rf) / σi.

RVOLi = (ri - rf) / ßi.

Example XVII.10: A U.S. investor considers foreign stock markets:

Market ri (%) σi ßWORLD RVAR RVOL

France .07 1.02 .89 .0196 .0225

Panama .15 2.20 .55 .0454 .1818

Korea .13 1.44 .51 .0556 .1569

U.S. .10 1.08 1.02 .0463 .0490

WORLD .12 1.01 1.00 .0693 .0700

rF .05

Using RVAR and RVOL, we can rank the foreign markets as follows:

Rank RVAR RVOL1 Korea Panama2 U.S. Korea3 Panama U.S.4 France France

Note: It is rare, but RVAR and RVOL can give different rankings. ¶

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2.C.1.iii Bankers Trust risk adjustment method

• Bankers Trust uses a modification of RVAR:

Risk-adjusted return on capital (RAROC) system.

• RAROC adjusts returns taking into account the capital at risk.

Capital at Risk: amount of capital needed to cover 99 percent of the maximum expected loss over a year.

Key: BT needs to hold enough cash to cover 99% of possible losses.

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Situation: Two traders I and II, dealing in different markets.

Trader I position (futures stock indices):Annualized profits: USD 3.3 million.Position: USD 45 million.Volatility: 21%

Trader II position (spot exchange rates):Annualized profits: USD 3 million.Position: USD 58 million. Volatility: 14%

Recall: BT needs to hold enough cash to cover 99% of possible losses.

(1) Calculate the worst possible loss in a 99% Confidence Interval.

Assuming a normal distribution: The 1% lower tail of the distribution lies 2.33 standard deviations below the mean.

BT determines the worst possible loss for both traders:

Trader I: 2.33 x 0.21 x USD 45,000,000 = USD 22,018,500.

Trader II:2.33 x 0.14 x USD 58,000,000 = USD 18,919,600.

(2) Calculate RAROC:

Trader I: RAROC = USD 3,300,000/USD 22,018,500

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= .1499.

Trader II:RAROC = USD 3,000,000/USD 18,919,600 = .1586.

Conclusion: Once adjusted for risk, Trader II provided a better return.

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2.C.1.iv Estimating the Jensen measure

Goal: measure the performance of managed fund i, with a return ri.

CAPM implies

(ri - rf) = αi + ßi (rm-rf) + εi.

If rm is the return of a market portfolio, then αi isequal to zero.

• Jensen measure: regression estimate of αi.

If tαi> 2, then the managed fund has a superior risk-adjusted performance than the market.

The Jensen measure reflects the selectivity ability of a manager.

• Sometimes, in the market regression we introduce other factors:

multi-factor market model.

Example: Analysts might include a portfolio of riskless assets denominated in each currency on the portfolio, with a return of rc,

(ri - rf) = αi + ßim (rm-rf) + ßic (rc-rf) + εi.

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Example XVII.13: Jensen measure for 14 US international funds.

TABLE XVII.3Jensen Measures (αi) and Tests of Performance (t-stats)

Variables in the regression: rm rm & rc

rm & D87

Alliance -0.191 -0.191 0.123(.476) (.480) (0.308)

GT Pacific -0.476 -0.465 0.005(.536) (.536) (0.009)

Kemper -0.182 -0.184 0.208(.471) (.486) (.563)

Keystone -0.389 -0.390 -0.280(1.420) (1.465) (.990)

Merrill 0.042 0.040 0.407(.088) (.085) (.862)

Oppenheimer -0.486 -0.486 -0.164(1.240) (1.232) (.424)

Price -0.083 -0.085 0.132(.259) (.277) (.410)

Putnam -0.030 -0.030 0.129(.110) (.112) (.467)

Scudder -0.150 -0.151 0.192(.453) (.468) (.694)

Sogen 0.198 0.199 0.484(.849) (.869) (2.285)

Templeton Global -0.054 -0.050 0.392(.124) (.164) (1.211)

Transatlantic -0.543 -0.544 -0.308(1.378) (1.446) (.771)

United 0.095 0.094 0.297

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(.419) (.416) (1.351)Vanguard 0.281 0.278 0.387

(.756) (.782) (1.005)

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Measured by the Jensen measure, the poor performance is surprising.

Possible explanation: Inclusion of an outlier (Crash of October 87).

One approach to separate the effects of a specific outlier in a regression is to include in the regression a Dummy variable.

Example XVII.14: For the mutual funds in Example V.8, we ran the following regression,

(ri - rf) = αi + ßim (rm-rf) + δi D87 + εi,

where D87 = 1 for t=October 1987. = 0 otherwise.

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3.A.1 Formation of Optimal Portfolios (OP)

● Approach based on a single-index model (like CAPM).

Using the RVOL to rank assets, we derive a number C*, a cut-off rate.

Very simple rules: i) Rank assets according to their RVOL from highest

to lowest.ii) OP: Invest in all stock for which RVOLi > C*.

● Calculation of C* (the cut-off rate)

Suppose we have a value for C*, Ci.

Assume that i securities belong to the optimal portfolio and calculate Ci.

If an asset belongs to the OP, all higher ranked assets also belong in OP.

It can be shown that for a portfolio of i asset C i is given by:

Ci = Cnum / Cden, Cnum = m

2 j=1I(rj - rf) (j/j

2),Cden = 1 + m

2 j=1I (j

2/j2),

where m2: variance of market index and j

2: asset i’s unsystematic risk.

Steps:

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1) Compute Ci (=Cnum / Cden) as if the first ranked asset in OP (i=1). 2) Compare C1 to RVOL1. If C1>RVOLi, then continue.3) Compute Ci as if the first and second ranked assets were in OP (i=2).4) Compare C2 to RVOL2. If C2 >RVOL2, then continue.....Stop the first time an asset i, Ci < RVOL . Then, Ci = C*

Example XVII.11: Mr. Krang wants to form an OP with foreign stocks:

Market (rI-rf) ßWORLD RVOL 2i (ri-

rf)ßi/2i

2mßi

2/2i Ci

Panama .10 .55 .1818 4.5314 .01214 0.06810 0.012

Korea .08 .51 .1569 1.8083 .02256 0.14674 0.029

U.S. .05 1.02 .0490 0.1051 .48525 10.09812

0.047

France .02 .89 .0225 0.2324 .07659 3.47686 0.041

Some calculations for Korea:

A. 2,Kor (From the Review Chapter, recall that 2

i = ßi2 2

m

+ 2i.)

2,Kor = (1.44)2 - (.51)2 (1.01)2 = 1.8083.

B. CKor

CKor = (1.01)2 [.01214 + .02256]/[ 1 + (0.06810 + 0.14674)] = 0.029.

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Now, we compare the Ci coefficients with the RVOLi. That is,

RVOLPan = .1818 > CPan = 0.012 (Panama should be included).RVOLKor = .1569 > CKor = 0.029 (Korea should be included).RVOLUS = .0490 > CUS = 0.047 (U.S. should be included).RVOLFra = .0225 < CFra = 0.041 (France should not be included).

From the above calculations, C*= CUS = .047. Then, Mr. Krang portfolio will include Panama, Korea, and the U.S. ¶