18

Sharpe ratio is defined as $\frac{(x - r)}{\sigma}$ where $x$ is return, $r$ is the risk free rate and $\sigma$ is volatility. Now levering up $n$ times multiplies both the return and volatility by $n$. But shouldn't the ratio change since $r$ stays the same? Ah, but remember, leverage isn't free. You have to fund leverage, and that cuts out of your return. ...


16

The key to this is to think about the enterprise value of a business separately from how it is financed. For simplicity sake, consider a business that comprises a sole gold bar (no workers, no extraction costs, etc). The value of the business is clearly just the value of the gold bar. If it were a listed company, with no debt, then the equity ...


8

The textbook academic answer is that Sharpe ratio is not impacted by leverage as explained by other answers. However, reality tells a different tale entirely: Imagine you lever up your investments by such amount that your future performance will critically hinge on the following conditions: That those who extended credit to you will not re-call their ...


6

Generally no. Sharpe ratio should vary linearly. Use leverage: the return increases, but so does volatility. De-lever" the return decreases but, so does volatility.


4

To answer your questions: 1) Yes, the above table is correct 2) Your results are correct except..... 1X loss = 9.6%. When you combine both positive and negative changes, it is the MEDIAN value that is of interest. Here are some links: http://www.futuresmag.com/Issues/2010/March-2010/Pages/Trading-with-leveraged-and-iinverse-ETFs.aspx http://...


4

Ciao, I'm studying this problem from a while. Let me post the graph obtained numerically. I've used the following parameters: $$ \left\{ \begin{array}{rcl} S &=& 2 \\ r &=& 0.01 \\ \sigma &=& 0.2 \\ K &\in& [0.1, 10] \\ T &=& 5 \\ t &=& 1 \end{array} \right. $$ This is not good since the function is always ...


4

Let us assume: a constant risk-free rate $r$ a risky asset with returns $X$ with expected value $\mathbb{E}(X)=\mu_X$ and variance $\text{Var}(X)=\sigma_X^2$ a portfolio investing $w$ in the risky asset and $(1-w)$ in the risk-free asset Then you can compute the expected value of the portfolio: $$\mu_P = \mathbb{E}(P) = w \mu_X + (1-w)r$$ and variance ...


3

There is a little known Greek called "lambda" or leverage which equals Delta times Stock price divided by option price $\lambda=\frac{\Delta . S}{c}$. So if $\lambda>1$ the option could be said to be leveraged, meaning the dollar value of a delta equivalent amount of stock is greater than the market value of the option.


3

You borrow that money from your broker. If you are retail client with for example IG they will offer this service to you. They will charge you some interest for the lending. There will be a margin account into which you need to deposit your cash. If the leveraged position you have loses money on the M2M and amount in margin account goes below a threshold, ...


3

Quite a good article can be found here: http://seekingalpha.com/article/3140956-investing-in-leveraged-etfs-theory-and-practice Just selling a pair of leveraged ETFs to harvest the "volatility decay" is comparable to a short straddle... highly skewed and therefore quite dangerous (from the article): There are no free lunches in the market. The apparent ...


3

I you take a long position in a stock the worst that can happen is the stock goes to zero, and you lose your investment. If you take a short position in a stock, you have the potential for unlimited loss (since there is no limit to how high the stock can go). Exactly the same principle pertains in FX. If your quote currency (or home currency) is EUR and ...


3

First of all you need a model to generate future returns, I assume you already have this. Since its just a model, there will be an unexplained component in the predictions made for every period $t$ and for every asset $i$. Let $\varepsilon_{t, i}$ denote this random innovation and $\mathrm{E}[r_{t, i}] = f(\varepsilon_{t, i})$ the expected asset return as ...


3

It is true that you don't change your risk/return ratio but you can scale the ingredients of this ratio, meaning that you can e.g. scale up the level of risk you are prepare to take to also lever up your returns. Through that mechanism you can make use of very small spreads.


3

It depends obviously on which specific leverage you attempt to measure but you can certainly build some sort of index from, for example, the below: Aggregate smoothed equity P/E ratio divergence from long term mean (in a sense it reflects how money is levered to buy stocks at multiples of their long term P/E mean). Broad money in circulation -> Money ...


3

It depends a lot on the structure of the ETF, it could be : * In the "terms and conditions" of the (highly possible) total return swap of the fund * Portfolio insurance * Option combination (or cap & floor) I think it's in the swap details, already saw that a few times.


3

Apologies in advance for being hyper-critical. I have somewhat strong feelings about this =P The purpose of risk parity is to improve portfolio efficiency via achieving better diversification. (We won't delve into philosophical debates about whether or not this is true here...) Mechanically, by leveraging up low risk assets (e.g., US Treasuries) and ...


2

cost of leverage for equity only long/short investing is a function of the margin deal you can negotiate with your broker, if you have a large amount of capital. If you don't have significant capital to start with, then it's likely you'll only be able to get 2x leverage with a loan rate between 4% and 10% (retail reg-t margin rates at most brokers) This ...


2

Even in a perfect world, a 3X leveraged ETF cannot achieve a compound return three times that of the underlying. In the case of periodic discrete rebalancing, we call this effect the "arithmetic of loss and recovery," but even in the limit of continuous rebalancing, this effect does not disappear. Ito's formula tells us that $$\mathrm d \log(S_\textrm{...


2

Your example could be correct but you're on the wrong track. Leveraged ETFs are designed for day trading, it isn't a leveraged 3x position that will return 3x the long term average of the name. The leverage is reweighted each day which will affect your performance. Eg if the market goes 100->99->100 the market is unchanged over 2 days. But a 3x ETF will go ...


2

High level Flow of funds comparative analysis for the U.S., Japan, and Euro Area by the bank of Japan. Country level report from the ECB. It is an 800+ page report so the link may take time to load (alternatively go to ECB data warehouse/reports/Euro Area accounts). Canadian financial flow accounts data.


2

The leverage is conceptual (as you're not borrowing something to buy more of something in the standard form of leverage). I think it'll become clear when you compare an equity tranche position to a position in the underlying index. An equity tranche on CDX IG, 0-3%, would incur a 26.6% loss if one of the constituents in the underlying index defaults. There ...


2

[Your] analysis is correct. The long trade is worth 100 - 100/FX, and the short trade is worth 100/FX -100, where FX=ending EURUSD exchange rate. The first expression has unlimited downside as FX -> 0, and the second expression is always > -100. ––– dm63 Feb 25, 2017 at 12:52


2

Sharpe ratio = $\frac{r_p - r_f}{\sigma_p}$, where: $r_p$ is the expected portfolio return $\sigma_p$ is the portfolio's standard deviation $r_f$ is the risk free rate. When you leverage '$n$' times: The leveraged portfolio return is $n r_p - (n-1) r_f$ (subtracting the cost of borrowing the money) The standard deviation increases to $n\sigma$ Hence: "...


2

One standard strategy is to short both "bull" and "bear" ETFs (usually called "double short"). A bit naive heuristics is that if you're loosing money holding a long position due to volatility, you can at the same time make money holding short positions.


2

In general any investment position is said to be leveraged, if it is financed by a debt position. This is with regards to options, stocks or any other security. Say you buy an option with maturity in one year at a premium of 100 USD, hold it to maturity and get a payoff of 120. You will have a profit of 20 USD, or 20% of your invested capital. Instead you ...


2

This would depend on whether the option you bought is cash- or physically-settled. Let $V_t$ be the intrinsic value of your option at time $t$, $T$ its maturity and $y$ the number of shares it gives right to. For example, for a call option of strike $K$ written on an underlying $S$ which price process is $(S_t)_{t \geq 0}$, the intrinsic value is $-$ ...


2

Leveraged ETF have negative gamma: the higher the volatility of the underlying index the bigger the negative drag. This is a big pitfall of those instruments because one can be correct with the overall forward direction of the market for say the next 1 year and still lose money with a LETF. For example if one bets the SPX will go down over next 12 months ...


2

I can see the genius is your investment thesis. I looked at the probability analysis in TOS and found that it is highly unlikely that the above options will go OTM on Jan 2019. Which means, this is the best alternative to a surefire bet. A leveraged bet at that. As for your concerns, dividends are priced into options anyway and if you are concerned about ...


2

Portfolio as it stands: 300% A -100% B Total: +200%. Ratio 1.5:0.5 Short additional 50% B. Now: 300% A -150% B 50% Cash Total: +200%. Ratio 1.5:0.75 The problem is that you have to rebalance vs the total value of the portfolio. To get back to having the amount of A being 2x the portfolio total, you need to buy more A, to get it up to 400%. I ...


2

Your leverage will be the amount of money you borrow to buy the risky portfolio P. Intuitively, the more you borrow money to buy P, the more you are exposed to market behaviour and so β will be high. As explained in the comment, if for instance your portfolio has a β of 2, you sell half of it and put your money at the risk free rate and get a general β of ...


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