Mathematical coincidence

In probability theory, the Chernoff bound, named after Herman Chernoff, gives exponentially decreasing bounds on tail distributions of sums of independent random variables. It is a sharper bound than the known first or second moment based tail bounds such as Markov's inequality or Chebyshev inequality, which only yield power-law bounds on tail decay. However, the Chernoff bound requires that the variates be independent – a condition that neither the Markov nor the Chebyshev inequalities require.

It is related to the (historically earliest) Bernstein inequalities, and to Hoeffding's inequality.

Example

Let X₁, ..., X_n be independent Bernoulli random variables, each having probability p > 1/2. Then the probability of simultaneous occurrence of more than n/2 of the events {X_k = 1} has an exact value S, where

$S=\sum _{i=\lfloor \frac{n}{2}\rfloor +1}^n(\genfrac{}{}{0ex}{}{n}{i})p^i(1-p{)}^{n-i}.$

The Chernoff bound shows that S has the following upper bound:

$S\le 1-{\mathrm{e}}^{-\frac{1}{2p}n{\left(p-\frac{1}{2}\right)}^2}.$

Indeed, noticing that μ = np, we get by the multiplicative form of Chernoff bound (see below or Corollary 13.3 in Sinclair's class notes),

$\mathbf{P}(S\le \frac{n}{2})=\mathbf{P}(S\le (1-(1-\frac{1}{2p}))\mu )\le e^{-\frac{\mu }{2}{(1-\frac{1}{2p})}^2}=e^{-\frac{1}{2p}n{(p-\frac{1}{2})}^2}$

This result admits various generalizations as outlined below. One can encounter many flavours of Chernoff bounds: the original additive form (which gives a bound on the absolute error) or the more practical multiplicative form (which bounds the error relative to the mean).

A motivating example

File:Chernoff bound.png

The simplest case of Chernoff bounds is used to bound the success probability of majority agreement for n independent, equally likely events.

A simple motivating example is to consider a biased coin. One side (say, Heads), is more likely to come up than the other, but you don't know which and would like to find out. The obvious solution is to flip it many times and then choose the side that comes up the most. But how many times do you have to flip it to be confident that you've chosen correctly?

In our example, let X_i denote the event that the ith coin flip comes up Heads; suppose that we want to ensure we choose the wrong side with at most a small probability ε. Then, rearranging the above, we must have:

$n\ge \frac{1}{(p-\frac{1}{2}{)}^2}\ln \frac{1}{\sqrt{\epsilon }}.$

If the coin is noticeably biased, say coming up on one side 60% of the time (p = .6), then we can guess that side with 95% (ε = .05) accuracy after 150 flips (n = 150). If it is 90% biased, then a mere 10 flips suffices. If the coin is only biased a tiny amount, like most real coins are, the number of necessary flips becomes much larger.

More practically, the Chernoff bound is used in randomized algorithms (or in computational devices such as quantum computers) to determine a bound on the number of runs necessary to determine a value by majority agreement, up to a specified probability. For example, suppose an algorithm (or machine) A computes the correct value of a function f with probability p > 1/2. If we choose n satisfying the inequality above, the probability that a majority exists and is equal to the correct value is at least 1 − ε, which for small enough ε is quite reliable. If p is a constant, ε diminishes exponentially with growing n, which is what makes algorithms in the complexity class BPP efficient.

Notice that if p is very close to 1/2, the necessary n can become very large. For example, if p = 1/2 + 1/2^m, as it might be in some PP algorithms, the result is that n is bounded below by an exponential function in m:

$n\ge 2^{2m}\ln \frac{1}{\sqrt{\epsilon }}.$

The first step in the proof of Chernoff bounds

The Chernoff bound for a random variable X, which is the sum of n independent random variables $X_1,...,X_n$, is obtained by applying e^tX for some well-chosen value of t. This method was first applied by Sergei Bernstein to prove the related Bernstein inequalities.

From Markov's inequality and using independence we can derive the following useful inequality:

For any t > 0,

$\mathbf{P}(X\ge a)=\mathbf{P}(e^{tX}\ge e^{ta})\le \frac{E[e^{tX}]}{e^{ta}}=\frac{E[\prod _i{e}^{t{X}_i}]}{e^{ta}}.$

In particular optimizing over t and using independence we obtain,

Similarly,

$\mathbf{P}(X\le a)=\mathbf{P}(e^{-tX}\ge e^{-ta})$

and so,

$\mathbf{P}(X\le a)\le {\min }_{t>0}{e}^{ta}\prod _{i}E[e^{-t{X}_i}].$

Precise statements and proofs

Theorem for additive form (absolute error)

The following Theorem is due to Wassily Hoeffding and hence is called Chernoff-Hoeffding theorem.

Assume random variables $X_1,\dots ,X_m$ are i.i.d. Let p = E[X_i], X_i ∈ {0, 1}, and ε > 0. Then

$\mathbf{P}(\frac{1}{m}\sum X_i\ge p+\epsilon )\le {\left({\left(\frac{p}{p+\epsilon }\right)}^{p+\epsilon }{\left(\frac{1-p}{1-p-\epsilon }\right)}^{1-p-\epsilon }\right)}^m=e^{-D(p+\epsilon \Vert p)m}$

and

$\mathbf{P}(\frac{1}{m}\sum X_i\le p-\epsilon )\le {\left({\left(\frac{p}{p-\epsilon }\right)}^{p-\epsilon }{\left(\frac{1-p}{1-p+\epsilon }\right)}^{1-p+\epsilon }\right)}^m=e^{-D(p-\epsilon \Vert p)m},$

where

$D(x\Vert y)=x\log \frac{x}{y}+(1-x)\log \left(\frac{1-x}{1-y}\right)$

is the Kullback-Leibler divergence between Bernoulli distributed random variables with parameters x and y respectively. If $p\ge 1/2$, then

$\mathbf{P}(X>mp+x)\le \exp (-\frac{x^2}{2mp(1-p)}).$

Proof

Let q = p + ε. Taking a = mq in (Template:EquationNote), we obtain:

$\mathbf{P}(\frac{1}{m}\sum X_i\ge q)\le {\inf }_{t>0}\frac{E[\prod e^{t{X}_i}]}{e^{tmq}}={\inf }_{t>0}{\left[\frac{E\left[e^{t{X}_i}\right]}{e^{tq}}\right]}^m.$

Now, knowing that $\mathbf{P}[X_i=1]=p,\mathbf{P}[X_i=0]=(1-p)$, we have

${\left[\frac{\mathbf{E}\left[e^{t{X}_i}\right]}{e^{tq}}\right]}^m={\left[\frac{p{e}^t+(1-p)}{e^{tq}}\right]}^m=[p{e}^{(1-q)t}+(1-p)e^{-qt}{]}^m.$

Therefore we can easily compute the infimum, using calculus:

$\frac{d}{dt}(p{e}^{(1-q)t}+(1-p)e^{-qt})=(1-q)p{e}^{(1-q)t}-q(1-p)e^{-qt}$

Setting the equation to zero and solving, we have

$\begin{array}{rl}(1-q)p{e}^{(1-q)t}& =q(1-p)e^{-qt}\\ (1-q)p{e}^t& =q(1-p)\end{array}$

so that

$e^t=\frac{(1-p)q}{(1-q)p}.$

Thus,

$t=\log \left(\frac{(1-p)q}{(1-q)p}\right).$

As q = p + ε > p, we see that t > 0, so our bound is satisfied on t. Having solved for t, we can plug back into the equations above to find that

$\begin{array}{rl}\log (p{e}^{(1-q)t}+(1-p)e^{-qt})& =\log (e^{-qt}(1-p+p{e}^t))\\ & =\log \left[e^{-q\log \left(\frac{(1-p)q}{(1-q)p}\right)}\right]+\log [1-p+p{e}^{\log \left(\frac{1-p}{1-q}\right)}{e}^{\log \frac{q}{p}}]\\ & =-q\log \frac{1-p}{1-q}-q\log \frac{q}{p}+\log [1-p+p\left(\frac{1-p}{1-q}\right)\frac{q}{p}]\\ & =-q\log \frac{1-p}{1-q}-q\log \frac{q}{p}+\log [\frac{(1-p)(1-q)}{1-q}+\frac{(1-p)q}{1-q}]\\ & =-q\log \frac{q}{p}+(1-q)\log \frac{1-p}{1-q}=-D(q\Vert p).\end{array}$

We now have our desired result, that

$\mathbf{P}(\frac{1}{m}\sum X_i\ge p+\epsilon )\le e^{-D(p+\epsilon \Vert p)m}.$

To complete the proof for the symmetric case, we simply define the random variable Y_i = 1−X_i, apply the same proof, and plug it into our bound.

Simpler bounds

A simpler bound follows by relaxing the theorem using $D(p+x\Vert p)\ge 2x^2$, which follows from the convexity of $D(p+x\Vert p)$ and the fact that

$\frac{d^2}{d{x}^2}D(p+x\Vert p)=\frac{1}{(p+x)(1-p-x)}\ge 4=\frac{d^2}{d{x}^2}(2x^2).$

This result is a special case of Hoeffding's inequality. Sometimes, the bound $D((1+x)p\Vert p)\ge x^{2}p/4$ for $-1/2\le x\le 1/2$, which is stronger for p < 1/8, is also used.

Theorem for multiplicative form of Chernoff bound (relative error)

Let random variables $X_1,\dots ,X_n$ be independent random variables taking on values 0 or 1. Further, assume that $\mathbf{P}(X_i=1)=p_i$. Then, if we let $X=\sum _{i=1}^n{X}_i$ and μ be the expectation of X, for any δ > 0

$\mathbf{P}(X>(1+\delta )\mu )<{\left(\frac{e^{\delta }}{(1+\delta {)}^{(1+\delta )}}\right)}^{\mu }.$

Proof

According to (Template:EquationNote),

$\begin{array}{rl}\mathbf{P}(X>(1+\delta )\mu )& \le {\inf }_{t>0}\frac{\mathbf{E}[\prod _{i=1}^n\exp (t{X}_i)]}{\exp (t(1+\delta )\mu )})\\ & ={\inf }_{t>0}\frac{\prod _{i=1}^n\mathbf{E}[\exp (t{X}_i)]}{\exp (t(1+\delta )\mu )}\\ & ={\inf }_{t>0}\frac{\prod _{i=1}^n[p_i\exp (t)+(1-p_i)]}{\exp (t(1+\delta )\mu )}\end{array}$

The third line above follows because $e^{t{X}_i}$ takes the value $e^t$ with probability p_i and the value 1 with probability $1-p_i$. This is identical to the calculation above in the proof of the Theorem for additive form (absolute error).

Rewriting $p_i{e}^t+(1-p_i)$ as $p_i(e^t-1)+1$ and recalling that $1+x\le e^x$ (with strict inequality if x > 0), we set $x=p_i(e^t-1)$. The same result can be obtained by directly replacing a in the equation for the Chernoff bound with (1 + δ)μ.

Thus,

$\mathbf{P}(X>(1+\delta )\mu )<\frac{\prod _{i=1}^n\exp (p_i(e^t-1))}{\exp (t(1+\delta )\mu )}=\frac{\exp ((e^t-1)\sum _{i=1}^n{p}_i)}{\exp (t(1+\delta )\mu )}=\frac{\exp ((e^t-1)\mu )}{\exp (t(1+\delta )\mu )}.$

If we simply set t = log(1+δ) so that t > 0 for δ > 0, we can substitute and find

$\frac{\exp ((e^t-1)\mu )}{\exp (t(1+\delta )\mu )}=\frac{\exp ((1+\delta -1)\mu )}{(1+\delta {)}^{(1+\delta )\mu }}={\left[\frac{\exp (\delta )}{(1+\delta {)}^{(1+\delta )}}\right]}^{\mu }$

This proves the result desired. A similar proof strategy can be used to show that

$\mathbf{P}(X<(1-\delta )\mu )<{\left[\frac{\exp (-\delta )}{(1-\delta {)}^{(1-\delta )}}\right]}^{\mu }.$

Using Chernoff bound, we can prove some simpler result:

$\mathbf{P}(X\ge (1+\delta )\mu )\le e^{-\frac{{\delta }^2\mu }{3}},0<\delta <1$,

$\mathbf{P}(X\le (1-\delta )\mu )\le e^{-\frac{{\delta }^2\mu }{2}},0<\delta <1$.

Better Chernoff bounds for some special cases

We can obtain stronger bounds using simpler proof techniques for some special cases of symmetric random variables.

Let $X_1,X_2,...,X_n$ be independent random variables,

$X=\sum _{i=1}^n{X}_i$.

(a) $\mathbf{P}(X_i=1)=\mathbf{P}(X_i=-1)=\frac{1}{2}$.

Then,

$\mathbf{P}(X\ge a)\le e^{\frac{-a^2}{2n}},a>0$,

and therefore also

$\mathbf{P}(|X|\ge a)\le 2e^{\frac{-a^2}{2n}},a>0$ .

(b) $\mathbf{P}(X_i=1)=\mathbf{P}(X_i=0)=\frac{1}{2},\mathbf{E}[X]=\mu =\frac{n}{2}$

Then,

$\mathbf{P}(X\ge \mu +a)\le e^{\frac{-2a^2}{n}},a>0$,

$\mathbf{P}(X\le \mu -a)\le e^{\frac{-2a^2}{n}},0<a<\mu$,

Applications of Chernoff bound

Chernoff bounds have very useful applications in set balancing and packet routing in sparse networks.

The set balancing problem arises while designing statistical experiments. Typically while designing a statistical experiment, given the features of each participant in the experiment, we need to know how to divide the participants into 2 disjoint groups such that each feature is roughly as balanced as possible between the two groups. Refer to this book section for more info on the problem.

Chernoff bounds are also used to obtain tight bounds for permutation routing problems which reduce network congestion while routing packets in sparse networks. Refer to this book section for a thorough treatment of the problem.

Matrix Chernoff bound

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

Rudolf Ahlswede and Andreas Winter introduced Template:Harv a Chernoff bound for matrix-valued random variables.

If M is distributed according to some distribution over d × d matrices with zero mean, and if $M_1,M_2,\dots ,M_t$ are independent copies of M then for any ε > 0,

$\mathbf{P}(\Vert \frac{1}{t}\sum _{i=1}^t{M}_i-\mathbf{E}[M]{\Vert }_2>\epsilon )\le d\exp (-C\frac{{\epsilon }^{2}t}{{\gamma }^2}).$

where $\Vert M{\Vert }_2\le \gamma$ holds almost surely and C > 0 is an absolute constant.

Notice that the number of samples in the inequality depends logarithmically on d. In general, unfortunately, such a dependency is inevitable: take for example a diagonal random sign matrix of dimension d. The operator norm of the sum of t independent samples is precisely the maximum deviation among d independent random walks of length t. In order to achieve a fixed bound on the maximum deviation with constant probability, it is easy to see that t should grow logarithmically with d in this scenario.

The following theorem can be obtained by assuming M has low rank, in order to avoid the dependency on the dimensions.

Theorem without the dependency on the dimensions

Let 0 < ε < 1 and M be a random symmetric real matrix with $\Vert \mathbf{E}[M]{\Vert }_2\le 1$ and $\Vert M{\Vert }_2\le \gamma$ almost surely. Assume that each element on the support of M has at most rank r. Set

$t=\mathrm{\Omega }\left(\frac{\gamma \log (\gamma /{\epsilon }^2)}{{\epsilon }^2}\right).$

If $r\le t$ holds almost surely, then

$\mathbf{P}(\Vert \frac{1}{t}\sum _{i=1}^t{M}_i-\mathbf{E}[M]{\Vert }_2>\epsilon )\le \frac{1}{\mathbf{p}\mathbf{o}\mathbf{l}\mathbf{y}(t)}$

where $M_1,M_2,\dots ,M_t$ are i.i.d. copies of M.

See also

• Bennett's inequality
• Bernstein inequalities (probability theory)
• Efron-Stein inequality
• Hoeffding's inequality
• Markov's inequality
• Chebyshev's inequality

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• Template:Cite arxiv
• Template:Cite web