The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsConstant proportion portfolio insurance

Constant proportion portfolio insurance

Step 1. Specify a target minimum “floor”

V floort = vfloor

tnowe∫ ttnow

Rrfs ds

, t ≥ tnow (9c.68)

risk-free rate

Step 2. Set the initial investment vstrattnow

> vfloortnow

Step 3. Compute the holdings

Hriskyt V risky

t ≡ mult× Cusht , mult ∈ [0, 1] (9c.71)

≡ V stratt −V floor

t excess cushion

⇓

dCushtCusht

= (1−mult)× dvrft

vrft

+ mult × dVriskyt

Vriskyt

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsConstant proportion portfolio insurance

Constant proportion portfolio insurance

Step 1. Specify a target minimum “floor”

V floort = vfloor

tnowe∫ ttnow

Rrfs ds

, t ≥ tnow (9c.68)

risk-free rate

Step 2. Set the initial investment vstrattnow

> vfloortnow

Step 3. Compute the holdings

Hriskyt V risky

t ≡ mult× Cusht , mult ∈ [0, 1] (9c.71)

≡ V stratt −V floor

t excess cushion

⇓

dCushtCusht

= (1−mult)× dvrft

vrft

+ mult × dVriskyt

Vriskyt

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsConstant proportion portfolio insurance

Constant proportion portfolio insurance

Step 1. Specify a target minimum “floor”

V floort = vfloor

tnowe∫ ttnow

Rrfs ds

, t ≥ tnow (9c.68)

risk-free rate

Step 2. Set the initial investment vstrattnow

> vfloortnow

Step 3. Compute the holdings

Hriskyt V risky

t ≡ mult× Cusht , mult ∈ [0, 1] (9c.71)

≡ V stratt −V floor

t excess cushion

⇓

dCushtCusht

= (1−mult)× dvrft

vrft

+ mult × dVriskyt

Vriskyt

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsConstant proportion portfolio insurance

Geometric Brownian motion

• vriskytnow

= $100, µ = 0.1, σ = 0.4

• vrftnow

= $100, rrf = 0.02

• vstrattnow

= $10, 000, vfloortnow

= $980, mult = 5

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsDrawdown control

Drawdown control

Step 1. Compute the high watermark, or running maximum

HWM t ≡ maxtstart≤s≤t

{V strats } (9c.75)

Step 2. Compute drawdown

DDt ≡ HWM t−V stratt (9c.76)

⇓ (V stratt > V floor

t )

DDt < HWM t−V floort (9c.77)

Step 3. Compute the holdings

Hriskyt V risky

t = mult×(V stratt −γ×HWM t), γ ∈ (0, 1) (9c.78)

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsDrawdown control

Drawdown control

Step 1. Compute the high watermark, or running maximum

HWM t ≡ maxtstart≤s≤t

{V strats } (9c.75)

Step 2. Compute drawdown

DDt ≡ HWM t−V stratt (9c.76)

⇓ (V stratt > V floor

t )

DDt < HWM t−V floort (9c.77)

Step 3. Compute the holdings

Hriskyt V risky

t = mult×(V stratt −γ×HWM t), γ ∈ (0, 1) (9c.78)

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update

The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsDrawdown control

Drawdown control

Step 1. Compute the high watermark, or running maximum

HWM t ≡ maxtstart≤s≤t

{V strats } (9c.75)

Step 2. Compute drawdown

DDt ≡ HWM t−V stratt (9c.76)

⇓ (V stratt > V floor

t )

DDt < HWM t−V floort (9c.77)

Step 3. Compute the holdings

Hriskyt V risky

t = mult×(V stratt −γ×HWM t), γ ∈ (0, 1) (9c.78)

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsDrawdown control

Geometric Brownian motion

• vriskytnow

= $100, µ = 0.1, σ = 0.4

• vrftnow

= $100, rrf = 0.02

• vstrattnow

= $10, 000, γ = 0.7, mult = 2

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy

Signal

Linear time invariant (LTI) filter

Lt =

∫ ttstart

b(t− s)dV riskys (9c.79)

⇓

dLt = L̃tdt+ b(0)dV riskyt (9c.81)

impulse response

≡∫ ttstart

b′(t− s)dV riskys

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy

Signal

Linear time invariant (LTI) filter

Lt =

∫ ttstart

b(t− s)dV riskys (9c.79)

⇓

dLt = L̃tdt+ b(0)dV riskyt (9c.81)

⇓ b(t− s) ∝ e−ln(2)τHL

(t−s)

Momentum

Smomt ≡ ln(2)

τHL

∫ ttstart

e− ln(2)τHL

(t−s)dV risky

s (9c.79)

⇓

dSmomt = ln(2)

τHL(−Smom

t dt+ dV riskyt ) (9c.81)

half-life

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy

Strategy

Consider the strategy

hrisky(It) ≡ hrisky(Lt) (9c.84)

⇓

Πtnow→thor︸ ︷︷ ︸P&L

= g(Lthor )− g(ltnow )︸ ︷︷ ︸option profile

−∫ thor

tnow

(g′(Lt)L̃t +1

2g′′(Lt)(b(0)σ(t, V risky

t ))2dt︸ ︷︷ ︸trading impact

(9c.86)

g(x) ≡ 1

b(0)

∫ x

c

hrisky(l)dl (9c.87)

volatility

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy

Strategy

Consider the strategy

hrisky(It) ≡ hrisky(Lt) (9c.84)

⇓

Πtnow→thor︸ ︷︷ ︸P&L

= g(Lthor )− g(ltnow )︸ ︷︷ ︸option profile

−∫ thor

tnow

(g′(Lt)L̃t +1

2g′′(Lt)(b(0)σ(t, V risky

t ))2dt︸ ︷︷ ︸trading impact

(9c.86)

signal = momentum of arithmetic

Brownian motion⇓Lt ≡ Smom

t , σ(t, V riskyt ) ≡ σ, hrisky(s) ≡ γs

Πtnow→thor = γ τHL2 ln(2)

((Smomthor

)2−(smomtnow

)2)+γ

∫ thor

tnow

((Smomt )2−σ

2

2ln(2)τHL

)dt (9c.87)

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy

Strategy

Consider the strategy

hrisky(It) ≡ hrisky(Lt) (9c.84)

⇓

Πtnow→thor︸ ︷︷ ︸P&L

= g(Lthor )− g(ltnow )︸ ︷︷ ︸option profile

−∫ thor

tnow

(g′(Lt)L̃t +1

2g′′(Lt)(b(0)σ(t, V risky

t ))2dt︸ ︷︷ ︸trading impact

(9c.86)

signal = value of risky instrument ⇓ constant filter b(·) ≡ 1⇒ L̃t = 0

Πtnow→thor = g(V riskythor

)−g(vriskytnow

)− 12

∫ thor

tnow

g′′(V riskyt ) (σ(t, V risky

t ))2 dt (9c.88)

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy

Strategy

Consider the strategy

hrisky(It) ≡ hrisky(Lt) (9c.84)

⇓

Πtnow→thor︸ ︷︷ ︸P&L

= g(Lthor )− g(ltnow )︸ ︷︷ ︸option profile

−∫ thor

tnow

(g′(Lt)L̃t +1

2g′′(Lt)(b(0)σ(t, V risky

t ))2dt︸ ︷︷ ︸trading impact

(9c.86)

Simple signal-induced strategy

Πtnow→thor = hrisky(vriskytnow

)×Πriskytnow→thor

+ γ × (Πriskytnow→thor

)2︸ ︷︷ ︸option profile

−γσ2

∫ thor

tnow

(V riskyt )2 dt︸ ︷︷ ︸

trading impact

signal = value reversal of risky

instrument under gBm⇓

{b(·) ≡ 1, σ(t, V risky

t ) ≡ σV riskyt

hrisky(V riskyt ) ≡ δ + 2γ(V risky

t − vtarget)

> 0 trend-following strategy< 0 mean-reverting strategy

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy

Simple signal-induced strategy

• V riskyt follows an Ornstein-Uhlenbeck process

• mean reverting half-life varies• vrisky

tnow= $100, µ = 0, σ = 0.01

• vstrattnow

= $10, 000, δ = 90, γ ∈ [−2, 2] varies

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