Copyright © 2018 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the Web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com.
B
F
THE BEHAVIORAL FINANCE PERSPECTIVE
T
B
F
P
Subjective Expected Utility E(U) = ∑ u(x i )p x( i ) Utility Calculation (Prospect Theory) U =w p( v1x) ( w +v x () 2 1 )p
()w2p + v x () n
() n
where: U = utility x = a particular outcome p = probability of x v = value of x w = probability-weighting function for outcome x; accounts for tendency to overreact to low probability events and underreact to other events
2
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P
W
M
MANAGING INDIVIDUAL INVESTOR PORTFOLIOS
M
I
I
P
Sharpe Ratio
Sharpe ratio is:
(expected return – risk-free rate) expected standard deviation
Capital gains tax payable = Price appreciation
× tCG ×
turnover rate
where: tCG = capital gains tax rate
Buy tax-free bonds when Rtax-free > Rtaxable × (1 − t) where: R = return t = applicable tax rate
4
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TAXES AND PRIVATE WEALTH MANAGEMENT IN A GLOBAL CONTEXT
T
P
W
M
G
C
Annual Accrual with a Single, Uniform Tax Rate rafter-tax
= rpre-tax (1 − tI ) n
FVIFpre-tax
= (1 +r
pre-tax
)
FVIFafter-tax
= [1r+
pre-tax t1
( − I )]n
where: = Future value interest (i.e., accumulation) factor = return t = tax rate n = number of periods
FVIF r
Deferral Method with a Sin gle, Uniform Tax Rate (Capital Gain) FVIFCG
= +(1 r CG)( −n t1)+
t
CG
CG
Taxable Gains When the Cost Basis Differs from Current Value Taxable gain = VT − Cost basis where: VT = terminal value Cost basis = amount paid for an asset
FVIFCG
= (1 +
rpre‐tax)
n
(1 − tCG) + tCGB
where: B
V0
=
Cost basis V0
= value of an asset when purchased
Accumulation Factor with Annual Wealth Tax
FVIFW
= (1 +r
)( − )
pre-tax t1
n
W
Effective Annual After‐Tax Return in a Blended Tax Regime
= −T (1P −t I I Pt−DDP tCGCG
r* r
)
where: P = proportion of return from income, dividends, and realized capital gains during the period
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5
TAXES AND PRIVATE WEALTH MANAGEMENT IN A GLOBAL CONTEXT
Effective Capital Gains Tax Rate
1 − PI − PD − PCG = tCG 1 − PI t −P tD D P− tCG CG
T*
The after-tax future value multiplier under this blended tax regime then becomes: FVIFafter-tax
= (1 +r
n (1+ *)− T
−
T *)t
−*B
CG
(1
)
If the portfolio is non‐dividend paying equity securities with no turnover (i.e., PD = PI = 0 and PCG = 1) held to the end of the horizon, the formula reduces to: FVIFafter-tax
= (−1 r )−n (t1 + CG t )
CG
If the portfolio is non‐dividend paying equity securities with 100 percent turnover and taxed annually, the formula reduces to:
FVIFafter-tax
= 1 +r (1t −
CG
)
n
Accrual Equivalent Return
Vn
= V0 (1 + rAE )n V
rAE
= n Vn − 1 0
where: Vn = value after
n
compounding periods
Accrual Equivalent Tax Rate rAE
= r (1 − TAE )
TAE
= 1−
rAE r
Tax‐Deferred Accounts FVIFTDA
6
= (1 +r)( n )1tt− B CG
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TAXES AND PRIVATE WEALTH MANAGEMENT IN A GLOBAL CONTEXT
Tax‐Exempt Accounts FVIFTEA
= (1 +r )n
FVIFTDA
=FVIFTEA
(1 −t )
Value Formula for a Tax Exempt Account
Vn
= V0 (1 − t0)(1 + r)n
Value Formula for a Tax‐Deferred Account n
Vn = V0(1 + r)
(1 − tn)
The Investor’s After‐Tax Risk
AT
= (1 −t )
where: σ = standard deviation of returns Ratio of Long-Term Capital Gains to Short-Term Capital Gains n
VLTG VSTG
= V0 (1 + r ) (1 − t LTGn) V0 1 + r (1 − tSTG )
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7
DOMESTIC ESTATE PLANNING: SOME BASIC CONCEPTS
D
E
P
:S
B
C
Core Capital kCore = PV of current lifestyle spending + emergency reserve
PV of Spending Need n
Liability0 =
∑
Expected spending (1 + r )t
t =0 n
=∑
p(Survival t ) × Spendingt
(1 + r )t
t =0
where: r = real risk-free rate
Joint Survival Probability Calculation p(Survival t ,C 1,C 2 ) = p(C1 ) + p(C 2 ) − p(C1 ) p(C 2 )
where: C1 = First spouse survives C2 = Second spouse survives Excess Capital Assets = House + Investments + Net employment capital Liabilities
=
Mortgage + Current lifestyle + Education needs + Retirement needs
KExcess = Assets − Liabilities
Relative Value of Tax-free Gifts
RVTax- free Gift =
FVGift FVBequest
n
=
1 + rg (1 − tig ) n [1 + re (1 − tie )(] 1 − Te )
where: FV = future value of the gift or bequest to the recipient n = expected number of years until donor’s death, at which time bequest transfers to recipient r = pre-tax returns to the gift recipient g or estate making the gift e ratetax onthat investments applies to gift g or estate making the gift e Tet = = tax estate applies tothat asset transfers at recipient donor’s death
8
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DOMESTIC ESTATE PLANNING: SOME BASIC CONCEPTS
Relative Value of Gifts Taxable to Recipient
RVRecipient Taxable Gift =
FVGift FVBequest
n
1 + rg (1 − tig ) (1 − Tg ) = n [1 + re (1 − tie )(] 1 − Te )
where: Tg = gift tax rate that applies to recipient Relative Value of Gifts Taxable to Donor But Not to Recipient
RVTaxableGift =
FVCharitable Gift FVBequest
=
n 1 + r(1 t) ig1 T(− TTg +ge/) g e × g − n [1 + re(1 − t)ie (1] −) Te
When the donor pays the gift tax and the recipient does not pay any tax, the rightmost numerator term in parentheses indicates the equivalent of a partial gift tax credit from reducing the estate by the amount of the gift. This formula assumes r g = re and tig = tie. Relative Value of Charitable Gratuitous Transfers
RVCharitable Gift =
FVCharitable Gift FVBequest
n
=
+(1 +rg )nT+ −oir [1t e −(1 iet ) ] (1 n
[1 + re (1 − tie )]
e)
(1 − Te )
where: Toi = tax on ordinary income (donor can increase the charitable gift amount)
Tax Code Relief Credit Method tCM =Max (tRC, tSC)
where: tRC = applicable tax rate in the residence country tSC = applicable tax rate in the source country Exemption Method t EM = t SC
Deduction Method t DM = t
t RC
+t SC t −
RC SC
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9
RISK MANAGEMENT FOR INDIVIDUALS
R
M
I
Human Capital
Human capital is calculated as follows: N
HC0
=
∑
p(st ) wt −1 (1 + gt )
(1 + r f + y)t
t 1 =
where: p(st) = probability of survival during a period, t wt-1 = income from employment in the previous year, t – 1 N = length of worklife in years rf = the risk-free rate y = an adjustment to rf for earnings volatility A Framework for Individual Risk Management
The formula for calculating the mortality-weighted net present value of the pension: N
mNPV0
=
p(s )b
∑ (1+t r )tt t 1 =
where: bt the future expected vested benefit p(st) the probability of surviving until year t r a discount rate reflecting higher required return for riskier benefit payments as well as whether nominal or real terms =
=
=
Gross and Net Life Insurance Premium Gross premium = Net Premium + Load representing insurance company overhead Net premium =
E (VDB )
1 + rp
E (VDB ) = DB × [1 − P ( St ) ]
where: E(VDB) = Expected value of the death benefit rP = return on the insurance company’s portfolio DB = death benefit P(St) = Probability of survival in period t
10
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RISK MANAGEMENT FOR INDIVIDUALS
Income Yield for an Immediate Fixed Annuity
YIncome =
CFAnnual P0
where: CFAnnual = guaranteed annual payment P0 = price of the immediate fixed annuity Policy Reserve Policy reserve = PV of future benefits – PV future net premium
Net Payment Cost Index
Net payment cost index =
Interest-adjusted annual payment cost
100 Interest-adjustment annual payment cost = Annuity due (20-year insurance cost, 5%, 20 years) 20-year insurance cost = FV annuity due (Premium, 5%, 20 years) –FV ordinary annuity (projected annual dividend, 5%, 20 years)
Note: Assumes policy owner will die at the end of the 20-year period. Surrender Cost Index
Surrender cost index =
Interest-adjusted annual surrender cost Policy face value/1000
Interest-adjustment annual surrender cost = Annuity due (20-year insurance cost, 5%, 20 years) 20-year insurance cost = FV annuity due (Premium, 5%, 20 years) –FV ordinary annuity (projected annual dividend, 5%, 20 years) –20-year projected cash value
Note: Assumes policy owner will receive projected cash value by surrendering the policy at the end of the period. Hint: The only differences in surrender cost index and net payment cost index are highlighted in the surrender cost formulas.
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11
RISK MANAGEMENT FOR INDIVIDUALS
Human Life Value: Growing Income Replacement (Life Insurance Need) VHL =P Va nnuityd ue( Y0,pretax, i,2 0y ears) i=
1+ r 1+ g
−1
where: VHL = human life value; i.e., amount of insurance required to replace insured’s income tax contribution Y0,pretax = The pretax income at time 0 required to replace the insured’s posttax contribution i = required return adjusted for a growing income r = return on investments g = growth rate of income
Note: Taxation of life insurance proceeds and annual annuities formed from life insurance proceeds differs by jurisdiction and should be considered in calculating pre-tax income replacement.
12
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P
M I
I
MANAGING INSTITUTIONAL INVESTOR PORTFOLIOS
M
I
I
P
Simple Spending Rule Spendingt = Spending rate × Ending market valuet−1 Rolling Three-year Average Spending Rule Spendingt = Spending rate × 1⁄3 [Ending market valuet−3 + Ending market valuet−2 + Ending market valuet−1] Geometric Smoothing Rule Spendingt = Smoothing rate × [Spendingt−1 × (1 + Inflationt−1)] + [(1 − Smoothing rate) × (Spending rate × Beginning market valuet−1)] Leverage-Adjusted Duration Gap Leverage-adjusted duration gap k
=
DA
−
kDL
L =
A
where: k = ratio of liabilities to assets, both at market value
14
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A
E P
M
A
CAPITAL MARKET EXPECTATIONS
C
M
E
Volatility Clustering 2 t
2 t −1
=
(1+ )− t2
where: σ2 = volatility β = decay factor (i.e., effect of prior volatility on future volatility) ε = error term Multifactor Regression Models Ri
= i+
b1F+ b2 F +…+ 1 2
bk+Fk
i
where: Fk = return to factor k bk = asset i’s return sensitivity to factork Quantitative Methods: Discounted Cash Flow Models ∞
V0
=∑
CFt
t =1 (1
+ r )t
where: CFt = cash flow in period t r = required return on investment
Dividend Discount Model (DDM)
Gordon (Constant) Growth Model
P0
=
D1 re
−g
=
D0 (1 + g) re
−g
where: P0 = current justified price D = dividend (in period specified by subscript t) g = long-run average growth rate re = required return on equity investments
16
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CAPITAL MARKET EXPECTATIONS
Expected Return, E(R) E ( R) =
D0 (1 + g) P0
+g=
D1 P0
+g
Grinold‐Kroner Model D E ( R) ≈ − P
∆%+ S INFL + + ∆gr
% PE
where: D/P = expected dividend yield S = number of shares outstanding (Note: % change in S is the opposite of the repurchase yield) INFL = inflation rate gr = real earnings growth PE = price-earnings ratio Build‐Up Approach E ( Ri ) = + RF
RP +1
+RP +2
...
RPk
where: RF = nominal risk-free rate interest rate RPk = risk premium k Fixed‐Income Premiums E ( Rb ) = +rrF
+RPINFL+
RPDefault +
RP +Liquidity
RPMaturity
RPTax
where: rrF = real risk-free rate INFL = inflation Equity Risk Premium E ( Re ) =+R= F
ERP
+ − year Treasury YTM10
ERP
where: ERP = equity risk premium YTM = yield-to-maturity
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17
CAPITAL MARKET EXPECTATIONS
International Capital Asset Pricing Model E ( Ri ) = R+ F
−
i [ E ( RMF)
R ]
where: βi = return sensitivity of asset i RM = global investable market (GIM) Asset Class Risk Premium Singer-Terhaar in a 100% fully integrated market COVi ,M σ i σ M ρi , M σ i βi = σ 2M = σ 2M = σ M ρi , M σ RPi = i (ρi , M )( RPM ) σM
where: COVi,M = covariance of asseti and GIM returns ρi,M = correlation of asset i and GIM returns
RPi
RP = M σ iρi , M σM
where: RPM/σM = Sharpe ratio for the market correlation, indicates degree of integration (Note that the correlation coefficient ρi,M = in a fully segmented market is equal to 1.0.) Singer‐Terhaar Approach for Expected Return including a Liquidity Risk Premium E ( Ri ) = RF
+ RPi* + RPLiquidity
where: RPi* = weighted average of completely segmented and perfectly integrated asset class risk premiums RPLiquidity = liquidity risk premium (primarily alternative investments including real estate) Gross Domestic Product (GDP) GDP = C +
+I + G −( X
M)
where: C = consumption I = investment spending G = government spending (X − M) = exports less imports (i.e., net exports)
18
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CAPITAL MARKET EXPECTATIONS
Taylor Rule ROptimal
= RNeutral +×
[0.5( GDP − gForecast +× GDP − gTrend )0
.5( I Forecast
I Target )]
where: ROptimal = short-term interest rate target RNeutral = interest rate under target growth and inflation GDPg = growth rates for GDP forecast and long-term trend I = inflation rate forecast and target Econometric Models %∆GDP= ∆%% +C ∆ +
∆I + %% ∆ G−
(X
M)
%∆C
= f (Disposable income and Interest rates) = f (Earnings and Interest rates) %∆ ( X − M ) = f (Foreigne xchanger ates) %∆I
Government Debt YTM Treas
= rrF + INFL
Corporate Debt Credit spread
YTM
=
YTM Corp
−
Treas
Inflation‐Linked Debt E ( INFL) = YTMTreas − YTM TIPS
where: YTMTIPS = yield on treasury inflation protected securities Capitalization Rate
(VRE = NOI/r) where: VRE = value of real estate NOI = net operating income
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19
CAPITAL MARKET EXPECTATIONS
Purchasing Power Parity Approach %∆FX f / d
≈ INFL f − INFLd
where: %FXf/d = foreign for domestic currency exchange rate INFL = foreign f and domestic d inflation
20
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A
A P
R M
D ( )
INTRODUCTION TO ASSET ALLOCATION
I
A
A
Cobb-Douglas
∆Y Y
≈
∆A A
+α
∆K K
+ (1 − α)
∆L L
where: Y = total real economic output. A = level of technology. K = level of capital. L = level of labor. α = output elasticity of capital. (1 – α) = output elasticity of labor. H-Model
V0
=
D0 r − gL
(1 + g ) + N ( g − g ) L S L 2
where: N = period of years from higher to lower linear growth rate. gS = short-term high growth rate. gL = long-term steady growth rate. Earnings-Based Models
Fed Model
E1 P0
=
r − ROE (1 − P ) p
=
yT
− yT (1)− p p
= yT
where: E1/P0 = Earnings yield p = dividend payout ratio yT = 10-year T-note yield ROE(1 – P) = sustainable growth rate Fed Model implicitly assumes that r = ROE = yT, which ignores the equity risk premium.
22
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INTRODUCTION TO ASSET ALLOCATION
Yardeni Model to Value an Equity Market
E1 P0
= yB − d × LTEG
where: yB = Moody’s A-rated corporate bond yield d = earnings projection weighting factor LTEG = Consensus S&P 500 5-year annual earning growth
Cyclically-Adjusted P/E Ratio
CAPE =
Real S&P500 Price Index 10-year MA Real S&P500 Reported Earning
where: MA = moving average Portfolio Asset Class Optimization max E [U (WT ) ] = f (W0 ,,wi ri , A) n
Subject to :
∑− w = 1 i
i 1
where: E[U(WT)] = Expected utility of wealth at time t W0 = Current wealth wi = weights of each asset class in the allocation ri = returns of each asset class in the allocation A = investor’s risk aversion For the case of a risky asset and a risk-free asset, the optimization becomes: * w =
1 µ − rf λ σ2
Where: w* = weight of the risk asset in the two-asset portfolio λ = investor’s risk aversion μ = risk asset return rf = risk-free asset return σ2 = risk asset variance
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23
PRINCIPLES OF ASSET ALLOCATION
P
A
A
Risk Objectives
UpE R=
2 p
( p−) 0.005 A× ×
where: U is the investor’s utility.
E(Rp) is the portfolio expected return. A is the investor’s risk aversion.
σ 2p is the variance of the portfolio. Roy’s Safety First Ratio (SF Ratio)
SF Ratio =
E ( R p ) − RL
σp
where: RL is the l owest acceptable return over a period of time.
E(Rp) is the portfolio’s expected return.
σp is t he portfolio’s standard deviation. Including International Assets • The Sharpe ratio of the proposed new asset class: SR[New] • The Sharpe ratio of the existing portfolio: SR[p] • The correlation between asset class return and portfolio return: Corr (R[New], R[p]) ( New [ ], SR[New ] > SR [p ] ×Corr R R p[ ]) Portfolio Risk Budgeting
Marginal contribution to total risk identifies the rate at which risk changes as asset i is added to the portfolio: MCTRi = βiP , Pσ
where: βi,P = beta of asset i returns with respect to portfolio returns σP = portfolio return volatility measure as standard deviation of asset i returns Absolute contribution to total risk identifies the contribution to total risk for asset i
ACTRi = wi × MCTRi % ACTRi to totalr isk =
24
ACTRi
σP
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PRINCIPLES OF ASSET ALLOCATION
The optimal portfolio occurs when:
ri − rf MCTRi
=
rj − r f MCTRj
==
rTP − r f
σ TP
where: σTP = standard deviation of the tangency portfolio Risk Parity
wi × cov ×1 Pi , = P n
2
where: wi = weight of asset i in the portfolio n = number of assets in the portfolio covi,P = covariance of asseti returns with portfolio returns σ2P = variance of portfolio returns
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25
A
A
R P
M
D ( )
ASSET ALLOCATION WITH REAL WORLD CONSTRAINTS
A
A
R
-W
C
After-tax Portfolio Optimization Optimizing a portfolio subject to taxes requires using the after-tax returns and risks on an ex-ante basis.
rat
= rpt
(1 − t )
where: rat = expected after-tax return rpt = expected pre-tax return t = expected tax rate Extending this to a portfolio with both income and capital gains:
rat =prd pt − t(1+d pr )a− ptt
cg(1
)
where: pd = proportion of return from dividend income pa = proportion of return from price appreciation (i.e., capital gain) td = tax rate on dividend income tcg = tax rate on capital gain This formula ignores the multi-period benefit from compounding capital gains rather than recognizing the annual capital gain. Taxes also affect expected standard deviation.
at
=
pt
(1 − t )
Taxes result in lower highs and higher lows, effectively reducing the mean return and muting dispersion. Equivalent After-Tax Rebalancing Range
Rat
= Rpt
(1 − t )
where: Rat = After-tax rebalancing range Rpt = Pre-tax rebalancing range Portfolio Value After Taxable Distributions
Vat
= Vpt
(1 −i t )
where: vat = after-tax portfolio value vpt = pre-tax portfolio value ti = tax rate on distributions as income
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27
CURRENCY MANAGEMENT: AN INTRODUCTION
C
M
:A I
Forward Exchange Rates
FFC/DC
= SFC/DC ×
FPC/BC
= SPC/BC ×
1 + (i FC × Actual360) 1 + (i DC × Actual360) 1 + (i PC × Actual360) 1 + (i BC × Actual360)
where: FFC/DC = forward rate for domestic currency in terms of foreign currency; same as “base currency in terms of price currency” SFC/DC = spot rate for domestic currency in terms of foreign currency iFC = interest rates in foreign currency country iDC = interest rates in domestic currency country Forward Premium/Discount
(i − i ) × Actual = S FC/DC FC DC Actual 360 1 + (i DC × 360) (i PC − i BC ) × Actual360 FPC/BC − SPC/BC = SPC/BC 1 + (i BC × Actual360) FFC/DC − SFC/DC
Domestic Return on Global Assets
R DC
(1 R
)(1 R
) 1
== +RFC +FC+RFX +− RFCFXFX R ≈ RFC + RFX
where: RDC = return in domestic currency terms RFC = return in foreign currency terms RFX = percentage change in SDC/FC (i.e., foreign currency in terms of domestic currency) Portfolio Return in Domestic Currency Terms
RDC
= [ w×1+
(1 R +FC11)(1 + ×R+FX ) w +2
(1− RFC 2 )(1 RFX 2 )] 1
where: wn = weight of asset in the portfolio
28
© Wiley 2018 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright.
CURRENCY MANAGEMENT: AN INTRODUCTION
Risk on Global Assets in Domestic Currency Terms 2
2
( RDC ) ≈
( RFC+)
2
( RFX + )× [2
×( RFC ) × ( RFX )(
R FC , R FX )]
Roll Yield
YRoll
=
( FP /B
− PSB /
)
SP / B
where || indicates absolute value. Positive roll yield occurs when a trader buys base currency at a forward discount or sells it at a forward premium. Minimum Variance Hedge
yt
=
+ xt
+t
where: yt = percentage change in asset to be hedged xt = percentage change in hedging instrument β = hedge ratio for minimum variance hedge ε = error term to be minimized Minimum Hedge Ratio
h=
=
( RDC ; RFX×)
σ ( RDC ) σ ( RFX )
where: h = hedge ratio ρ = correlation of return in domestic currency terms and return on conversion to domestic currency.
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29
MARKET INDEXES AND BENCHMARKS
M
I
B
Factor-Model-Based
Rportfolio = ap + b1f1 + b2f2 … bkfk + εp where: aP = expected portfolio return if all sensitivities equal 0 bk = sensitivity to systematic factors fk = systematic factors εp = residual return from non-systematic factors
30
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F
-I
P
M
( )
INTRODUCTION TO FIXED INCOME PORTFOLIO MANAGEMENT
I
F
-I
P
M
Expected Return Decomposition
E ( R) ≈ Yield income +Rolldown return +E(∆P based on investor’s yield and spread view) –E(Credit losses) +E(Currency gains or losses)
where: Yield income = Annual coupon payment/Current bond price = Current yield Rolldown return = (B1 − B0) / B1 = % change in bond price due to changing time to maturity E(ΔP based on investor’s yield and spread view) = −( DM × ΔY%) + (0.5 × C × ΔY%) Note: Credit losses and currency gains or losses are discussed elsewhere. Effect of Leverage on the Portfolio
rP
=
Portfolio return Portfolio equity
= r1 +
VB VE
=
[r1 ×V−( −V× E
)(V B
r
)
B
B
]
VE
(r1 − rB )
where: r 1 = Return on the unlevered portfolio
rB = Borrowing costs VE = Equity VB = Borrowed amount The remainder of the equation indicates the return effect of leverage such that r 1 > rB results in a positive contribution from leverage. Leverage Using Futures
Leverage Futures=
Notional value – Margin Margin
Securities Lending
Rebate rate = Collateral earnings rate
32
−
Security lending rate
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INTRODUCTION TO FIXED INCOME PORTFOLIO MANAGEMENT
Total Return Analysis 1
Total future dollars n Semiannual total return = −1 Full price of the bond Dollar Duration Dollar duration = Duration × Bond value1 × 0.01
1Note:
Bond value is market valuenot par value
Spread Duration
Three major types of spreads: •
•
•
Nominal spread is the difference between the portfolio yield and the treasury yield for the same maturities. Zero-volatility spread (Z-spread) is the constant spread over all the Treasury spot rates at all maturities that forces equality between the bond’s price and the present value of the bond’s cash flows. Option-adjusted spread (OAS) is the spread over the treasury or the benchmark after incorporating the effects of any embedded options in the bond.
Economic Surplus
Economic Surplus = MVAssets − PVLiabilities where: MVAssets = market value of assets PVLiabilities = present value of liabilities Derivatives Overlay
Nf Futures BPV
Liability portfolio BPV – Asset portfolio BPV =
≈
Futures BPV
BPVCTD CFCTD
where: Nf = number of futures contract to immunize portfolio BPV = Basis point value CTD = Cheapest to deliver CF = Conversion factor
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33
INTRODUCTION TO FIXED INCOME PORTFOLIO MANAGEMENT
NP
=
Liability portfolio BPV – Asset portfolio BPV Swap BPV
×
100
where: NP = Notional principal of the swap Index Matching to a Fixed-Income Portfolio Active return = Portfolio return – Benchmark index return
34
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F
‐I
P
M
( )
YIELD CURVE STRATEGIES
Y
C
S
Yield Curve Measurement
Butterfly spread = –(Short-term yield) + ( × 2 Medium-term yield) – (Long-term yield Higher spread values indicate greater curvature.
36
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FIXED INCOME ACTIVE MANAGEMENT: CREDIT STRATEGIES
F
-I
A
M
:C
S
Excess Return on Credit Securities
XR ≈ (s×) (t− ∆ × )s
SD
where: s = spread at the beginning of the measurement period t = holding period (i.e., fractional portion of the year) SD = spread duration of the bond Expected Excess Return on Credit Securities
EXR ≈ (s× ) t– ∆ ( × s −)SD (× × t ) p
L
where: p = annual probability of default L = expected severity of loss
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37
E
P
M
EQUITY PORTFOLIO MANAGEMENT
E
P
M
Manager Active Return
Information ratio =
Active return Active risk
where: Active return = Portfolio return – Portfolio’s benchmark return Active risk = Tracking risk (i.e., annualized standard deviation of active returns) Manager’s “true” active return = Manager’s return − Manager’s normal benchmark return Manager’s “misfit” active return = Manager’s normal benchmark − Investor’s benchmark Total active risk =
true active risk 2 − misfit active risk 2
Portfolio Information Ratio (Fundamental Law of Active Management) IRP ≈ IC ×
BR
where: IR = information ratio for the portfolio IC = information coefficient (i.e., correlation between forecast return and active return; investment insight) BR = breadth (i.e., number of independent active management decisions made each year
Portfolio Active Return n
ARP =
∑h
Ai rAi
i =1
n
Aσ P =
∑h
2 Ai
2
σ Ai
i =1
where: ARP = Active return for a portfolio of managers rAi = each manager’s active return hAi = weight assigned for each manager’s active return AσP = Standard deviation of active returns for a portfolio of managers (assumes zero correlation of their returns)
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39
A
I P
M
ALTERNATIVE INVESTMENTS FOR PORTFOLIO MANAGEMENT
A
I
P
M
Spot Return E ( ∆F ) = E ( ∆S )
where: F = forward price S = spot price Roll Return (or Roll Yield) rRoll =
Ft −1 − FS tt − (
− St −1 )
Ft −1
=
∆ F − ∆S Ft −1
Total return on commodity index = Spot return + Roll return + Collateral return where: Collateral return = risk-free rate times the cash held as collateral over holding period Fund Returns r=
NAVt − NAVt −1 NAVt −1
=
NAVt NAVt−1
−1
Rolling Return
ri RRn ,t = +
r+ t −1 + ri −(n −1) /n
Downside Deviation (Semideviation)
∑ i=1[ min(ri − r*,0)] n
Downside deviation =
2
n −1
where: ri = return on asset i r* = specified return
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41
ALTERNATIVE INVESTMENTS FOR PORTFOLIO MANAGEMENT
Drawdown Drawdown is a period where portfolio value is less than at some previous high water mark. Maximum drawdown = max( HWMt − Lt +n ) where: HWMt = high-water mark at time t Lt+n = low after the same high-water mark Performance Appraisal Sharpe Ratio Sharpe ratio j =
(rj − rF )
σj
where: rj = return on asset j rF = annualized risk-free rate σj = standard deviation of asset j returns Sortino Ratio Sortino ratio =
rj − rF DD
where: DD = Downside deviation Gain‐to‐Loss Ratio G/L ratio =
42
#monthly gains #monthly losses
×
Average gain Average loss
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R
M
RISK MANAGEMENT
R
M
Portfolio Theory rP = w1r1 + w2r2
2
P=
2
2
w1 +1
2
2
w2+ 2
2w1w 2
1
2
where: r P = portfolio return σ 2P w
= portfolio variance = portfolio weights for assets 1 and 2 r = returns for assets 1 and 2 σ = standard deviation for assets 1 and 2 ρ = correlation between assets 1 and 2
Value at Risk (VAR)
Miniumum $ VaR = V×P [ E (−RP ) Zα
P
]
where: VP = portfolio value E(RP) = expected portfolio return zα = the number of standard deviations at the selected confidence level σP = standard deviation of portfolio returns (e.g., 1.65 for 5% and 2.33 for 1%) Note: To convert to daily values, divide annual expected return by 250 trading days and annual expected standard deviation by (250)0.5, the square root of 250. (Hint: we expect that the number of standard deviations for a given probability well be provided on the exam.)
44
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R
M
A
D
RISK MANAGEMENT APPLICATIONS OF FORWARD AND FUTURES STRATEGIES
R
M
A
F
F
S Managing Equity Risk by Beta Adjustment
=S S Nf+f
TS
Nf
f
β −β S = T S βf f
βS βT
is the current beta of an equity portfolio. is the target beta of an equity portfolio: the desired level of beta after hedging. S is the market value of a current equity portfolio. β f is the beta of the index futures. It is often close to one, but may not be exactly equal to one. f is the futures price of market index futures. Nf is the number of futures contracts needed to hedge the equity portfolio in order to achieve the target beta after hedging is established. Creating Synthetic Equity or Cash Positions
Long stock+Short futures=Long risk‐free bond Long stock=Long risk‐free bond+Long futures Creating a Synthetic Index Fund
V is the amount of money to be invested. f is the futures price of market index futures. q is the price multiplier of the futures contract (e.g., the S&P Index futures is $250). T is the time to maturity of the futures contract. r is the risk‐free interest rate. δ is the dividend yield of the market index. St is the level of stock index at time t. N f is the number of futures contracts. N *f is the rounded (whole number) number of futures contracts.
Futures payoff= N *f q (ST – f)
V
=
N *f
( N *f qf ) (1 + r )T
=
V (1 + r )T fq
46
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RISK MANAGEMENT APPLICATIONS OF FORWARD AND FUTURES STRATEGIES
Creating Cash Out of Equity
Unit of stock = N f q
N *f
=−
1 (1 + δ) T
V (1 + r )T qf
Due to rounding, the amount converted to cash is: V*
=
− N f qf (1 + r )T
Adjusting Duration Using Futures
N bf
DUR T M M − DUR = MDUR f
Nsf
β −β = T S βf
B
B
f B
S fs
The same concepts can be used to adjust the allocation to different types of equities or bonds, or preinvesting in an asset class using equity or bond futures. Currency Forward Contracts (Foreign Exchange)
If desiring to hedge an amount of money to be received in a foreign currency, the hedger can sell the value received in foreign currency (FC) forward for the desired currency (DC) to lock in the value received in DC:
VDC
= V FC × F DC FC
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47
RISK MANAGEMENT APPLICATIONS OF SWAP STRATEGIES
R
M
A
O
S
Notation for Options
S = The value of the underlying stock X = The exercise price of an option c = A call option premium p = A put option premium V = Value of a position π = Profit of a transaction r = Risk‐free rate of interest T = Option expiration S*T = Breakeven (profit equals 0) Put and call subscripts go from 1 being the lowest exercise price to 3 being the highest exercise price, with 2 between 1 and 3 in exercise price. Covered Calls
Value at initiation
=
V0 = S0 + c0
=
VT = ST − cT = ST − max(ST − X, 0)
Value at option expiration
Profit at option expiration = VT − V0 = [ST − max(ST − X, 0)] − [S0 − c0] Max profit
=
X – [S0 – c0 ] when ST ≥ X
Max loss
=
S0 – c0 when ST = 0
Breakeven point
=
S*T = S0 − c0
=
V0 = S0 − p0
Protective Put Value at initiation
Value at option expiration = VT = ST + pT = ST + max[X − ST, 0) Profit at option expiration = VT − V0 = [ST + max[X − ST, 0)] − [S0 + p0] Max profit
=∞
when ST approaches ∞
Max loss
=
[S0 + p0 ] − X when ST ≤ X
Breakeven point
=
S*T = S0 + p0
Bull Spreads Value at initiation Value at option expiration
=
V0 = c1 − c2
=
VT = max(ST − X1, 0) − max(ST − X2, 0)
Profit at option expiration = VT − V0 = [max(ST – X1, 0) − max(ST − X2, 0)] − [c1 − c2]
48
Max profit
=
[X2 – X1] − [c1 − c2] when ST ≥ X2
Max loss
=
c1 − c2 when ST ≤ X1
Breakeven point
=
S*T = X1 + [c1 − c2]
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RISK MANAGEMENT APPLICATIONS OF OPTION STRATEGIES
Bear Spreads Value at initiation Value at option expiration
=
V0 = p2 – p1
=
max[X2 –ST, 0) – max(X1 – ST, 0)
Profit at option expiration = VT − V0 = [max(X2 − ST, 0) – max( X1 – ST , 0)] – [p2 – p1] Max profit
=
[X2 − X1] – [p2 − p1] when ST ≥ X1
Max loss
=
p2 – p1 when ST
Breakeven point
=
S*T = X2 – [p2 – p1]
=
V0 = c1 – 2c2 + c3
=
VT = max(sT – X1, 0) – 2max(ST − X2, 0) + max(ST − X3, 0)
≥
X2
Butterfly Spreads Value at initiation Value at option expiration
Profit at option expiration = VT − V0 = [max[ST – X1, 0) – 2max[ST − X2, 0) + max(ST − X3, 0)] – [ c1 – 2c2 + c3] [X2 – X1] – [c1 – 2c2 + c3] when ST = X2
Max profit
=
Max loss
=
c1 – 2c2 + c3 when ST ≥ X3 or ST ≤ X1,
Breakeven points
=
S*T = X1, + [c1 – 2c2 + c3] and 2X2 – X1, – [c1 – 2c2 + c3]
Collars Value at initiation
=
V0 = S0 + [p1 − c2] = S0 + 0 = S0
Value at option expiration = VT = ST + max(X1 − ST, 0) – max(ST – X2, 0) Profit at option expiration = VT − V0 = [ST + max(X1 − ST, 0)] − max(ST − X2, 0) − [S0] Max profit
=
X2 − S0 when ST ≥ X2
Max loss
=
S0 − X1 when ST ≤ X1
Breakeven point
=
S*T = S0
Straddles Value at initiation Value at option expiration
=
V0 = c0 + p0
=
VT = max(ST
−
X, 0) + max(X − ST, 0)
Profit at option expiration = VT − V0 = [max(ST − X, 0)] + max(X − ST, 0) − [c0 + p0] Max profit
=
∞ when ST → ∞
Max loss
=
c0 + p0 when ST = X
Breakeven point
=
S*T = X +[c0 + p0] and X − [c0 + p0 ]
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49
RISK MANAGEMENT APPLICATIONS OF SWAP STRATEGIES
Option Strap Strip = c + 2p Strap = 2c + p Box Spread Value at initiation Value at option expiration
=
V0 = [c1 – c2 ] + [p2 – p1]
=
VT = X2 – X1
Profit at option expiration = VT – V0 = [X2 – X1] – [[c1 – c2 ] + [p2 – p1]] =
=
Profitof terminal [X2 – X1]stock – [[c1price – c2] regardless
Max profit Max loss
=
None
Breakeven point
=
None
+
[p2 – p1]]
Interest Rate Option Strategies Call option payoff
= Notional principal × max(Realized spot rate − Exercise rate,0) ×
Put option payoff
Days in underlying rate 360
= Notional principal × max(Exercise rate − Realized spot rate,0 ) ×
Days in underlying rate 360
Combining Caplets with a Floating Rate Loan Because the first rate is usually already set, there are usually (n – 1) caplets to protect a floating rate loan. There may be another caplet if taken out prior to borrowing the money.
iFRN
= VL ( Libort −1 + S L )
m
360
where: iFRN = loan interest on the floating rate note VL = loan value; the amount of the loan Libort-1 = Libor on the previous reset date SL = spread over Libor m = actual days in the settlement period
PayoffCaplet
= VL × max(0, Lt −1 − rx )
m 360
where: rX = exercise rate for the cap 50
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RISK MANAGEMENT APPLICATIONS OF OPTION STRATEGIES
Combining Floorlets with a Floating Rate Loan Loan interest is calculated as with caplets.
PayoffFloorlet
= V×L
max(0, − r=x Lt−1 )
m 360
Collar on a Floating Rate Loan A borrower’s collar consists of long cap and short floor positions with the intent of zero cost from the strategy (i.e., exercise rates are selected such that floor premium equals cap premium). A lender’s collar consists of long floor and short cap positions. Loan interest is calculated as before. Payoffs from the cap and floor positions are calculated as before. Effective interest = interest due – caplet payoff – floorlet payoff Note: Effective interest applies to both borrower and lender, but whether a receipt or payment depends on whether the cap/floor has been bought/sold. Risk Management of an Option Portfolio
Option delta =
Option gamma
Option vega
=
Change in option price Change in the underlying stock price
=
∆c ∆S
in option delta = ChangeChange in the underlying stock price
Change in option price Change in annualized stock return volatility
Delta Hedge
Delta Hedge Ratio =
Nc NS
1 = − ∆c / ∆S
where: N = number of call options c; number of shares S ∆c = change in call option price ∆S = change in share price ∆c/∆S = option delta
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51
RISK MANAGEMENT APPLICATIONS OF SWAP STRATEGIES
R
M
A
S
S
Convert a Floating Rate to a Fixed Rate
[Pay fixed and receive floating interest rate swap] = Floating-rate bond – Fixed-rate bond Floating-rate bond = Fixed-rate bond + [Pay fixed and receive floating interest rate swap]
− Floating-rate − =bond
−Fixed-rate bond
[Pay fixed and receive floating interest rate swap]
Fixed-rate bond = Floating-rate bond − [Pay fixed and receive floating interest rate swap]
− Fixed-rate − =bond
+ Floating-rate bond [Pay fixed and receive floating interest rate swap]
Change the Duration of a Fixed‐Income Portfolio
[Pay fixed and receive floating interest rate swap] = Floating-rate bond − Fixed-rate bond
Decrease Portfolio Duration
Duration of [Pay fixed and receive Duration of Duration of = − floating interest rate swap] (floating‐rate bond) (fixed‐rate bond)
Increase Portfolio Duration
Duration of [Pay floating and receive = fixed interest rate swap]
Duration of Duration of − (fixed‐rate bond) (floating‐rate bond)
Duration Management
NP = VP VP ( MDUR ) P(
$ DURP + S
×
MDURT
− MDURP
MDURS
= VP) MDURT − NP ( MDURS ) = $ DURT
where: NP = notional principal of the swap VP = value of the portfolio MDUR = target, portfolio, and swap modified durations $DUR = dollar duration *Note: To decrease duration MDUR ( T < MDURP) the denominator needs to be negative so the investor must be the fixed rate payer.
52
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RISK MANAGEMENT APPLICATIONS OF SWAP STRATEGIES
Floating Rate Notes
Leveraged Floating-rate Notes CFLeveraged floater = − kL ( FP )
= (ci ) k ( FP ) ( ) swap = Lk FP CFFixed side of swap = − ( FS ) k ( FP ) CFNet = kFPc( i − FS ) CFBond
CFFloating side of
where: CF = net cash flow from the perspective of leveraged floater issuer k = leverage ratio L = receive floating rate on the swap FP = face value/principal amount of the floating-rate note FS = pay fixed rate on the swap ci = coupon interest rate (i.e., fixed interest rate on a coupon-paying bond) Inverse Floating-rate Notes
CFNet = FP (FS + ci − b)
where: b = base interest rate against which Libor is drawn Swapping Foreign Currency Cash Flows for Domestic Currency Cash Flows
CFF
= NPF (iF )
NPF
=
NPD
=
CFD
= NPD (iD )
t 360
CFF t 360 NPF × S D / F or NPF S F / D
(iF )
t 360
where: CFF = Foreign cash flow receipt to be swapped NPF = Notional principal of foreign currency to be swapped at iF to achieve periodic CFF iF = interest rate applicable to notional principal of foreign currency to be swapped t = number of days between cash flows to be received NPD = Notional principal of domestic currency swapped for foreign currency D/F = spot rate for foreign currency priced in domestic currency S SF/D = spot rate for domestic currency priced in foreign currency CFD = Domestic cash flow received in exchange for swapped foreign cash flow
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53
RISK MANAGEMENT APPLICATIONS OF SWAP STRATEGIES
Creating Dual‐Currency Bonds
Dual currency bond = Conventional bond + Currency swap (without exchanging NP) Swaptions
Receiver swaption = Call option on a bond Payer swaption = Put option on a bond Add or Remove Embedded Options in a Bond
Receiver swaption = Call option on bond Payer swaption = Put option on bond Callable bond = Straight bond − Call option on bond Putable bond = Straight bond + Put option on bond Callable bond = Straight bond − Receiver swaption Putable bond = Straight bond + Payer swaption Monetizing an Embedded Call
(2a) −Callable bond = −Straight bond + Receiver swaption (2b) −Straight bond = −Callable bond − Receiver swaption NCF
= NP ×− {[ F+(+t , T ) s]− +LIBOR − X F (t , T ) LIBOR} = NP×−[ −X= s] × −NP− −[ =(r0 s) s] NP(r0 )
where: NCF = Non-callable bond cash flows NP = notional principal F(t,T) = prevailing swap rate s = credit spread over Libor X = strike rate r0 = interest rate on t he non-callable bond with credit spread s over Libor Creating a Synthetic Embedded Call
(2a) −Callable bond = −Straight bond + Receiver swaption NCF − =× NP [ Noncallable −bond fixed + − rate ( LIBOR Strike rate Prevailing swap rate+ LIBOR ) = NP×− −[ r0 LIBOR +− X+ F (t , T=−) +LIBOR] −− r0 (r0 s ) F (t , T )
= ×NP−
54
[(F+t ,)T
s]
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T
,M
,
R
EXECUTION OF PORTFOLIO DECISIONS
E
P
Effective spread = 2 × Execution price −
D (Bid price + Ask price)
2
Implementation Shortfall Costs for purchases Explicit Costs Expicit costs =
Commissions, taxes, and fees SH
× PH
Realized Profit/Loss RPL =
S × ( PE SH
− PR ) × PH
Slippage (Delay Costs)
Delay =
S × ( PR SH
− PH ) × PH
Unrealized Profit/Loss (Missed Trade Opportunity Cost)
UPL =
( S H − S) × ( PL − SH
PH )
× PH
Implementation Shortfall Implementation shortfall =
Commissions + −S (+PE SH
P−L ) × PH
S H ( PL
PH)
where: S = shares executed SH = hypothetical shares executed PH = hypothetical price; benchmark price PE = execution price PR = relevant price when shares are not purchased at the hypothetical price P L
56
= last available valuation price
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M
R
The Perold‐Sharpe Analysis of Rebalancing Strategies VPortfolio
= VRisk-free + VStocks
Buy‐and‐Hold Strategies
m
=
VStocks
(VPortfolio − Floor )
=
VStocks
(
VPortfolio
− VRisk-free
0
)
Constant Mix Strategies
m
=
VStocks VPortfolio
Constant Proportion Portfolio Insurance (CPPI) Investment in Stock = M(Vportfolio − Floor)
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57
P
E
EVALUATING PORTFOLIO PERFORMANCE
E
P
P
The Holding Period Return
rt
=
MV
t
−MV
0
MV0
=
MVt MV0
−1
where: MVt is the market value of the account at time t; and rt is the holding period return over the investment period. Cash Inflow Occurs at the Beginning of the Reporting Period
rt
=
MV t −MV (
+
0 CF
0)
MV0 + CF0
Cash Inflow Occurs at the End of the Reporting Period
rt
=
MV (
t
CF −
−
t )MV
0
MV0
Total return = Income yield + Capital gains yield The Time‐Weighted Rate of Return
rtwr
= (1 r+ t r 1+ )(1t … r 2+ ) tn−(1
)1
where: rti is the holding period return of sub‐period i, and there are a total of n sub‐periods. Annualizing Returns
rannual
= +nr(1 r yr .1+)(1 r yr … +.2 ) −r(1=yrnr+ .
) r+ 1
… (1 +
yr .1 )(1
yr .2 )
(1
yr .n )
1/ n
− 1
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59
EVALUATING PORTFOLIO PERFORMANCE
Benchmarks
P=P P = B + (P − B) P = M + (B − M) + (P − B) = M + S + A P B M A S
is the return performance of a portfolio under management. is the return on a benchmark that matches the portfolio’s investment style. is the return on the market portfolio. is the portfolio manager’s active return: A = P − B. is the return contribution due to style selection: S = B − M.
Systematic Bias
P = M + (B − M) + (P − B) = M + S + A Correlation(A, S) = 0 Correlation(E, S) = Correlation(P − M, B − M) > 0 Tracking Error Tracking error = Volatility(A) < Volatility(E) = Volatility(P − B) < Volatility(P − M)
where: Volatility(E) = Excess volatility of returns from the managed portfolio over returns from the market portfolio Hedge Fund Benchmarks
MV
−MV
rt
=
rv
= rP − rB
t
0
MV0
where: rv is the value‐added return. rP is the portfolio return. rB is the benchmark return.
60
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EVALUATING PORTFOLIO PERFORMANCE
Sector Weighting Micro Attribution S
Selection effect =
∑= [w
Bi
× (rPi − rBi )]
i 1
S
Sector allocation effect =
∑= [(w−
Pi
w× Bi− r) rBi (
B
)]
i 1
S
Interaction effect =
∑= [(w− w× Pi
−r)
Bi
r(Pi
Bi
)]
i 1
Ex Post Alpha Also known as Jensen’s alpha
Rt
−ftr = +
−
( R ft t r Mt
)+
where for period t: Rt = the portfolio return rft = the risk-free return RMt = the return on the market index α = the intercept of the regression β = the beta of the portfolio return relative to the market index return ε = the random error term Treynor Measure
TA
=
R A − rf
βA
Sharpe Ratio
Sharpe ratio A
=
R A − rf
σA
M‐Squared (M2)
M2
R −r = rf + A f σ M σA
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61
EVALUATING PORTFOLIO PERFORMANCE
Information Ratio
IRA
=
R A − RB
σ A− B
where: σA–B = standard deviation of the differential returns of the asset over the benchmark
62
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G
I
P
S
OVERVIEW OF THE GLOBAL INVESTMENT PERFORMANCE STANDARDS
O
G P
I S
Return Calculation Methodologies
Total Return rt
=
V1 − V0 V0
Time‐Weighted Return rtw
= (1 +r r+1 )(1
r 2+) −(1
n)
1
Modified Dietz Method rmodDietz =
V V1 − CF 0 − V0
+ ∑ t =CF ( wi × 1 n
i)
Weighting Formula wi
=
CD − Di CD
where: wi weight of cash flow i =
CD Di
=
=
calendar the the period calendar days days in since beginning of the period to receipt of cash flow i
Modified Internal Rate of Return Approach n
V1
= ∑CF
(1 ir
+
)w V
i
t =1
+r
0 (1
+
)
Original Dietz Method V V− C0 − F rDietz = 1 V0 + 0.5CF
64
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OVERVIEW OF THE GLOBAL INVESTMENT PERFORMANCE STANDARDS
Composite Returns
Beginning Assets Method
rc
V = ∑ rpi n 0, pi ∑ V0, pi pi =1
where: rc composite return rpi individual portfolio returns V ,pi individual portfolio beginning value =
=
=
0
Beginning Assets Plus Weighted Cash Flows Method
rc
V = ∑ rpi pi ∑ Vpi
Presentation and Reporting
∑ = (r − r )
2
n
SC
=
i
i 1
C
n −1
where: Sc = standard deviation of returns in the composite ri = individual portfolio returns rc = composite returns n
SCaw ,
=
∑r=r (
−
i Cw aw i 1
= wi −
2
, i
)
V0,i
∑=V = ∑w × r = n i 1 0, i n
= rCaw,
i i i 1
where: SC,aw asset-weighted standard deviation wi beginning-of-period weight for portfolio i in the composite V0,i beginning-of-period value of portfolio i =
=
=
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65
OVERVIEW OF THE GLOBAL INVESTMENT PERFORMANCE STANDARDS
Ly
=
y
100
(n + 1)
rY = r A− L−yn
(
−
r, ALyrA B
)(
)
where: Ly location of the portfolio in the yth percentile n number of observations r linear interpolation of return between observations around the yth percentile =
=
=
Portfolio returns Less:
Trading costs
Equals: Gross-of-fee returns Less:
Management fees
Equals: Net-of-fee returns
n
rann
= n ∏r(+11)t − = +r r +(r1 )(1+r (1…()) +)2
1
3
1
t =1
1/ n
n
− 1
where: n number of compounding periods during the annual period =
Internal Rates of Return Since Inception (SI‐IRR) Using Quarterly or More Frequent Cash Flows
rann
66
4
= (1 + rQuarterly ) − 1
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