Leverage, Exposure & Risk
Starting point definition of return r(0,T) on beginning capital A0 between 0 and T...
{ A(0) + Inc(0,T) + C(0,T) } / A(0) = 1 + {Inc(0,T) + C(0,T) } / A(0) = 1 + r(0,T)
Inc(0, T)... income
C(0,T)... capital gains
Gains and Losses = Inc(0, T) + C(0, T)
Leverage L defined as a gains and losses multiplier...
{ A(0) + L*{Inc(0,T) + C(0,T) }} / A(0)
= 1 + L * {Inc(0,T) + C(0,T) } / A(0)
= 1 + L * r(0,T)
Exposure E defined as notional amount which corresponds to given leverage l, return r(0,T) and beginning capital A0, i.e. expressing leverage as an "exposure / capital multiplier".
Leverage as a capital multiplier...
a(T) = A(0) * r(0,T) = Inc(0,T) + C(0,T)
e(T) = A(0) * L * r(0,T) = E(0) * r(0,T)
E(0) > A(0) if L > 1
Result: Exposure is a positive function of leverage and vice versa.
Expressing leverage as a $ exposure in addition to initial capital invested...
L * A(0) = A(0) + X(0)
X(0) = A(0)*{L-1} respectively L = 1 + X(0)/A(0)
Calculating leverage from exposures
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(Market) Risk exposure = exposure to market risk = exposure to "non-cash assets"
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Long market risk exposure = market risk exposure of long positions
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Short market risk exposure = market risk exposure of short positions
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Net Market Risk exposure = Long market risk exposure - Short market risk exposure.
Alternative formulation: Leverage = Net Market Risk Exposure > Net Asset Value.
Typical Quoting Conventions for Leverage by Hedge Fund Strategy...
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Convertible Arbitrage: Gross Longs / Net Assets
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Equity Hedge = (Gross Longs + Gross Shorts)/Net Assets
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Event Driven = (Gross Longs + Gross Shorts)/Net Assets, with “hedges” not included (e.g. Credit Default Swaps)
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Distressed = Gross Longs/Net Assets
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Merger Arbitrage = (Gross Longs + Gross Shorts)/Net Assets
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Equity Market Neutral = (Gross Longs + Gross Shorts)/Net Assets
Exposure Based Leverage=(Longs+Shorts)/NAV
Liquidation Based Leverage=(Longs)/NAV
Netting issues
...
Market Risk is the uncertainty about gains and losses over the time period 0 to T. Risk is usually measured as the resulting volatility in returns, calculated as the annualized standard deviation of returns.
Risk of leveraged returns...
var[ 1 + L * r(0,T)) ] = L^2 * var[r(0,T)] = L^2 * v^2
v(L)= L * v
Result: "risk is proportional to leverage", market risk as a positive linear function of leverage
Result: Leverage is a deterministic concept, risk a probabilistic one.
Leverage allows to magnify exposures and, as a direct consequence, magnify risks. The term leverage can be defined in balance-sheet terms, in which case it refers to the ratio of assets to net worth. Alternatively, leverage can be defined in terms of risk, in which case it is a measure of economic risk relative to capital.
Default Risk defined as a situation in which gains and losses exceed the initial capital...
Default | e(T) < A(0)
A(0) < A(0) * L * r(0,T)
1 < L * r(0,T)
Default | r(0,T) < 1 / L
Result: Leverage can be interpreted as the adverse market move of minus 100 / L percent necessary to drive a portfolio into "default".
Result: A Leverage > 1 increases the likelihood of default.
Situations r(0, T) < -100% are not zero sum games anymore and can result in problems for the financial system as a whole: Leverage has the potential to increase "systemic risks" (and is therefore permanently on the radar of regulators).
Leverage, Beta & Delta: Certain instruments (especially derivatives) may not constitute "balance-sheet" leverage, but might represent "economic" leverage (i.e., they display heightened price sensitivity to market fluctuations). Difference between Beta > 1 and Risk Exposure > 1: Beta measures gains and losses relative to a "market" or similar broad risk factors. Leverage measures gains and losses relative to the capital dedicated. Delta measures gains and losses relative to the underyling risk factor (-> Leverage of a call option is equal to 1 / Delta).
Financial Gearing as a synonym for leverage: 1. The ratio of a company's debt to equity. Gearing is an indicator of a company's ability to service its debt. A company with a high proportion of debt to equity (high gearing) is more vulnerable to fluctuations in business activity. It therefore represents higher risk for equity holders and offers greater return. Whether gearing is acceptable or not is often judged by comparisons with companies in similar industries or sectors. 2. In derivatives markets, gearing is the measure of the amount of cash spent purchasing an option or a futures contract, compared to the actual value of the underlying position. The more highly geared the trading position the greater the risk that a small change in market prices will completely wipe out the cash investment. That also means that a small change in market prices in the right direction can produce large profits in relation to the size of the cash investment.
Leverage Implemention
Economic leverage can be obtained in various ways, such as through the use of repurchase agreements, short positions, and derivatives.
Two basic mechansims: a) borrowing/lending, b) unfunded instrument.
a) Leverage with Borrowing
Return of the resulting portfolio...
A(0) * rp = {A(0) + X(0)}*r(0,T) -X(0)*rf
...substituting X(0) = A(0)*{L-1}...
A(0) * rp = {A(0) + A(0)*{L-1}}*r(0,T) - A(0)*{L-1}*rf
rp = L*r(0,T) -{L-1}*rf
Risk of the resulting portfolio...
var[rp] = var[L*r(0,T) -{L-1}*rf] = L^2 * var[r(0,T)]
Related: Short Selling (a very simplistic case)...
A(0) * rp = {A(0) - X(0)}*r(0,T) + X(0)*rf = {A(0) - A(0)*{L-1}}*r(0,T) + A(0)*{L-1}*rfrp = (2-L)*r(0,T) + {L-1}*rf
...
b) Leverage with Unfunded Instruments
Case of a linear unfunded instrument...
A(0) * rp = {A(0) + X(0)}*r(0,T)
More realistic: Margin requirements (initial + maintencance), collateral etc. Example initial margin...
A(0) * rp = {A(0) + X(0) - D(0)}*r(0,T) + D(0)*rf
...with initial margin requirement D(0) = m*X(0)...
A(0) * rp = {A(0) + X(0) *{1- m}}*r(0,T) +m*X(0)*rf
A(0) * rp = {A(0) + A(0)*{L-1}*{1- m}}*r(0,T) +m*A(0)*{L-1}*rf
rp = {L-m*{L-1}}*r(0,T) +m*{L-1}*rf
...
Non-linear instruments: delta-adjustments
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Case study: UCITS III product directive.
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