2018 cfa Level 3 Wiley Formula Sheet

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B�� F��

THE BEHAVIORAL FINANCE PERSPECTIVE

T�� B�� F�� P�� Subjective Expected Utility E(U) = ∑ u( xi ) p( xi ) Utility Calculation (Prospec (Prospectt Theory) U

=

w( p1 )v( x1 ) + w( p2 ) v( x 2 ) + w( p n ) v(x n )

where: = utility U = x = a particular outcome p = probability of x value of x v = value w = probability-weighting function for outcome x ; accounts for tendency to overreact to low probability events and underreact to other events

2

© Wiley 2018 All Rights Reserved. Any unauthorized copying or distribution will constitute an infringement of copyright.

P�� P�� W�� M�� M��

MANAGINGG INDIVIDUAL INVESTOR PORTFOLIOS MANAGIN

M�� I�� I�� P�� Sharpe Ratio

Shar Sharpe pe rati ratio o is: is:

(expe expect cted ed retur eturn n – riskisk-ffree ree rate rate)) expecte expected d standar standard d deviatio deviation n

Capital gains tax payable = Price appreciation × t CG × turnover rate where: t CG = capital gains tax rate

Buy tax-free bonds when R tax-free > Rtaxable × (1 − t ) where: R = return t = = applicable tax rate

4


TAXES AND PRIVATE WEALTH MANAGEMENT IN A GLOBAL CONTEXT

T�� P�� W�� M�� G�� C�� Annual Accrual with a Single, Uniform Tax Tax Rate rafter-tax

= rpre-tax (1 − t I ) n

FVIFpre-tax FVIFafter-tax

= (1 + rpre -tax )

= [1 + rpre-tax (1 − t I )] n

where: = Future value interest (i.e., accumulation) factor FVIF = = return r = = tax rate t = n = number of periods Deferral Method with a Single, Sin gle, Uniform Tax Tax Rate (Capital Gain) FVIFCG

= (1 + rCG )n (1 − tCG ) + tCG

Taxable Gains When the Cost Basis Differs from Current Value Taxable gain = V T − Cost basis where: V T =

terminal value Cost basis = amount paid for an asset FVIF CG =

(1 +

r pre‐tax)n (1 − t CG)

+ t CG B

where: B

=

V 0 =

Cost Cost basis basis V 0

value of an asset when purchased

Accumulation Factor with Annual Wealth Tax FVIFW

= (1 + rpre-tax )(1 − tW ) 

n

Effective Annual After‐Tax Return in a Blended Tax Regime r*

= rT (1 − PI tI − PD tD − PCG tCG )

where: P = proportion of return from income, dividends, and realized capital gains during the period


5


Effective Capital Gains Tax Rate

 1 − P I − PD − PCG  = t CG     1 − P I t − PDt D − PCGtCG

T*

The after-tax future value multiplier under this blended tax regime then becomes:

= (1 + r* )n (1 − T * ) + T * − tCG (1 − B)

FVIFafter-tax

If the portfolio is non‐dividend paying equity securities with no turnover t urnover (i.e., P D = P I = 0 and PCG = 1) held to the end of the horizon, the formula reduces to: FVIFafter-tax

= (1 − r )n (1 − tCG ) + tCG

If the portfolio is non‐dividend paying equity securities with 100 percent turnover and taxed annually, the formula reduces to:

FVIFafter-tax

= 1 + r (1 − tCG ) 

n

Accrual Equivalent Return

Vn

= V0 (1 + r AE )n =n

r AE

V n V 0

−1

where: V n = value after n compounding periods Accrual Equivalent Tax Rate r AE

= r (1 − T AE )

T AE

= 1−

r AE r

Tax‐Deferred Accounts FVIFTDA

6

= (1 + r )n (1 − t )tCG B



Tax‐Exempt Accounts FVIFTEA

= (1 + r ) n

FVIFTDA

= FVIFTEA (1 − t )

Value Formula for a Tax Exempt Account

V n = V 0 (1

− t 0)(1 + r )n

Value Formula for a Tax‐Deferred Account V n = V 0(1 + r )n

(1 − t n)

The Investor’s After‐Tax Risk

σ AT = σ (1 − t ) where: σ = standard deviation of returns Ratio of Long-T Long-Term erm Capital Gains to Short-Term Short-Term Capital Gains

V LTG V STG

+ r )n (1 − t LTG ) = n V0  1 + r (1 − t STG )  V0 (1


7

DOMESTIC ESTATE PLANNING: SOME BASIC CONCEPTS

D�� E�� P��: S�� B�� C�� Core Capital k Core = PV of current lifestyle spending + emergency reserve

PV of Spending Need

Liability 0 =

n

∑

Expecte Expected d spending spending (1 + r )t

t = 0 n

=∑

urviva vall t ) × Spen pending dingt p(Survi

t = 0

(1 + r )t

where: = real risk-free rate r =

Joint Survival Probability Calculation p(Survival t ,C 1,C 2 ) = p(C 1) + p(C 2 ) − p(C 1) p(C 2 )

where: C 1 = First spouse survives C 2 = Second spouse survives Excess Capital Assets = House + Investments + Net employment capital Liabilities = Mortgage + Current lifestyle + Education needs + Retirement needs K Excess = Assets − Liabilities

Relative Value of Tax-free Gifts n

1 + rg (1 − t ig )  RV Tax- free Gift = = n FV (1 − T e ) Bequest [1 + re (1 − tie ) ] (1 FV Gift

where: = future value of the gift or bequest to the recipient FV = n = expected number of years until donor’s death, at which time bequest transfers to recipient = pre-tax returns to the gift recipient g or estate making the gift e r = = tax rate on investments that applies to gift recipient g or estate making the gift e t = T e = estate tax that applies to asset transfers at donor’s death

8


DOMESTIC ESTATE PLANNING: SOME BASIC CONCEPTS

Relative Value Value of Gifts Taxable to Recipient n

1 + rg (1 − tig )  (1 − T g ) RV Recipient = Recipient Taxable Taxable Gift = n FV (1 − T e ) Bequest [1 + re (1 − tie ) ] (1 FV Gift

where: T g = gift tax rate that applies to recipient Relative Value of Gifts Taxable to Donor But Not to Recipient

RV TaxableGift =

FV Charitable Charitable Gift FV Bequest

1 + rg (1 − tig )n 1 − Tg + (Tg Te × g/ e) = n [1 + re (1 − tie )] (1 − T e )

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 Value of Charitable Gratuitous Transfer Transferss

RV Charitable Gift =

FV Charitable Gift FV Bequest

n

=

(1 + rg )n + Toi [1 + re (1 − tie ) ] (1 − te ) n

[1 + re (1 − tie ) ]

(1 − T e )

where: T oi = tax on ordinary income (donor can increase the charitable gift amount) Tax Code Relief Credit Method t CM =Max (t RC , t SC )

where: t RC = applicable tax rate in the residence country t SC = applicable tax rate in the source country Exemption Method t EM = t SC

Deduction Method t DM = t RC + t SC − t RC t SC


9

RISK MANAGEMENT FOR INDIVIDUALS

R�� M�� I�� Human Capital

Human capital is calculated as follows: N

HC 0

=

∑

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 r f = 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

mNPV 0

=

p(s )b

∑ (1+t r )t t 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 Gros Grosss prem premiu ium m = Net Net Prem Premiu ium m + Load Load repr repres esen enti ting ng insu insura ranc ncee comp compan any y over overhe head ad Net premiu premium m=

E (V DB )

1 + r p

E (V DB ) = DB × [1 − P (S t )]

where: E (V DB) = Expected value of the death benefit r P = return on the insurance company’s portfolio DB = death benefit P(S t ) = Probability of survival in period t

10



Income Yield Yield for an Immediate Fixed Annuity

Y Income Income =

CF Annual P0

where: CF Annual = guaranteed annual payment P0 = price of the immediate fixed annuity Policy Reserve Poli Policy cy rese reserrve = PV of futu futurre bene benefi fits ts – PV futur uturee net net prem premiu ium m

Net Payment Cost Index

Net Net paym paymen entt cost cost inde index x=

Interes Interest-ad t-adjust justed ed annual annual payment payment cost

100 Inter Interes estt-adj adjus ustm tment ent annual annual paymen paymentt cost cost = Annuit Annuity y due (20(20-yea yearr insur insuranc ancee cost, cost, 5%, 5%, 20 years years)) 20-y 20-yea earr insu insura ranc ncee cost cost = FV annu annuit ity y due due (Pre (Premi mium um,, 5%, 5%, 20 year years) s) –FV –FV ordin ordinary ary annui annuity ty (proje (projecte cted d annual annual divide dividend, nd, 5%, 5%, 20 year years) s)

Note: Assumes policy owner owner will die at the end of the 20-year period. Surrender Cost Index

Surre Surrende nderr cost cost index index =

Interes Interest-a t-adjus djusted ted annual annual surrend surrender er cost Polic Policy y face face value/ value/100 1000 0

Inter Interes estt-adj adjus ustme tment nt annual annual surr surrend ender er cost cost = Annuit Annuity y due (20(20-yea yearr insur insuranc ancee cost, cost, 5%, 5%, 20 years years)) 20-y 20-yea earr insu insura ranc ncee cost cost = FV annu annuit ity y due due (Pre (Premi mium um,, 5%, 5%, 20 year years) s) –FV ordina ordinary ry annuit annuity y (pro (projec jected ted annua annuall divide dividend, nd, 5%, 5%, 20 year years) s) –20-y –20-year ear proje projecte cted d cash cash value 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.


11


Human Life Value: Value: Growing Income Replacement (Life Insurance Need) V PV an a nnuity du d ue (Y0,pretax , i, 20 2 0 ye y ears) HL = PV i=

1 + r 1+ g

−1

where: V HL = human life value; i.e., amount of insurance required to replace insured’s income tax contribution Y 0, pretax 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 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


P�� M�� I�� I�� I��

MANAGINGG INSTITUTIONAL INVESTOR PORTFOLIOS MANAGIN

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 Leverag Leverage-a e-adjus djusted ted duratio duration n gap k

=

D A

−

kDL

L =

A

where: k = = ratio of liabilities to assets, both at market value

14


A�� E�� A�� P�� M��

CAPITAL MARKET EXPECTATIONS

C�� M�� E�� E�� Volatility Clustering

σ t2 = βσ βσ t2−1 + (1− β)εt 2 where: σ2 = volatility β = decay factor (i.e., effect of prior volatility on future volatility) ε = error term Multifactor Regression Models Ri

= α i + b1F1 + b2 F2 + … + bk F k + εi

where: F k = return to factor k bk = asset i’s return sensitivity to factor k Quantitative Methods: Discounted Cash Flow Models ∞

V 0

=∑

CF t

t =1 (1

+ r )t

where: CF t = 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 r e = required return on equity inve i nvestments stments

16



R) Expected Return, E ( R E ( R) =

D0 (1 + g) P0

+g=

D1 P0

+g

Grinold‐Kroner Model E ( R ) ≈

D P

− %∆S + INFL + gr + %∆PE

where: D/P = expected dividend 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

+ RP1 + RP2 + ... + RPk

where: RF = nominal risk-free rate interest rate RPk = risk premium k Fixed‐Income Premiums E ( Rb ) = rrF

+ RPINFL + RPDefault + RPLiquidity + RPMaturity + RPTax

where: rr F = real risk-free rate INFL = inflation Equity Risk Premium E ( Re ) = RF

+ ERP = YTM10 − year Treasury + ERP

where: ERP = equity risk premium YTM = = yield-to-maturity


17


International Capital Asset Pricing Model E ( Ri ) = RF

+ βi [E (RM ) − RF ]

where: βi = return sensitivity of asset i R M = global investable market (GIM) Asset Class Risk Premium Singer-T Singer-Terhaar in a 100% fully integrated market

βi =

COV i ,M

=

σ i σ M ρi ,M  σ i  =   ρi ,M 2 σ M  σ M 

σ σ RPi = i (ρi , M )( RPM ) σ M 2 M

where: COV i,M = covariance of asset i and GIM returns ρi,M = correlation of asset i and GIM returns

RPi

 RP  =  M  σ iρi , M  σ M 

where: RP M / σ M = Sharpe ratio for the market ρi,M = correlation, indicates degree of integration (Note that the correlation coefficient in a fully segmented market is equal to 1.0.) Singer‐Terhaar Singer‐T erhaar Approach Approach for Expected Return including a Liquidity Risk Premium E ( Ri ) = RF

+ RPi* + RPLiquidity

where: RPi* = weighted average of completely segmented segmented and perfectly integrated asset class risk premiums RP Liquidity = liquidity risk premium (primarily alternative investments investments including real estate) Gross Domestic Product (GDP) GDP = C + I + G + ( X − M )

where: C = = consumption I = = investm investment ent spending G = governm government ent spending X − M ) = exports less imports (i.e., net exports) ( X

18



Taylor Rule ROptimal

(GDPgForecast − GDPgTrend ) + 0. 0.5 × (I Forecast − I Target ) ] = RNeutral + [0.5 × (G

where: ROptimal = short-term interest rate target R Neutral = 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) %∆ I = f (Earnings and Interest rates) %∆ ( X − M ) = f (Foreign ex exchange ra rates) Government Debt YTM Treas

= rrF + INFL

Corporate Debt Credit Credit spread spread = YTM Corp − YTMTr eas

Inflation‐Linked Debt E (INFL ) = YTMTreas

− YTM TIPS

where: YTM TIPS = yield on treasury inflation protected securities Capitalization Rate

(V RE = NOI/r ) where: V RE = value of real estate NOI = = net operating income


19


Purchasing Power Parity Approach %∆FX f / d

≈ INFL f − INFLd

where: %FX f/d = foreign for domestic currency exchange rate INFL = foreign f and and domestic d inflation inflation

20


A�� A�� A�� R�� D�� P�� M�� (�)

INTRODUCTION TO ASSET ALLOCATION

I�� A�� A� �� A�� A�� Cobb-Douglas

∆Y ∆ A ∆K ∆ L ≈ +α + (1− α) Y

A

K

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

V 0

=

(1 + g ) + N ( g − g )  L S L  2 r − g L   D0

where: N = = period of years from higher to lower linear growth rate. gS = short-term high growth rate. g L = long-term steady growth rate. Earnings-Based Models

Fed Model E 1 P0

=

r − ROE (1 − P ) p

=

yT

− yT (1 − p) p

= yT

where: E 1 / P0 = Earnings yield p = dividend payout ratio yT = 10-year T-note yield ROE (1 (1 – P) = sustainable growth rate Fed Model implicitly assumes that r = = ROE = = yT , which ignores the equity risk premium.

22


INTRODUCTION TO ASSET ALLOCATION

Yardeni Model to Value an Equity Market E 1 P0

= y B − d × LTEG

where: y B = Moody’s A-rated corporate bond yield d = earnings projection weighting factor LTEG = Consensus S&P 500 5-year annual earning growth Cyclically-Adjusted Cyclically-Adjust ed P/E Ratio CAPE = =

Real Real S&P50 S&P500 0 Pric Pricee Index Index 10-y 10-yea earr MA Real Real S&P5 S&P500 00 Repo Report rted ed Earn Earnin ing g

where: MA = moving average Portfolio Portf olio Asset Class Optimization max E [U (WT ) ] = f (W0 , wi , ri , A) n

Subject to :

∑− w = 1 i

i 1

where: E [U (W T )] = Expected utility of wealth at time t W 0 = Current wealth wi = weights of each asset class in the allocation r i = 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 − r f 

  λ  σ 2 

Where: w* = weight of the risk asset in the two-asset portfolio λ = = investor’s risk aversion μ = risk asset return r f = risk-free asset return σ2 = risk asset variance


23

PRINCIPLES OF ASSET ALLOCATION

P�� A�� A�� A�� Risk Objectives U p = E ( R p ) − 0.005 × A × σ 2p

where: U is is the investor’s utility. E ( R R p) is the portfolio expected return. A is the investor’s risk aversion.

σ p2 is the variance of the portfolio. Roy’s Safety First Ratio (SF Ratio)

SF Rati Ratio o=

E ( R p ) − RL

σ p

where: R L is the lowest l owest acceptable return return over a period of time. E ( R R p) is the portfolio’s expected expected return.

t he portfolio’s standard deviation. σ p is the Including International Assets The Sharpe ratio of the proposed new asset class: SR[New] • The Sharpe ratio of the existing portfolio: SR[ p p] • • The correlation between asset class return and portfolio return: Corr ( R R[ New New], R[ p p]) SR[ New] > SR[ p] × Corr ( R[ New], R[ p])

Portfolio Portf olio Risk Budgeting

Marginal contribution contribution to total risk identifies the rate at which risk changes as asset i is added to the portfolio: MCTR i = βi ,P σ P

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 % ACTRi to total ri risk =

σP

24


PRINCIPLES OF ASSET ALLOCATION

The optimal portfolio occurs when: ri − r f MCTRi

=

r j − r f MCTR j

==

rTP − r f

σ TP

where: σTP = standard deviation of the tangency portfolio Risk Parity

wi × covi ,P =

1 n

× σ 2P

where: wi = weight of asset i in the portfolio n = number of assets in the portfolio covi,P = covariance of asset i returns with portfolio returns σ2P = variance of portfolio returns


25

A�� A�� A�� R�� D�� P�� M�� (�)

ASSET ALLOCATION WITH REAL�WORLD CONSTRAINTS

A�� A�� A�� R��-W 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: r at = expected after-tax return r pt = expected pre-tax return = expected tax rate t = Extending this to a portfolio with both income and capital gains: rat

= pd rpt (1 − td ) + parpt (1 − tcg )

where: pd = proportion of return from dividend income pa = proportion of return from price appreciation (i.e., capital gain) t d = tax rate on dividend income t cg = 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 a ffect expected standard deviation.

σ at = σ pt ( 1 − t ) Taxes result in lower highs and a nd higher lows, effectively reducing the mean return a nd muting dispersion. Equivalent After-Tax Rebalancing Range Rat

= Rpt (1− t )

where: Rat = After-tax rebalancing range R pt = Pre-tax rebalancing range Portfolio Value After Taxable Distributions Vat

= Vpt (1 − t i )

where: vat = after-tax portfolio value vpt = pre-tax portfolio value ti = tax rate on distributions as income


27

CURRENCY MANAGEMENT: AN INTRODUCTION

C�� M��: A� I�� Forward Exchange Rates

FFC/DC

= SFC/DC ×

FPC/BC

= SPC/BC ×

1 + (i FC × Actual 360) 1 + (i DC × Actual 360) 1 + (i PC × Actual 360) 1 + (i BC × Actual 360)

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  = SFC/DC  FC DC Actual 360   1 + (i DC × 360)   (i PC − i BC ) × Actual 360  FPC/BC − SPC/BC = SPC/BC   1 + (i BC × Actual 360)   FFC/DC − SFC/DC

Domestic Return on Global Assets R DC

= (1 + RFC )(1 + RFX ) − 1 = RFC + RFX + RFC RFX ≈ RFC + RFX

where: R DC = 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 R DC

= [ w1 × (1 + RFC1)(1 + RFX1) + w 2 × (1 + RFC 2)(1 + RFX 2)] − 1

where: wn = weight of asset in the portfolio

28


CURRENCY MANAGEMENT: AN INTRODUCTION

Risk on Global Assets in Domestic Currency Terms

σ 2 ( R DC ) ≈ σ 2 ( RFC ) + σ 2 ( RFX ) + [2 × σ( RFC ) × σ( RFX ) × ρ( RFC , RFX )] Roll Yield

Y Roll

=

( FP / B − S P / B ) S P / B

where || indicates absolute value. value. Positive roll yield yield occurs when a trader buys base currency at a forward discount or sells it at a forward premium. Minimum Variance Hedge yt

= α +β x t + εt

where: yt = percentage change in asset to be hedged x t = percentage change in hedging instrument β = hedge ratio for minimum variance hedge ε = error term to be minimized Minimum Hedge Ratio

 σ( R DC )    σ ( RFX ) 

h = β = ρ ( R DC ; RFX ) × 

where: h = hedge ratio ρ = correlation of return in domestic currency terms and return on conversion to domestic currency.


29

MARKET INDEXES AND BENCHMARKS

M�� I�� B�� Factor-Model-Based

R portfolio = a p + b1 f 1 + b2 f 2 … bk f k + ε p

where: aP = expected portfolio return if all sensitivities equal 0 bk = sensitivity to systematic factors f k = systematic factors ε p = residual return from non-systematic factors

30


F��-I�� P�� M�� (�)

INTRODUCTION TO FIXED�INCOME PORTFOLIO MANAGEM MANAGEMENT ENT

I�� I�� F��-I�� P�� M�� Expected Return Decomposition E ( R ) ≈ Yield income

+Rolldown +Rolldown return return +E( +E(∆P base based d on inv investor’ or’s yiel ield and sprea pread d view view)) –E(Cred –E(Credit it losses) losses) +E(Cur +E(Curre rency ncy gains gains or losses losses))

where: Yield income = Annual 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

r P

=

Portfolio return Portfolio equity

= r 1+

V B V E

=

[r1 × (V

E

− VB

) − (VB

× rB

) ]

V E

(r1 − r B )

where: r 1 = Return on the unlever unlevered ed portfolio r B = Borrowing costs V E = Equity V B = Borrowed amount The remainder of the equation indicates the return effect of leverage such that r1 > rB results in a positive contribution from leverage. Leverage Using Futures

Leverage Futures =

Notional value – Margin Margin

Securities Lending

Rebate rate = Collateral earnings rate − Security lending rate

32


INTRODUCTION TO FIXED�INCOME PORTFOLIO MANAGEMENT

Total Return Analysis 1

 Total Total future future dollar dollarss  n Semi Semian annu nual al tota totall retu return rn =  −1 Full pric pricee of the the bond bond   Full  Dollar Duration Dollar duration = Duration × Bond value1 × 0.01 1

Note: Bond value is market value not par par value

Spread Duration

Three major types of spreads: •

•

•

Nominal spread is is the difference between the portfolio yield and the treasury yield for the same maturities. Zero-volatility spread spread (Z-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) (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 Liability portfolio BPV – Asset portfolio BPV

N f

=

Futures BPV

≈

Futures BPV BPV CTD CF CTD

where: N f = number of futures contract to immunize portfolio BPV = Basis point value CTD = Cheapest to deliver CF = Conversion factor


33

INTRODUCTION TO FIXED�INCOME PORTFOLIO MANAGEM MANAGEMENT ENT

NP

=

Liability portfolio BPV – Asset portfolio BPV Swap BPV

×

100

where: NP = Notional principal of the swap Index Matching to a Fixed-Income Portfolio Acti Active ve retu return rn = Port Portfo foli lio o retu return rn – Benc Benchm hmar ark k inde index x retu return rn

34


F��‐I�� P�� M�� (�)

YIELD CURVE STRATEGIES

Y�� C�� C� �� S�� Yield Curve Measurement

Butterfly Butterfly spread spread = –(Short-term –(Short-term yield) yield) + ( × 2 Medium-term Medium-term yield) yield) – (Long-term (Long-term yield Higher spread values indicate greater curvature.

36


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 = holding period (i.e., fractional portion of the year) t = 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


37

E�� P�� M��

EQUITY PORTFOLIO MANAGEM MANAGEMENT ENT

E�� P�� M�� M�� Manager Active Return

Info Inform rmati ation on ratio ratio =

Active Active return return Active Active risk risk

where: Active return = Portfolio return – Portfolio’ Portfolio’ss 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 Portf olio 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 r Ai i =1

n

Aσ P

=

∑h

2 Ai

2

σ Ai

i =1

where: ARP = Active return for a portfolio of managers r Ai = each manager’s active return h Ai = 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)


39

A�� I�� P�� M��

ALTERNATIVE INVESTMENTS FOR PORTFOLIO MANAGEMENT

A�� A�� I�� P�� M�� Spot Return E ( ∆F ) = E ( ∆S )

where: = forward price F = = spot price S = Roll Return (or Roll Yield) r Roll =

Ft −1 − Ft − ( St − St −1 ) ∆ F − ∆S

=

F t −1

F t −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 − NAV t −1 NAV t −1

=

NAV t NAV t −1

−1

Rolling Return RRn,t = ri + rt −1 +  + ri − (n−1)  / n

Downside Deviation (Semideviation)

∑ i=1[ min(ri − r *, 0)] n

Downsid Downsidee deviation deviation =

2

n −1

where: r i = return on asset i r * = specified return


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: HWM t = high-water mark at time t Lt+n = low after the same high-water mark Performance Appraisal Sharpe Ratio Sharpe Sharpe ratio ratio j =

(r j − r F )

σ j

where: r j = return on asset j r F = annualized risk-free rate σ j = standard deviation of asset j returns Sortino Ratio Sortino Sortino ratio ratio =

r j − r F DD

where: DD = Downside deviation Gain‐to‐Loss Ratio G/L ratio atio =

42

#monthly #monthly gains gains #monthly losses losses

×

Averag Averagee gain Average Average loss


R�� M��

RISK MANAGEMENT

R�� M�� Portfolio Theory rP = w1r1 + w2r2

2

2

2

2

2

σ P = w1 σ1 + w2 σ 2 + 2w1w2 ρσ1σ 2

where: r P = portfolio return 2

σ P = portfolio variance w =

portfolio weights for assets 1 and 2 = returns for assets 1 and 2 r = σ = standard deviation for assets 1 and 2 ρ = correlation between assets 1 and 2

Value at Risk (VAR)

Miniumum $ VaR = VP × [ E ( RP ) − Zα σ P ] where: V P = portfolio value E ( R RP) = expected portfolio return level zα = the number of standard deviations at the selected confidence level returns (e.g., 1.65 for 5% and 2.33 for 1%) σP = standard deviation of portfolio returns Note: To 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


R�� M�� M�� A�� A�� D�� D��

RISK MANAGEMENT APPLICATIONS OF FORWARD AND FUTURES STRATEGIES

R�� M�� A�� F�� F�� S�� Managing Equity Risk by Beta Adjustment Adjustment

βT S = β S S + N f β f f N f

 β − β   S  =  T S     β f   f 

β S βT

is the current beta of an equity portfolio. is the target beta of an equity portfolio: the desired level level of beta after hedging. S is the market value of a current equity portfolio. equal to β f is the beta of the index futures. It is often close to one, but may not be exactly equal one. f is the futures price of market index futures. N f is the number of futures contracts 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

is the amount of money to be inve invested. sted. is the futures price of market index futures. is the price multiplier multiplier of the futures futures contract contract (e.g., the S&P Index futures is $250). is the time to maturity of the futures contract. is the risk‐free interest rate. δ is the dividend yield of the market index. S t is the level of stock index at time t . N f is the number of futures contracts. V f q T r

* N f is the rounded (whole number) number number of futures contracts.

Futures payoff = N f * q (ST – f )

V =

*

N f

46

( N f * qf ) (1 + r )T

=

V (1 + r )T fq


RISK MANAGEMENT APPLICATIONS OF FORWARD AND FUTURES STRATEGIES

Creating Cash Out of Equity

Unit of stoc tock = 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

 MDURT − MDURB   B  =   MDUR f     f B   βT − β S  S   β f  f s

Nsf = 

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:

V DC

= VFC × F DC FC


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 = Risk‐free rate of interest r = 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 = V 0 = S 0 + c0 Value at option expiration = V T = S T − cT = S T − max(S T − X , 0) Profit at option expiration

=

V T − V 0 = [S T − max(S T − X , 0)] − [S 0 − c0]

Max profit = X – – [S 0 – c0 ] when ST ≥ X Max loss = S 0 – c0 when S T = 0 Breakeven point = S *T = S 0 − c0 Protective Put Value at initiation = V 0 = S 0 − p0 Value at option expiration = V T = S T + pT = S T + max[ X − S T , 0) X − Profit at option expiration = V T − V 0 = [S T + max[ X − S T , 0)] − [S 0 + p0] X − Max profit = ∞ when S T approaches ∞ Max loss = [S 0 + p0 ] − X when when S T ≤ X Breakeven point = S *T = S 0 + p0

Bull Spreads Value at initiation = V 0 = c1 − c2 Value at option expiration = V T = max(S T − X 1, 0) − max(S T − X 2, 0) Profit at option expiration = V T − V 0 = [max(S T – X 1, 0) − max(S T − X 2, 0)] − [c1 − c2] Max profit = [ X X 2 – X 1] − [c1 − c2] when S T ≥ X 2 Max loss = c1 − c2 when S T ≤ X 1 Breakeven point = S *T = X 1 + [c1 − c2]

48


RISK MANAGEMENT APPLICATIONS OF OPTION STRATEGIES

Bear Spreads Value at initiation = V 0 = p2 – p1 Value at option expiration = max[ X X 2 –S T , 0) – max( X 1 – S T , 0) Profit at option expiration

=

V T − V 0 = [max( X X 2 − S T , 0) – max( X 1 – S T , 0)] – [ p2 – p1]

Max profit = [ X X 2 − X 1] – [ p p2 − p1] when S T ≥ X 1 Max loss = p2 – p1 when S T ≥ X 2 Breakeven point = S *T = X 2 – [ p p2 – p1] Butterfly Spreads Value at initiation = V 0 = c1 – 2c2 + c3 Value at option expiration = V T = max(sT – X 1, 0) – 2max( S T − X 2, 0) + max(S T − X 3, 0) Profit at option expiration

=

V T − V 0 = [max[S T – X 1, 0) – 2max[ S T − X 2, 0) + max(S T − X 3, 0)] – [ c1 – 2c2 + c3]

Max profit = [ X X 2 – X 1] – [c1 – 2c2 + c3] when S T = X 2 Max loss = c1 – 2c2 + c3 when ST ≥ X 3 or ST ≤ X 1 , Breakeven points = S *T = X 1, + [c1 – 2c2 + c3] and 2 X 2 – X 1, – [c1 – 2c2 + c3] Collars Value at initiation = V 0 = S 0 + [ p p1 − c2] = S 0 + 0 = S 0 Value at option expiration = V T = S T + max( X X 1 − S T , 0) – max( S T – X 2, 0) Profit at option expiration = V T − V 0 = [S T + max( X X 1 − S T , 0)] − max(S T − X 2, 0) − [S 0] Max profit = X 2 − S 0 when S T ≥ X 2 Max loss = S 0 − X 1 when S T ≤ X 1 Breakeven point = S*T = S 0 Straddles Value at initiation = V 0 = c0 + p0 Value at option expiration = V T = max(S T − X , 0) + max( X X − − S T , 0) Profit at option expiration = V T − V 0 = [max(S T − X , 0)] + max( X X − − S T , 0) − [c0 + p0] Max profit = ∞ when S T → ∞ Max loss = c0 + p0 when S T = X Breakeven point = S*T = X +[c0 + p0] and X − [c0 + p0 ]


49


Option Strap Strip = c + 2 p Strap = 2c + p Box Spread Value at initiation = V 0 = [c1 – c2 ] + [ p p2 – p1] Value at option expiration = V T = X 2 – X 1 Profit at option expiration = V T – V 0 = [ X X 2 – X 1] – [[ c1 – c2 ] + [ p p2 – p1]] Max profit = Profit = [ X X 2 – X 1] – [[c1 – c2] + [ p p2 – p1]] regardless of terminal stock price Max loss = None Breakeven point = None Interest Rate Option Strategies Cal Call optio ption n pay payoff

tionaal pri princip cipal × max(R ax(Rea eali lize zed d spot pot rate − Exerci ercisse rate,0) te,0) = Notion

× Put option tion payof ayofff

Days Days in under underlyi lying ng rate rate 360

otiona nall princi incip pal × max(E x(Exercise rate ate − Real Realiized zed spot pot rate,0 ate,0)) = Notio

×

Days Days in under underlyi lying ng rate 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

 m  = VL ( Libort −1 + SL )  360 

where: iFRN = loan interest on the floating rate note V L = loan value; the amount of the loan Libor t -1 -1 = Libor on the previous reset date S L = spread over Libor m = actual days in the settlement period

PayoffCaplet

= VL × max(0, Lt −1 − r x )

m

360

where: r X = exercise rate for the cap 50


RISK MANAGEMENT APPLICATIONS OF OPTION STRATEGIES

Combining Floorlets with a Floating Rate Loan Loan interest is calculated as with caplets.

PayoffFloorlet

= VL × max(0, rx − 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 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. Effectivee interest = interest due – caplet Effectiv c aplet 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

Opti Option on delt deltaa =

Option Option gamma gamma

Optio Option n vega vega

=

Chan Change ge in opti option on pric pricee Chan Change ge in the the unde underrlyin lying g stoc stock k pric pricee

=

=

∆ c ∆S

Chan Change ge in opti option on delt deltaa Chan Change ge in the the unde underl rlyi ying ng stoc stock k pric pricee

Chan Change ge in opti option on pric pricee Change Change in annual annualize ized d stock stock retur return n volati volatilit lity y

Delta Hedge

Delta Delta Hedge Hedge Ratio Ratio =

 1  = −   ∆c / ∆S  N S N c

where: = number of call options c; number of shares S N = ∆c = change in call option price ∆S = = change in share price ∆c /∆S = = option delta


51


R�� M�� A�� S�� S�� Convert Conve rt a Floating Rate to a Fixed Rate

[Pay fixed and receive floating interest rate swap] = Floating-rate bond – Fixed-rate bond Floa Floati ting ng--rate rate bond bond = Fixe Fixedd-ra rate te bond bond + [Pay [Pay fixed ixed and and recei eceive ve floa floati ting ng inte intere rest st rate rate swap swap]]

− Float loatin ingg-rrate ate bond bond = − Fixed ixed--rate bond bond − [Pay [Pay fixed ixed and recei eceive ve floa loating ting inte interrest est rate swa swap] Fixe Fixedd-rrate ate bond bond = Floa Floati ting ng--rate rate bond bond − [Pay [Pay fixed ixed and and rece receiv ivee floa floati ting ng inte interrest est rate rate swap swap]]

− Fixed ixed--rate bond bond = − Float loatin ing g-rate bond bond + [Pay [Pay fixed ixed and recei eceive ve floa loating ting inte interrest est rate swa swap] Change the Duration of a Fixed‐Income Portfolio

[Pay [Pay fixe fixed d and and rece receiv ivee floa floati ting ng inte intere rest st rate ate swap swap]] = Float loatin ingg-rrate ate bond bond − Fixe Fixedd-rrate ate bond 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 (fixed‐rate bond)

−

Duration of (floating‐rate bond)

Duration Management NP = V P

×

MDURT

− MDURP

MDURS

VP ( MDURP ) = VP ( MDURT ) − NP ( MDUR S )

$ DURP + S

= $DURT

where: NP = notional principal of the swap V P = value of the portfolio M DUR = target, portfolio, and swap modified durations $ DUR = dollar duration

*Note: To decrease duration ( MDURT < MDURP) the denominator needs needs to be negative negative so the investor must be the fixed rate payer.

52



Floating Rate Notes

Leveraged Lever aged Floating-rate Notes CF Leveraged floater = − kL ( FP ) CF Bond = ( ci ) k ( FP )

= Lk ( FP ) ) ( ) ( CFFixe Fixed d side side of swap swap = − FS k FP CF Net = kFP (ci ci − FS )

CFFloa Floati ting ng side side of

swap swap

where: = net cash flow from the perspective of leveraged floater issuer CF = = leverage ratio k = L = receive floating rate on the swap FP = face value/principal 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 CF Net = 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 =

 360  t

CF F  t

  360  NP D = NPF × S D / F or NPF  t  CF D = NPD (iD )  360  (iF )

S F / D

where: CF F = Foreign cash flow receipt to be swapped NPF = Notional principal of foreign currency to be swapped at iF to achieve periodic CF F iF = interest rate applicable to notional principal of foreign currency to be swapped t = = number of days between cash flows to be received NP D = Notional principal of domestic currency swapped for foreign currency S D/F = spot rate for foreign currency priced in domestic currency S F/D = spot rate for domestic currency priced in foreign currency CF D = Domestic cash flow received in exchange for swapped foreign cash flow


53


Creating Dual‐Currency Bonds

Dual currency bond = Conventional bond + Currency swap (without exchanging NP) Swaptions

Receiver Receiv er swaption = Call option on a bond Payer swaption = Put option on a bond Add or Remove Embedded Options in a Bond

Receiver Receiv er 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(r 0 )

where: = Non-callable bond cash flows NCF = NP = notional principal F(t,T) = prevailing swap rate s = credit spread over Libor = strike rate X = t he non-callable bond with credit spread s over Libor r 0 = interest rate on the Creating a Synthetic Embedded Call

(2a) −Callable bond = −Straight bond + Receiver swaption NCF = NP × [− Noncallable bond fixed rate −

ike rate − Prevailing swap rate+ LIBOR ) ( LIBOR + Strike

= NP × [−r0 − LIBOR + X − F (t , T ) + LIBOR] = −r0 + (r0 − s) − F(t, T ) = NP × −[F (t , T ) + s]

54


T��, M��, �� R��

EXECUTION OF PORTFOLIO DECISIONS

E�� P�� D��  

Effective spread = 2 × Execution price −

(Bi (Bid price + Ask price)  2



Implementation Implementat ion Shortfall Costs for purchases Explicit Costs Expicit Expicit costs costs =

Comm Commis issi sion ons, s, taxe taxes, s, and and fees fees S H

× PH

Realized Profit/Loss

RPL

=

S

× ( P E − PR ) S H × PH

Slippage (Delay Costs)

Delay =

S

× ( P R − PH ) S H × PH

Unrealized Profit/Loss (Missed Trade Opportunity Cost)

UPL

=

( S H − S ) × ( PL − PH ) S H

× PH

Implementation Shortfall Implementation Implementation shortfall shortfall =

Commissions + S (P E − PL ) + SH ( PL − PH ) S H

× PH

where: = shares executed S = S H = hypothetical shares executed P H = hypothetical price; benchmark price P E = execution price P R = relevant price when shares are not purchased at the hypothetical price P L = last available valuation price

56


M�� R�� The Perold‐Sharpe Analysis of Rebalancing Strategies VPort Portfo foli lio o

= VRisk Risk-f -fre reee + V Stoc Stocks ks

Buy‐and‐Hold Strategies

m

=

V Stocks

(V Portfolio − Floor)

=

V Stocks

(V

Port Portfo foli lio o − V Risk Risk-f -fre reee0

)

Constant Mix Strategies

m

=

V Stocks V Portfolio

Constant Proportion Portfolio Insurance (CPPI) Investment Inve stment in Stock = M (V portfolio − Floor)


57

P�� E� E��

EVALUATING PORTFOLIO PERFORMANCE

E�� P�� P�� The Holding Period Return MV MVt − MV0 r t = MV 0

=

MV t MV 0

−1

where: MV t is the market value of the account at time t ; and r t is the holding period return over the investment period. Cash Inflow Occurs at the Beginning of the Reporting Period MV MVt − ( MV0 + CF0 ) r t = MV0 + CF 0

Cash Inflow Occurs at the End of the Reporting Period r t =

( MVt

− CFt ) − MV0 MV 0

Total return = Income yield + Capital gains yield The Time‐Weighted Rate of Return rtwr

= (1 + rt1 )(1 + rt 2 ) … (1 + rtn ) − 1

where: r ti is the holding period return of sub‐period i, and there are a total of n sub‐periods. Annualizing Returns rannual

1/ n

= n (1 + ryr .1 )(1 + ryr .2 )…(1 + ryr .n ) − 1 = (1 + r yr .1)(1 + ryr .2) …(1 + ryr .n )  −1


59


Benchmarks P = P P = B + (P − B)

+ ( B + S + + A P = M + B − M ) + (P − B) = M + P B M A S

is the return return performance of a portfolio under management. is the return on on a benchmark that matches the portfolio’ portfolio’ss investment investment style. is the return 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 + ( B + S + + A P = M + B − M) + (P − B) = M + Correlation( A, A, S ) = 0 Correlation( E E , S ) = Correlation(P − M , B − M) > 0 Tracking Error Tracking error = Volatility( A) < Volatility( E ) = Volatil Volatility( ity(P − B) < Volatility(P − M ) where: Volatility( E E ) = Excess volatility of returns from the managed portfolio over returns from the market portfolio Hedge Fund Benchmarks M MV Vt − MV0 r t = MV 0 rv

= rP − r B

where: r v is the value‐added return. r P is the portfolio return. r B is the benchmark return.

60



Sector Weighting Micro Attribution S

Selection effect =

∑= [w

Bi

× (rPi − r Bi )]

i 1

S

Sector allocation effect =

∑= [(w

Pi

− wBi ) × (rBi − rB ) ]

i 1

S

Interaction effect =

∑= [(w

Pi

− wBi ) × (rPi − rBi ) ]

i 1

Ex Post Alpha Also known as Jensen’s alpha Rt

− r ft = α + β( RMt − r ft ) + εt

where for period t : Rt = the portfolio return r ft = the risk-free return R Mt = 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

T A

=

R A − r f

β A



Sharpe Ratio

Shar Sharpe pe ratio ratio A

=

R A − r f

σ A



M‐Squared ( M 2)

M 2

 R − r  = r f +  A f  σ M  σ A  




61


Information Ratio

IR A

=

R A − RB

σ A− B



where: σ A–B = standard deviation of the differential returns of the asset over the benchmark

62


G�� I�� P�� S��

OVERVIEW OF THE GLOBAL INVESTMENT PERFORMANCE STANDARDS

O�� G�� I�� P�� S�� Return Calculation Methodologies

Total Return r t =

V1 − V 0 V 0

Time‐Weighted Return rtw

= (1 + r1 )(1 + r2 )(1 + rn ) − 1

Modified Dietz Method r modDietz

=

V1 − V0

− CF V0 + ∑ (CFi × wi ) t =1 n

Weighting Formula wi

=

CD − Di CD

where: wi weight of cash flow i CD calendar days in the period Di calendar days since the beginning of the period to receipt of cash flow i =

=

=

Modified Internal Rate of Return Approach n

V1

= ∑ CFi (1 + r )w  + V0 (1 + r ) i

t =1

Original Dietz Method r Dietz =

64

V1 − V0 V0

CF − CF

+ 0.5CF



Composite Returns

Beginning Assets Method

 V  0, pi  rc = ∑ r pi   ∑ n V 0, pi   pi =1 where: r c composite return r pi individual portfolio returns V 0 ,pi individual portfolio beginning value =

=

=

Beginning Assets Plus Weighted Cash Flows Method

rc

 V  = ∑ r pi  pi   ∑V pi 

Presentation and Reporting

∑ = (r − r )

2

n

S C =

i 1

i

C

n −1

where: S c = standard deviation of returns in the composite r i = individual portfolio returns r c = composite returns n

SC ,aw

=

∑= (r − r i

2

C ,aw

)

wi

i 1

= wi −

V 0,i

∑ = V = ∑ w × r = n

i 1 0, i

= rC ,aw

n

i

i

i 1

where: S C,aw asset-weighted standard deviation wi beginning-of-period weight for portfolio i in the composite V 0 ,i beginning-of-period value of portfolio i =

=

=


65


L y =

rY

y

100

(n + 1)

= rA − ( L y − n A, Ly )(r A − rB ) 

where: L y 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: Equal s: Gross Gross-of-of-fee fee returns returns Less:

Management fees

Equals Equ als:: Ne Net-o t-of-f f-fee ee ret return urnss

rann

=

n

n

∏= (1 + r ) − 1 = (1 + r )(1 + r )(1 + r )…(1 + r ) 

1/ n

t

1

2

3

n

−1

t 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 + r Quarterly ) − 1


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2018 cfa Level 3 Wiley Formula Sheet

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