ผู้ใช้:Mopza/กระบะทราย5

ในวิชาการเงิน เส้นอัตราผลตอบแทนพันธบัตร (อังกฤษ: Yield Curve) เป็นเส้นโค้งที่แสดงถึงอัตราผลตอบแทน หรืออัตราดอกเบี้ยระหว่างพันธบัตรที่มีอายุแตกต่างกัน (เช่น พันธบัตรอายุ 2 เดือน, 2 ปี, 20 ปี ฯลฯ) เส้นโค้งนี้จะแสดงถึงความสัมพันธ์ระหว่างอัตราดอกเบื้ย (หรือค่าใช้จ่ายในการกู้ยืมเงิน) กับระยะเวลาที่ตราสารครบกำหนดอายุ ซึ่งก็คือหนี้ที่ให้กับผู้กู้ในจำนวนเงินแต่ละสกุลเงิน เช่น อัตราดอกเบี้ยสำหรับสกุลเงินดอลลาร์สหรัฐ ของพันธบัตรรัฐบาลสหรัฐ ในแต่ละอายุของพันธบัตรจะถูกติดตามอย่างใกล้ชิดจากนักลงทุน ซึ่งอัตราผลตอบแทนพันธบัตรอายุต่างๆ นี้จะถูกเขียนบนกราฟ จนเรียกว่า "เส้นโค้งอัตราผลตอบแทนพันธบัตร" สำหรับลักษณะทางคณิตศาสตร์ที่เป็นทางการ จะเรียกว่า "โครงสร้างของอัตราดอกเบี้ย"

รูปทรงของเส้นอัตราผลตอบแทนพันธบัตรจะแสดงให้เห็นถึงความสำคัญของผู้ให้กู้กับผู้กู้โดยเฉพาะ (เช่นพันธบัตรรัฐบาลสหรัฐ) หรือความสำคัญระหว่างผู้ให้กู้ที่เกี่ยวข้องเพียงรายเดียว กับผู้กู้ที่เกี่ยวข้องทุกราย พร้อมกับองค์ประกอบอื่นๆ ที่เกิดขึ้นอย่างเท่าเทียมกัน ซึ่งผู้ให้กู้มักจะขายพันธบัตรให้กับกองทุนต่างๆ มากกว่าการขายพันธบัตรให้กับบุคคลที่สาม (เช่น บุคคลธรรมดา ห้างร้านต่างๆ) เนื่องจากอัตราดอกเบี้ยถือเป็นค่าใช้จ่ายของผู้ให้กู้ที่จะต้องใช้เมื่อมีการกู้ยืมเกิดขึ้น หากอัตราดอกเบื้ยปรับตัวสูงขึ้น จะทำให้ผู้ให้กู้ได้รับดอกเบี้ยมากขึ้นด้วย นอกจากนี้ ผู้ให้กู้อาจพิจารณาถึงความเสี่ยงในอนาคตที่อาจเกิดขึ้นกับผู้กู้ เช่น การผิดนัดชำระหนี้ หรืออัตราเงินเฟ้อปรับตัวสูงขึ้น นักลงทุนส่วนใหญ่ต้องการอัตราดอกเบื้ยที่สูงในพันธบัตรระยะยาวมากกว่าพันธบัตรระยะสั้นซึ่งให้อัตราดอกเบี้ยที่ต่ำกว่า แต่มักจะมีความเสียงที่สูงกว่าพันธบัตรระยะสั้น ซึ่งในบางครั้ง เมื่อผู้ให้กู้เห็นว่าการถือพันธบัตรระยะยาวจะป้องกันความเสี่ยงได้ดีกว่าพันธบัตรระยะสั้น เส้นอัตราผลตอบแทนพันธบัตรมักจะลาดลงหรือกลับทิศ (อังกฤษ: Inverted Yield Curve) พร้อมกับอัตราผลตอบแทนพันธบัตรระยะยาวที่ลดลง จนต่ำกว่าอัตราผลตอบแทนพันธบัตรระยะสั้น เนื่องจากผู้กู้สามารถหาวิธีการกู้ยืมเงินในระยะยาวที่ง่ายกว่าวิธีอื่นๆ

เส้นอัตราผลตอบแทนของตราสารหนี้เป็นเครื่องมือที่ใช้สำหรับกำหนดอัตราผลตอบแทนในการลงทุนตราสารหนี้ ซึ่งโดยทั่วไปแล้ว ผลตอบแทนที่ได้รับในแต่ละปีจะขึ้นกับระยะเวลาที่ลงทุน เช่น ธนาคารจะให้ดอกเบี้ยสำหรับผู้ที่ฝากเงินในบัญชีออมทรัพย์ มากกว่าผู้ที่ฝากเงินในบัญชีกระแสรายวัน หากผู้ฝากเงินไม่ถอนเงินจากบัญชีดังกล่าวเลยเป็นเวลา 5 ปี ซึ่งอาจกล่าวโดยใช้ฟังก์ชันทางคณิตศาสตร์ได้ว่า R=Y(t) โดยให้ R คือผลตอบแทน Y คืออัตราผลตอบแทน และ t คือระยะเวลาที่ลงทุน โดยนักวิเคราะห์การลงทุนในตราสารหนี้จะใช้เส้นผลตอบแทนนี้ในการวิเคราะห์พันธบัตร และหลักทรัพย์ที่เกี่ยวข้อง เพื่อความเข้าใจในเงื่อนไขต่างๆ ในตลาดการเงิน และเพื่อหาจังหวะการลงทุน ส่วนนักเศรษฐศาสตร์ก็ใช้เส้นอัตราผลตอบแทนนี้ในการทำนายสภาพเศรษฐกิจ

The yield curve function Y is actually only known with certainty for a few specific maturity dates, while the other maturities are calculated by interpolation (see Construction of the full yield curve from market data below).

The yield curve

โดยปกติแล้วเส้นอัตราผลตอบแทนพันธบัตรจะมีลักษณะชันขึ้นอย่างไม่สมมาตร นั่นคือ พันธบัตรที่มีอายุยาวมากขึ้นเท่าไร ก็จะมีผลตอบแทนที่มากกว่าพันธบัตรอายุสั้นเท่านั้น มีคำอธิบายอยู่สองอย่างสำหรับเส้นผลตอบแทนที่ชันขึ้น กล่าวคือ Yield curves are usually upward sloping asymptotically: the longer the maturity, the higher the yield, with diminishing marginal increases (that is, as one moves to the right, the curve flattens out). There are two common explanations for upward sloping yield curves. First, it may be that the market is anticipating a rise in the risk-free rate. If investors hold off investing now, they may receive a better rate in the future. Therefore, under the arbitrage pricing theory, investors who are willing to lock their money in now need to be compensated for the anticipated rise in rates—thus the higher interest rate on long-term investments.

Another explanation is that longer maturities entail greater risks for the investor (i.e. the lender). A risk premium is needed by the market, since at longer durations there is more uncertainty and a greater chance of catastrophic events that impact the investment. This explanation depends on the notion that the economy faces more uncertainties in the distant future than in the near term. This effect is referred to as the liquidity spread. If the market expects more volatility in the future, even if interest rates are anticipated to decline, the increase in the risk premium can influence the spread and cause an increasing yield.

The opposite position (short-term interest rates higher than long-term) can also occur. For instance, in November 2004, the yield curve for UK Government bonds was partially inverted. The yield for the 10-year bond stood at 4.68%, but was only 4.45% for the 30-year bond. The market's anticipation of falling interest rates causes such incidents. Negative liquidity premiums can also exist if long-term investors dominate the market, but the prevailing view is that a positive liquidity premium dominates, so only the anticipation of falling interest rates will cause an inverted yield curve. Strongly inverted yield curves have historically preceded economic recessions.

The shape of the yield curve is influenced by supply and demand: for instance, if there is a large demand for long bonds, for instance from pension funds to match their fixed liabilities to pensioners, and not enough bonds in existence to meet this demand, then the yields on long bonds can be expected to be low, irrespective of market participants' views about future events.

The yield curve may also be flat or hump-shaped, due to anticipated interest rates being steady, or short-term volatility outweighing long-term volatility.

Yield curves continually move all the time that the markets are open, reflecting the market's reaction to news. A further "stylized fact" is that yield curves tend to move in parallel (i.e., the yield curve shifts up and down as interest rate levels rise and fall: a "parallel shift").

Types of yield curve

There is no single yield curve describing the cost of money for everybody. The most important factor in determining a yield curve is the currency in which the securities are denominated. The economic position of the countries and companies using each currency is a primary factor in determining the yield curve. Different institutions borrow money at different rates, depending on their creditworthiness. The yield curves corresponding to the bonds issued by governments in their own currency are called the government bond yield curve (government curve). Banks with high credit ratings (Aa/AA or above) borrow money from each other at the LIBOR rates. These yield curves are typically a little higher than government curves. They are the most important and widely used in the financial markets, and are known variously as the LIBOR curve or the swap curve. The construction of the swap curve is described below.

Besides the government curve and the LIBOR curve, there are corporate (company) curves. These are constructed from the yields of bonds issued by corporations. Since corporations have less creditworthiness than most governments and most large banks, these yields are typically higher. Corporate yield curves are often quoted in terms of a "credit spread" over the relevant swap curve. For instance the five-year yield curve point for Vodafone might be quoted as LIBOR +0.25%, where 0.25% (often written as 25 basis points or 25bps) is the credit spread.

Normal yield curve

U.S. Treasury yield curves for different dates. The July 2000 yield curve (red line, top) is inverted.

From the post-Great Depression era to the present, the yield curve has usually been "normal" meaning that yields rise as maturity lengthens (i.e., the slope of the yield curve is positive). This positive slope reflects investor expectations for the economy to grow in the future and, importantly, for this growth to be associated with a greater expectation that inflation will rise in the future rather than fall. This expectation of higher inflation leads to expectations that the central bank will tighten monetary policy by raising short-term interest rates in the future to slow economic growth and dampen inflationary pressure. It also creates a need for a risk premium associated with the uncertainty about the future rate of inflation and the risk this poses to the future value of cash flows. Investors price these risks into the yield curve by demanding higher yields for maturities further into the future. In a positively sloped yield curve, lenders profit from the passage of time since yields decrease as bonds get closer to maturity (as yield decreases, price increases); this is known as rolldown and is a significant component of profit in fixed-income investing (i.e., buying and selling, not necessarily holding to maturity), particularly if the investing is leveraged.

However, a positively sloped yield curve has not always been the norm. Through much of the 19th century and early 20th century the US economy experienced trend growth with persistent deflation, not inflation. During this period the yield curve was typically inverted, reflecting the fact that deflation made current cash flows less valuable than future cash flows. During this period of persistent deflation, a 'normal' yield curve was negatively sloped.

Steep yield curve

Historically, the 20-year Treasury bond yield has averaged approximately two percentage points above that of three-month Treasury bills. In situations when this gap increases (e.g. 20-year Treasury yield rises much higher than the three-month Treasury yield), the economy is expected to improve quickly in the future. This type of curve can be seen at the beginning of an economic expansion (or after the end of a recession). Here, economic stagnation will have depressed short-term interest rates; however, rates begin to rise once the demand for capital is re-established by growing economic activity.

In January 2010, the gap between yields on two-year Treasury notes and 10-year notes widened to 2.92 percentage points, its highest ever.

Flat or humped yield curve

A flat yield curve is observed when all maturities have similar yields, whereas a humped curve results when short-term and long-term yields are equal and medium-term yields are higher than those of the short-term and long-term. A flat curve sends signals of uncertainty in the economy. This mixed signal can revert to a normal curve or could later result into an inverted curve. It cannot be explained by the Segmented Market theory discussed below.

Inverted yield curve

An inverted yield curve occurs when long-term yields fall below short-term yields.

Federal Funds Rate compared to U.S. Treasury interest rates

Under unusual circumstances, investors will settle for lower yields associated with low-risk long term debt if they think the economy will enter a recession in the near future. For example, the S&P 500 experienced a dramatic fall in mid 2007, from which it recovered completely by early 2013. Investors who had purchased 10-year Treasuries in 2006 would have received a safe and steady yield until 2015, possibly achieving better returns than those investing in equities during that volatile period.

Campbell R. Harvey's 1986 dissertation showed that an inverted yield curve accurately forecasts U.S. recessions. An inverted curve has indicated a worsening economic situation in the future 7 times since 1970. The New York Federal Reserve regards it as a valuable forecasting tool in predicting recessions two to six quarters ahead.

In addition to potentially signaling an economic decline, inverted yield curves also imply that the market believes inflation will remain low. This is because, even if there is a recession, a low bond yield will still be offset by low inflation. However, technical factors, such as a flight to quality or global economic or currency situations, may cause an increase in demand for bonds on the long end of the yield curve, causing long-term rates to fall. Falling long-term rates in the presence of rising short-term rates is known as "Greenspan's Conundrum".

Relationship to the business cycle

The slope of the yield curve is one of the most powerful predictors of future economic growth, inflation, and recessions. One measure of the yield curve slope (i.e. the difference between 10-year Treasury bond rate and the 3-month Treasury bond rate) is included in the Financial Stress Index published by the St. Louis Fed. A different measure of the slope (i.e. the difference between 10-year Treasury bond rates and the federal funds rate) is incorporated into the Index of Leading Economic Indicators published by The Conference Board.

An inverted yield curve is often a harbinger of recession. A positively sloped yield curve is often a harbinger of inflationary growth. Work by Arturo Estrella and Tobias Adrian has established the predictive power of an inverted yield curve to signal a recession. Their models show that when the difference between short-term interest rates (they use 3-month T-bills) and long-term interest rates (10-year Treasury bonds) at the end of a federal reserve tightening cycle is negative or less than 93 basis points positive, a rise in unemployment usually occurs. The New York Fed publishes a monthly recession probability prediction derived from the yield curve and based on Estrella's work.

All the recessions in the US since 1970 (up through 2018) have been preceded by an inverted yield curve (10-year vs 3-month). Over the same time frame, every occurrence of an inverted yield curve has been followed by recession as declared by the NBER business cycle dating committee. The yield curve became inverted in the first half of 2019, for the first time since 2007.

Event	Date of Inversion Start	Date of the Recession Start	Time from Inversion to Recession Start	Duration of Inversion	Time from Recession Start to NBER Announcement	Time from Disinversion to Recession End	Duration of Recession	Time from Recession End to NBER Announcement	Max Inversion
			Months	Months	Months	Months	Months	Months	Basis Points
1970 Recession	Dec-68	Jan-70	13	15	NA	8	11	NA	−52
1974 Recession	Jun-73	Dec-73	6	18	NA	3	16	NA	−159
1980 Recession	Nov-78	Feb-80	15	18	4	2	6	12	−328
1981–1982 Recession	Oct-80	Aug-81	10	12	5	13	16	8	−351
1990 Recession	Jun-89	Aug-90	14	7	8	14	8	21	−16
2001 Recession	Jul-00	Apr-01	9	7	7	9	8	20	−70
2008–2009 Recession	Aug-06	Jan-08	17	10	11	24	18	15	−51
TBD	May-19	TBD	TBD	TBD	TBD	TBD	TBD	TBD	TBD
Average since 1969			12	12	7	10	12	15	−147
Std Dev since 1969			3.83	4.72	2.74	7.50	4.78	5.45	138.96

Table Note: The New York Federal Reserve recession prediction model uses the month average 10 year yield vs the month average 3 month bond equivalent yield to compute the term spread. Therefore, intra-day and daily inversions do not count as inversions unless they lead to an inversion on a monthly average basis. In December 2018 portions of the yield curve inverted for the first time since the 2008–2009 Recession. However the 10-year vs 3-month portion did not invert until March 22 2019 and it reverted to a positive slope by April 01 2019 (i.e. only 8 days later). The month average of the 10-year vs 3-month (bond equivalent yield) difference reached zero basis points in May-2019. Both March and April 2019 had month-average spreads greater than zero basis points despite intra-day and daily inversions in March and April. Therefore, the table shows the 2019 inversion beginning from May-2019. Likewise, daily inversions in Sep-1998 did not result in negative term spreads on a month average basis and thus do not constitute a false alarm.

Estrella and others have postulated that the yield curve affects the business cycle via the balance sheet of banks (or bank-like financial institutions). When the yield curve is inverted, banks are often caught paying more on short-term deposits (or other forms of short-term wholesale funding) than they are making on new long-term loans leading to a loss of profitability and reluctance to lend resulting in a credit crunch. When the yield curve is upward sloping, banks can profitably take-in short term deposits and make new long-term loans so they are eager to supply credit to borrowers. This eventually leads to a credit bubble.

ดูเพิ่ม: Net interest margin

Theory

There are three main economic theories attempting to explain how yields vary with maturity. Two of the theories are extreme positions, while the third attempts to find a middle ground between the former two.

Market expectations (pure expectations) hypothesis

บทความหลัก: Expectations hypothesis

This hypothesis assumes that the various maturities are perfect substitutes and suggests that the shape of the yield curve depends on market participants' expectations of future interest rates. It assumes that market forces will cause the interest rates on various terms of bonds to be such that the expected final value of a sequence of short-term investments will equal the known final value of a single long-term investment. If this did not hold, the theory assumes that investors would quickly demand more of the current short-term or long-term bonds (whichever gives the higher expected long-term yield), and this would drive down the return on current bonds of that term and drive up the yield on current bonds of the other term, so as to quickly make the assumed equality of expected returns of the two investment approaches hold.

Using this, futures rates, along with the assumption that arbitrage opportunities will be minimal in future markets, and that futures rates are unbiased estimates of forthcoming spot rates, provide enough information to construct a complete expected yield curve. For example, if investors have an expectation of what 1-year interest rates will be next year, the current 2-year interest rate can be calculated as the compounding of this year's 1-year interest rate by next year's expected 1-year interest rate. More generally, returns (1+ yield) on a long-term instrument are assumed to equal the geometric mean of the expected returns on a series of short-term instruments:

$(1+i_{lt})^{n}=(1+i_{st}^{{\text{year }}1})(1+i_{st}^{{\text{year }}2})\cdots (1+i_{st}^{{\text{year }}n}),$

where i_st and i_lt are the expected short-term and actual long-term interest rates (but $i_{st}^{{\text{year}}1}$ is the actual observed short-term rate for the first year).

This theory is consistent with the observation that yields usually move together. However, it fails to explain the persistence in the shape of the yield curve.

Shortcomings of expectations theory include that it neglects the interest rate risk inherent in investing in bonds.

Liquidity premium theory

The liquidity premium theory is an offshoot of the pure expectations theory. The liquidity premium theory asserts that long-term interest rates not only reflect investors' assumptions about future interest rates but also include a premium for holding long-term bonds (investors prefer short term bonds to long term bonds), called the term premium or the liquidity premium. This premium compensates investors for the added risk of having their money tied up for a longer period, including the greater price uncertainty. Because of the term premium, long-term bond yields tend to be higher than short-term yields and the yield curve slopes upward. Long term yields are also higher not just because of the liquidity premium, but also because of the risk premium added by the risk of default from holding a security over the long term. The market expectations hypothesis is combined with the liquidity premium theory:

$(1+i_{lt})^{n}=rp_{n}+((1+i_{st}^{\mathrm {year} 1})(1+i_{st}^{\mathrm {year} 2})\cdots (1+i_{st}^{\mathrm {year} n}))$

Where $rp_{n}$ is the risk premium associated with an ${n}$ year bond.

Preferred habitat theory

The preferred habitat theory is a variant of the liquidity premium theory, and states that in addition to interest rate expectations, investors have distinct investment horizons and require a meaningful premium to buy bonds with maturities outside their "preferred" maturity, or habitat. Proponents of this theory believe that short-term investors are more prevalent in the fixed-income market, and therefore longer-term rates tend to be higher than short-term rates, for the most part, but short-term rates can be higher than long-term rates occasionally. This theory is consistent with both the persistence of the normal yield curve shape and the tendency of the yield curve to shift up and down while retaining its shape.

Market segmentation theory

This theory is also called the segmented market hypothesis. In this theory, financial instruments of different terms are not substitutable. As a result, the supply and demand in the markets for short-term and long-term instruments is determined largely independently. Prospective investors decide in advance whether they need short-term or long-term instruments. If investors prefer their portfolio to be liquid, they will prefer short-term instruments to long-term instruments. Therefore, the market for short-term instruments will receive a higher demand. Higher demand for the instrument implies higher prices and lower yield. This explains the stylized fact that short-term yields are usually lower than long-term yields. This theory explains the predominance of the normal yield curve shape. However, because the supply and demand of the two markets are independent, this theory fails to explain the observed fact that yields tend to move together (i.e., upward and downward shifts in the curve).

Historical development of yield curve theory

On 15 August 1971, U.S. President Richard Nixon announced that the U.S. dollar would no longer be based on the gold standard, thereby ending the Bretton Woods system and initiating the era of floating exchange rates.

Floating exchange rates made life more complicated for bond traders, including those at Salomon Brothers in New York City. By the middle of the 1970s, encouraged by the head of bond research at Salomon, Marty Liebowitz, traders began thinking about bond yields in new ways. Rather than think of each maturity (a ten-year bond, a five-year, etc.) as a separate marketplace, they began drawing a curve through all their yields. The bit nearest the present time became known as the short end—yields of bonds further out became, naturally, the long end.

Academics had to play catch up with practitioners in this matter. One important theoretic development came from a Czech mathematician, Oldrich Vasicek, who argued in a 1977 paper that bond prices all along the curve are driven by the short end (under risk neutral equivalent martingale measure) and accordingly by short-term interest rates. The mathematical model for Vasicek's work was given by an Ornstein–Uhlenbeck process, but has since been discredited because the model predicts a positive probability that the short rate becomes negative and is inflexible in creating yield curves of different shapes. Vasicek's model has been superseded by many different models including the Hull–White model (which allows for time varying parameters in the Ornstein–Uhlenbeck process), the Cox–Ingersoll–Ross model, which is a modified Bessel process, and the Heath–Jarrow–Morton framework. There are also many modifications to each of these models, but see the article on short rate model. Another modern approach is the LIBOR market model, introduced by Brace, Gatarek and Musiela in 1997 and advanced by others later. In 1996 a group of derivatives traders led by Olivier Doria (then head of swaps at Deutsche Bank) and Michele Faissola, contributed to an extension of the swap yield curves in all the major European currencies. Until then the market would give prices until 15 years maturities. The team extended the maturity of European yield curves up to 50 years (for the lira, French franc, Deutsche mark, Danish krone and many other currencies including the ecu). This innovation was a major contribution towards the issuance of long dated zero coupon bonds and the creation of long dated mortgages.

Construction of the full yield curve from market data

See also: Bootstrapping (finance); Fixed-income attribution#Modeling the yield curve.

Typical inputs to the money market curve

Type	Settlement date	Rate (%)
Cash	Overnight rate	5.58675
Cash	Tomorrow next rate	5.59375
Cash	1m	5.625
Cash	3m	5.71875
Future	Dec-97	5.76
Future	Mar-98	5.77
Future	Jun-98	5.82
Future	Sep-98	5.88
Future	Dec-98	6.00
Swap	2y	6.01253
Swap	3y	6.10823
Swap	4y	6.16
Swap	5y	6.22
Swap	7y	6.32
Swap	10y	6.42
Swap	15y	6.56
Swap	20y	6.56
Swap	30y	6.56
A list of standard instruments used to build a money market yield curve.
The data is for lending in US dollar, taken from October 6, 1997

The usual representation of the yield curve is in terms of a function P, defined on all future times t, such that P(t) represents the value today of receiving one unit of currency t years in the future. If P is defined for all future t then we can easily recover the yield (i.e. the annualized interest rate) for borrowing money for that period of time via the formula

$Y(t)=P(t)^{-1/t}-1.$

The significant difficulty in defining a yield curve therefore is to determine the function P(t). P is called the discount factor function or the zero coupon bond.

Yield curves are built from either prices available in the bond market or the money market. Whilst the yield curves built from the bond market use prices only from a specific class of bonds (for instance bonds issued by the UK government) yield curves built from the money market use prices of "cash" from today's LIBOR rates, which determine the "short end" of the curve i.e. for t ≤ 3m, interest rate futures which determine the midsection of the curve (3m ≤ t ≤ 15m) and interest rate swaps which determine the "long end" (1y ≤ t ≤ 60y).

The example given in the table at the right is known as a LIBOR curve because it is constructed using either LIBOR rates or swap rates. A LIBOR curve is the most widely used interest rate curve as it represents the credit worth of private entities at about A+ rating, roughly the equivalent of commercial banks. If one substitutes the LIBOR and swap rates with government bond yields, one arrives at what is known as a government curve, usually considered the risk free interest rate curve for the underlying currency. The spread between the LIBOR or swap rate and the government bond yield, usually positive, meaning private borrowing is at a premium above government borrowing, of similar maturity is a measure of risk tolerance of the lenders. For the U. S. market, a common benchmark for such a spread is given by the so-called TED spread.

In either case the available market data provides a matrix A of cash flows, each row representing a particular financial instrument and each column representing a point in time. The (i,j)-th element of the matrix represents the amount that instrument i will pay out on day j. Let the vector F represent today's prices of the instrument (so that the i-th instrument has value F(i)), then by definition of our discount factor function P we should have that F = AP (this is a matrix multiplication). Actually, noise in the financial markets means it is not possible to find a P that solves this equation exactly, and our goal becomes to find a vector P such that

$AP=F+\varepsilon \,$

where $\varepsilon$ is as small a vector as possible (where the size of a vector might be measured by taking its norm, for example).

Note that even if we can solve this equation, we will only have determined P(t) for those t which have a cash flow from one or more of the original instruments we are creating the curve from. Values for other t are typically determined using some sort of interpolation scheme.

Practitioners and researchers have suggested many ways of solving the A*P = F equation. It transpires that the most natural method – that of minimizing $\epsilon$ by least squares regression – leads to unsatisfactory results. The large number of zeroes in the matrix A mean that function P turns out to be "bumpy".

In their comprehensive book on interest rate modelling James and Webber note that the following techniques have been suggested to solve the problem of finding P:

1. Approximation using Lagrange polynomials
2. Fitting using parameterised curves (such as splines, the Nelson-Siegel family, the Svensson family, or the Cairns restricted-exponential family of curves). Van Deventer, Imai and Mesler summarize three different techniques for curve fitting that satisfy the maximum smoothness of either forward interest rates, zero coupon bond prices, or zero coupon bond yields
3. Local regression using kernels
4. Linear programming

In the money market practitioners might use different techniques to solve for different areas of the curve. For example, at the short end of the curve, where there are few cashflows, the first few elements of P may be found by bootstrapping from one to the next. At the long end, a regression technique with a cost function that values smoothness might be used.

Recent practice

Post the financial crisis of 2008 swap valuation is typically under a "multi-curve" framework; the preceding, by contrast, describes the "self discounting" approach. The reason for the change is that post-crisis, the overnight index swap rate is considered a better proxy than Libor for the "risk-free rate", and "OIS-discounting" is now standard (sometimes, referred to as "CSA-discounting"). The result is that, in practice, curves are built as a "set" and not individually: "forecast curves" are constructed for each floating-leg LIBOR tenor; and discounting is on a single, common OIS curve which must simultaneously be constructed. See แม่แบบ:Sectionlink for the math, and for context, แม่แบบ:Sectionlink.

Effect on bond prices

There is a time dimension to the analysis of bond values. A 10-year bond at purchase becomes a 9-year bond a year later, and the year after it becomes an 8-year bond, etc. Each year the bond moves incrementally closer to maturity, resulting in lower volatility and shorter duration and demanding a lower interest rate when the yield curve is rising. Since falling rates create increasing prices, the value of a bond initially will rise as the lower rates of the shorter maturity become its new market rate. Because a bond is always anchored by its final maturity, the price at some point must change direction and fall to par value at redemption.

A bond's market value at different times in its life can be calculated. When the yield curve is steep, the bond is predicted to have a large capital gain in the first years before falling in price later. When the yield curve is flat, the capital gain is predicted to be much less, and there is little variability in the bond's total returns over time.

Rising (or falling) interest rates rarely rise by the same amount all along the yield curve—the curve rarely moves up in parallel. Because longer-term bonds have a larger duration, a rise in rates will cause a larger capital loss for them, than for short-term bonds. But almost always, the long maturity's rate will change much less, flattening the yield curve. The greater change in rates at the short end will offset to some extent the advantage provided by the shorter bond's lower duration.

Long duration bonds tend to be mean reverting, meaning that they readily gravitate to a long-run average. The middle of the curve (5–10 years) will see the greatest percentage gain in yields if there is anticipated inflation even if interest rates have not changed. The long-end does not move quite as much percentage-wise because of the mean reverting properties.

The yearly 'total return' from the bond is a) the sum of the coupon's yield plus b) the capital gain from the changing valuation as it slides down the yield curve and c) any capital gain or loss from changing interest rates at that point in the yield curve.

See also

• Short-rate model
• Zero-coupon bond
• Zero interest rate policy

References

Books

• J H M Darbyshire (2017). Pricing and Trading Interest Rate Derivatives (2nd ed. 2017 ed.). Aitch and Dee Ltd. ISBN 978-0995455528.
• Leif B.G. Andersen; Vladimir V. Piterbarg (2010). Interest Rate Modeling. Atlantic Financial Press. ISBN 978-0-9844221-0-4. {{cite book}}: ไม่รู้จักพารามิเตอร์ |lastauthoramp= ถูกละเว้น แนะนำ (|name-list-style=) (help)
• Jessica James; Nick Webber (2001). Interest Rate Modelling. John Wiley & Sons. ISBN 978-0-471-97523-6. {{cite book}}: ไม่รู้จักพารามิเตอร์ |lastauthoramp= ถูกละเว้น แนะนำ (|name-list-style=) (help)
• Riccardo Rebonato (1998). Interest-Rate Option Models. John Wiley & Sons. ISBN 978-0-471-97958-6.
• Nicholas Dunbar (2000). Inventing Money. John Wiley & Sons. ISBN 978-0-471-89999-0.
• N. Anderson, F. Breedon, M. Deacon, A. Derry and M. Murphy (1996). Estimating and Interpreting the Yield Curve. John Wiley & Sons. ISBN 978-0-471-96207-6.{{cite book}}: CS1 maint: multiple names: authors list (ลิงก์)
• Andrew J.G. Cairns (2004). Interest Rate Models – An Introduction. Princeton University Press. ISBN 978-0-691-11894-9.
• John C. Hull (1989). Options, Futures and Other Derivatives. Prentice Hall. ISBN 978-0-13-015822-2. See in particular the section Theories of the term structure (section 4.7 in the fourth edition).
• Damiano Brigo; Fabio Mercurio (2001). Interest Rate Models – Theory and Practice. Springer. ISBN 978-3-540-41772-9.
• Donald R. van Deventer; Kenji Imai; Mark Mesler (2004). Advanced Financial Risk Management, An Integrated Approach to Credit Risk and Interest Rate Risk Management. John Wiley & Sons. ISBN 978-0-470-82126-8.

Articles

• Ruben D Cohen (2006) "A VaR-Based Model for the Yield Curve [download]" Wilmott Magazine, May Issue.
• Lin Chen (1996). Stochastic Mean and Stochastic Volatility – A Three-Factor Model of the Term Structure of Interest Rates and Its Application to the Pricing of Interest Rate Derivatives. Blackwell Publishers.
• Paul F. Cwik (2005) "The Inverted Yield Curve and the Economic Downturn [download]" New Perspectives on Political Economy, Volume 1, Number 1, 2005, pp. 1–37.
• Roger J.-B. Wets, Stephen W. Bianchi, "Term and Volatility Structures" in Stavros A. Zenios; William T. Ziemba (2006). Handbook of Asset and Liability Management, Volume 1. North-Holland. ISBN 978-0-444-50875-1. {{cite book}}: ไม่รู้จักพารามิเตอร์ |lastauthoramp= ถูกละเว้น แนะนำ (|name-list-style=) (help)
• Hagan, P.; West, G. (June 2006). "Interpolation Methods for Curve Construction" (PDF). Applied Mathematical Finance. 13 (2): 89–129. CiteSeerX 10.1.1.529.9594. doi:10.1080/13504860500396032.
• Rise in Rates Jolts Markets – Fed's Effort to Revive Economy Is Complicated by Fresh Jump in Borrowing Costs author = Liz Rappaport. Wall Street Journal. May 28, 2009. p. A.1

External links

• Euro area yield curves – European Central Bank website
• Dynamic Yield Curve – This chart shows the relationship between interest rates and stocks over time.
• NYFed Current Issue – Current Issue of New York Federal Reserve outlining their view of inverted yield curve
• Yield curve: 10-Year Treasury Constant Maturity Minus 2-Year Treasury Constant Maturity, daily since June 1976, via FRED
• US yield curve

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Avicii
อาวีชี่ ในปี 2014
ข้อมูลพื้นฐาน
ชื่อเกิด	Tim Bergling
รู้จักในชื่อ	Tim Berg Tom Hangs Timberman
เกิด	(1989-09-08) 8 กันยายน ค.ศ. 1989 สตอกโฮล์ม ประเทศสวีเดน
ที่เกิด	Östermalm, Stockholm, Sweden
แนวเพลง	EDM progressive house electro house
อาชีพ	Electronic musician DJ remixer record producer
เครื่องดนตรี	กีตาร์ เปียโน เครื่องสังเคราะห์เสียง FL Studio Logic Pro
ช่วงปี	2006–ปัจจุบัน
ค่ายเพลง	Island LE7ELS Vicious Big Beat Size Armada Spinnin' Ultra Joia UMG PRMD
เว็บไซต์	avicii.com

ทิม เบิร์กลิง หรือชื่อที่รู้จักในวงการคือ อาวีชี่ (อังกฤษ: Avicii) เป็นดีเจและโปรดิวเซอร์เพลงเฮาส์ชาวสวีดิช เขาได้รับการจัดอันดับในอันดับที่ 3 ในนิตยสาร ดีเจ แม็กกาซีน ในปี 2012 และ 2013และเขายังได้รับการเสนอชื่อเข้าชิง รางวัลแกรมมี ถึง 2 ครั้ง, ครั้งหนึ่งในเพลง "ซันไชน์" ซึ่งร่วมกับ เดวิด เกตตา ในปี 2012 และอีกครั้งในเพลง "เลเวล" ในปีต่อมา โดยมีเพลงที่ได้รับการตอบรับที่ดี เช่น "I Could Be the One" with Nicky Romero, "Wake Me Up", "You Make Me", "X You", "Hey Brother", "Addicted to You", "The Days", "The Nights", "Levels" and "Waiting for Love".

He released his debut studio album, True, in 2013. It received generally positive reviews from music critics, peaked within the top ten in more than fifteen countries, and topped the Australian, Swedish, Danish and US charts. In 2015, Avicii released his second studio album, Stories.

Early life

Tim Bergling was born on 8 September 1989 in Stockholm, Sweden. In May 2007, Avicii signed on with the Dejfitts Plays label. Then, in 2010, Avicii released the hit song "Seek Bromance", which reached the top 20 in several countries including Belgium, France, the Netherlands, United Kingdom, and Sweden. He also remixed Nadia Ali's classic single "Rapture" for her album Queen of Clubs Trilogy: Onyx Edition. In October 2010, Avicii signed with the European A&R team with EMI Music Publishing.

Career

2011–12: "Levels" and international breakthrough

ไฟล์:Avicii New Logo.png

Avicii wordmark, 2008–

In 2011, Avicii released "Levels", which launched him into the mainstream. The song was produced and written by Tim Bergling and contains a vocal sample from the 1962 gospel-inspired "Something's Got a Hold on Me" by Etta James. The same vocal sample was used first by Pretty Lights in his 2006 song "Finally Moving". This sample was also used by Drum and Bass producer Logistics on "Call Me Back" and by Flo Rida in his single "Good Feeling", which was produced by DJ Prak and Cirkut. "Levels" reached the top ten in Austria, Belgium, Bosnia, Croatia, Denmark, Finland, Germany, Greece, Ireland, Italy, the Netherlands, Slovenia and the United Kingdom, whilst topping the charts in Hungary, Norway and Sweden.

In 2012, his collaboration track "Sunshine" with David Guetta was nominated for a Grammy award under the category for Best Dance Recording. His track "Fade into Darkness" was sampled by Leona Lewis on her single "Collide". The sampling was not accredited and led to controversy as Avicii attempted to block the single's release. However, the matter was resolved outside of court with representatives stating "that Leona Lewis and Avicii will work together on the forthcoming single of Collide".

On 23 March 2012, Avicii's unsigned single "Last Dance" was previewed on Pete Tong's show on BBC Radio 1. The song was later released on 27 August 2012. At Ultra Music Festival 2012 in Miami, he premiered two tracks, "Girl Gone Wild" (Avicii's UMF Remix) with Madonna and "Superlove" with Lenny Kravitz. Avicii's UMF Remix of "Girl Gone Wild" was released on 20 April 2012, and "Superlove" with Kravitz was released on 29 May 2012. After reaching two million followers on Facebook, Avicii released a new song titled "Two Million". It was put out as a free download on his official Soundcloud page. On 27 April 2012, Avicii released "Silhouettes". The song featured vocals from Salem Al Fakir and peaked at number 5 on the UK Dance charts and number 4 on the Billboard Hot Dance Club Songs.

Avicii was a featured performer on 4 August 2012 at Lollapalooza festival in Chicago's Grant Park. On 12 August 2012, he released "Dancing in My Head" (Avicii's 'Been Cursed' Mix) on Beatport. The track features vocals from Eric Turner. A radio edit of the track, titled "Dancing in My Head" (Tom Hangs Mix) was released on 14 August 2012 on iTunes, and a remix EP was later released on 30 October 2012 featuring remixes from Charlie Bernardo and Michael Woods. On 26 September 2012, Avicii made history by being the first DJ to headline the world famous Radio City Music Hall in New York City. He performed two sold out shows on 26 and 27 September. He was supported by Matt Goldman and Cazzette at the two shows. At his Radio City Music Hall shows, he played a preview of his new track with Mike Posner titled "Stay with You".

On 29 December 2012, Avicii premiered many new songs at Lights All Night, Dallas Convention Center, some of which made it into his first album, True. These unreleased songs included "I'll Be Gone" and "Let It Go". The instrumental of "Let It Go" was mixed with the a capella of "Addicted to You" to become "Addicted to You (Avicii by Avicii)".

On 29 December 2012, Avicii released "I Could Be the One" with Nicky Romero. After first being debuted at his shows almost a year earlier, the track finally got released via Avicii's record label LE7ELS. The new vocal version was released along with an instrumental version, a radio edit, and remixes from Audrio and Didrick. On 9 January 2013, Avicii launched the Avicii X You project, a partnership with Ericsson designed to create the world's first "crowdsourced" hit song. The project enabled fans to send in basslines, effects, melodies, rhythms and vocals to Avicii as sound files over the Internet. The song features sequences from Kian Sang (melody), Naxsy (bassline), Martin Kupilas (beat), ВАНЯ ХАКСИ (break), Jonathan Madray, Mateusz Kolata, and Christian Westphalen (effects). Avicii acted as executive producer and created the finished song officially titled X You, which was released on 26 February 2013. On 30 January 2013, Avicii released "Three Million" featuring Negin to mark the three millionth fan on his Facebook page.

Avicii was nominated for a Grammy for Best Dance Recording with "Levels" at the 2013 Grammy Awards. He was nominated alongside Calvin Harris and Ne-Yo, Skrillex, Swedish House Mafia and John Martin, and Al Walser. The award show took place on 10 February 2013. From late February to early March 2013, Avicii toured Australia as one of the headline acts in the Future Music Festival alongside The Prodigy and The Stone Roses. In late February 2013, Avicii made his first South American tour, with concerts in Venezuela, Argentina, Colombia, Brazil and Chile.

2013: True

บทความหลัก: True (Avicii album)

Avicii at Inox Festival, 2011.

In March 2013, Avicii announced and premiered many new tracks from his new album True, which would later be set to be released in September, during his Main Stage set at Ultra Music Festival in Miami. The new tracks were experimental in nature. For example, Avicii brought out a stomping band to play through the new bluegrass-tinged song "Wake Me Up". Many of these new songs, however, received mixed critical reviews after the concert.

On 11 April 2013, Avicii released his new album promo mix on SoundCloud. It contained some songs from True and some of his non-album singles. It also contained some of his ID's such as "Black and Blue" and "Enough is Enough (Don't Give Up On Us)".

The EBU and SVT announced on 15 April that Avicii, along with ex-ABBA members Björn Ulvaeus and Benny Andersson, had composed the anthem for the Eurovision Song Contest 2013. The song was performed for the first time in the Final on 18 May.

On 14 June 2013, the world premiere of Avicii's new single, "Wake Me Up", was previewed by Pete Tong on BBC Radio 1, featuring vocals from Aloe Blacc. The song was later released on iTunes and radio on 25 June 2013. It is the first single from Avicii's album True, which was released on 16 September 2013. "Wake Me Up" was number 1 on the Spotify Global Chart and Avicii was at 2 in most streamed artist worldwide. "Wake Me Up" later went on to set a then record of 14 weeks as the number one hit on Billboards Dance/Electronic Songs list. The Official Charts Company announced on 21 July that "Wake Me Up" had become the UK's fastest selling single of 2013 after overtaking Robin Thicke's "Blurred Lines", having sold 267,000 copies in its first week on sale in the UK. "Wake Me Up" subsequently became a major hit, topping the charts in over 20 countries including Australia, Germany, Ireland, Italy, Sweden, the Netherlands, and New Zealand.

On 19 October 2013, the DJ Mag 2013 Top 100 DJs List was announced, with Avicii ranking number 3 on the list. Hardwell became the new World No.1 displacing Armin Van Buuren. On 28 October 2013, Avicii released the single "Hey Brother" with vocals by Dan Tyminski. On 10 November, Avicii won his first award "Best Electronic" at the MTV EMA's.

On 24 November 2013 he won the American Music Award for favorite Electronic Dance Music Artist.

In December 2013, Avicii released his fourth single off the album, "Addicted To You", which reached number 1 in Australia, with vocals from Audra Mae, who also sings on Shame on Me and Long Road to Hell, both tracks on the album. Avicii also released the single, Lay Me Down featuring vocals from Adam Lambert, and guitar playing from Nile Rodgers, who plays on many songs including Daft Punk's album Random Access Memories, and Adam Lambert's album Trespassing. On 29 December 2013, Avicii debuted his new track "Dreaming of Me", featuring vocals from Audra Mae, via episode 19 of his LE7ELS podcast. It is unknown whether it will be released in the future.

In January 2014, the French hit music radio station NRJ released a new web radio station called "NRJ AVICII", which plays songs only by Avicii.

On 8 March 2014, Avicii's Instagram account confirmed a collaboration with Madonna. The extent of the collaboration is unknown. On 21 March 2014, Avicii released a remixed edition of his album True titled True: Avicii By Avicii containing remixes by himself of all the tracks, excluding "Heart Upon My Sleeve" for unknown reasons. The promotion of this album was supposed to begin at the 2014 Ultra Music Festival, but unfortunately Avicii announced he had been hospitalised on 28 March, and was unable to play his closing set at the festival. On 28 March 2014, FIFA and Sony Music Entertainment announced that Avicii would be collaborating with Carlos Santana, Wyclef Jean and Alexandre Pires for the official FIFA World Cup Anthem titled "Dar um Jeito (We Will Find a Way)". The anthem was performed at the FIFA World Cup Closing Ceremony on 13 July 2014.

In 2014, following the FIFA World Cup, Avicii released his single "Lay Me Down". He also produced and collaborated with Chris Martin from Coldplay, co-writing and co-producing the track "A Sky Full of Stars" from the band's sixth studio album Ghost Stories, released on 19 May 2014. He also played and recorded the piano parts on the track. "A Sky Full of Stars" was released on 3 May as the second single from Ghost Stories. "Lovers on the Sun", a track Avicii co-produced with David Guetta, was released on 30 June 2014. He has also worked with Wyclef Jean on a single titled "Divine Sorrow" off his upcoming album Clefication.

2014–2015: Stories and major collaborations

บทความหลัก: Stories (Avicii album)

In July 2014, Avicii told Rolling Stone that he had worked on 70 songs for his next album Stories, which would include collaborations with Jon Bon Jovi, Serj Tankian of System of a Down, Chris Martin, Wyclef Jean and Matisyahu. Describing the album, Avicii said: "It's going to be a lot more song-oriented."

During his tour for True, he also performed his upcoming single "No Pleasing A Woman" with vocals from Billie Joe Armstrong of American rock band Green Day. It has a similar instrumental to "Wake Me Up" albeit different chord progressions, along with "No Pleasing A Woman." Avicii also performed other upcoming songs like "In Love With Your Ghost" with Daniel "Danne" Adams-Ray, "Love To Fall" with Tom Odell and "Million Miles" with LP which is the demo version of "Trouble", a song from "Stories" with vocals from Wayne Hector.

"Lose Myself", a collaboration between Avicii and Chinese singer Wang Leehom, was released on 1 September 2014. On 8 September 2014, his 25th birthday, Avicii decided to cancel all of his remaining performances of 2014 due to health concerns. The following day, he announced through Denim & Supply that he would release his new single "The Days" later in 2014; a video was also featured with a preview of the track. "The Days" is a collaboration between Avicii and Robbie Williams, and was released on 3 October 2014 via PRMD. On 16 September 2014, it was announced through EA Sports that Avicii was debuting a brand new track called "The Nights" exclusively on FIFA 15. On 17 November 2014, it was officially announced that the track is to be released as a part of an EP together with "The Days".

On 17 November 2014, Wyclef Jean released a track titled "Divine Sorrow" featuring Avicii.

On 2 March 2015, Avicii performed live at Australia's Future Music Festival his upcoming single "Heaven", a collaboration with Coldplay frontman Chris Martin on vocals. However, the vocals were later resung by singer Simon Aldred from Cherry Ghost; Chris Martin will receive writing credits on "Heaven". A lot of songs were leaked into the internet via Avicii's UMF set 2015. These songs include "Heaven", "Waiting For Love," and some of his ID's "For A Better Day", "City Lights", "Can't Catch Me", "True Believer", "What Would I Change It To", "Can't Love You Again" (previously leaked on the internet under the name "Don't Call") and "Attack".

On 25 April 2015, Avicii announced on episode 35 of his LE7ELS podcast that he would be playing some of his unreleased songs on his show. He also previewed his bootleg of Kings of Tomorrow's song "Finally" and one of his old songs that he did with ASH, titled "Papa Was a Rolling Stone".

On 6 May 2015, Avicii released his rework of Nina Simone's version of "Feeling Good", composed in 1964 by the English songwriters Anthony Newley and Leslie Bricusse. On 8 May 2015, Avicii previewed his song "I'll Be Gone" via episode 422 of Tiësto's Club Life Podcast which has a very similar instrumental to Liar Liar (Avicii by Avicii) albeit different vocals and chord progressions. It was leaked into the internet between 2013–2014 via the name "Stars" and is yet to be released. It also shares a similar drop to one of Avicii's ID's "Black and Blue". On 22 May 2015, Avicii premiered another single from Stories, "Waiting for Love". The track was co-produced by fellow DJ and producer Martin Garrix, and featured vocals from Simon Aldred of Cherry Ghost. On 25 May 2015, Avicii previewed three old tracks on episode 36 of his LE7ELS podcast: "Tracks of My Tears", "Sorry Mr. Atari" and "Time to Get lll", all of which are songs that Avicii had made years ago. "Tracks of My Tears" is the original version of Avicii's "All You Need is Love". On 27 May 2015, it was confirmed by a Swedish newspaper through an interview with Avicii that his sophomore album Stories would be released in fall 2015.

On 3 July 2015, Avicii previewed two tracks from his album Stories on episode 37 of his LE7ELS podcast: "Broken Arrows" featuring Zac Brown Band and "Can’t Catch Me" featuring Wyclef Jean and Matisyahu. He also played a full version of his track "Heaven" and also revealed that he was doing another track with Chris Martin on vocals called "True Believer". Later, Avicii also revealed that he is also singing on that track. On 18 July 2015, it was announced by Avicii that he had finally finished production on Stories after 2 years of work.

A couple of weeks later, on 4 August 2015, it was announced that the final singles before the release of Stories would be "For a Better Day" featuring American singer Alex Ebert and "Pure Grinding" featuring vocals from Kristoffer Fogelmark and Earl St. Clair. On 27 August, Avicii released a teaser video on Instagram with the song "Pure Grinding" playing. The tracks "For a Better Day" and "Pure Grinding" were released the following day through Spotify and iTunes. On 26 September, Avicii announced "Stories – Megamix" on Spotify.

Stories was released on 2 October 2015 alongside 3 promotional singles: "Broken Arrows" with Zac Brown, "Ten More Days" with Zak Abel and "Gonna Love Ya" with Sandro Cavazza.

2016: New music and upcoming third studio album

On 15 January 2016, Avicii released his remix of Morten's "Beautiful Heartbeat". Coke partners with Avicii for global campaign anthem featuring Conrad Sewell. The song was released on 19 January. On 25 January, Avicii teamed up again with Coldplay to co-produce the band's single "Hymn for the Weekend", which was released as the second single from their album A Head Full of Dreams. In 2016, according to a report by Inc magazine, Avicii was 6th fastest growing company in Europe, with a revenue of €7.7 million in 2014.

On 19 March, Avicii performed live at Ultra Music Festival and premiered new IDs (Including a song called "Without You" with Sandro Cavazza). On 29 March, Avicii announced on Facebook that he would be retiring from performing live and touring in 2016.

Influences

Daft Punk (left) and Swedish House Mafia (right) have both been influences on Avicii.

Avicii's influences include Daft Punk, former trio Swedish House Mafia (Sebastian Ingrosso, Steve Angello, Axwell) and Eric Prydz.

Avicii is named in the song "I Took a Pill in Ibiza" by American singer and songwriter Mike Posner. The song was made in the week they wrote a song together in Sweden called “Stay With You”. Posner went to a show of Avicii in Ibiza and got a mystery pill from someone. The song starts with:

I took a pill in Ibiza. To show Avicii I was cool.

Charitable work

Since achieving widespread commercial success, Avicii began working with his manager and executive producer Ash Pournouri to start House for Hunger in 2011, a charity dedicated to alleviating global hunger. The pair wanted to showcase the giving spirit fostered by the house music community. Avicii explained, "You have to give something back. I am so fortunate to be in the position where I can actually do that. I feel lucky every day when I wake up and am able to do what I love and make a living." In addition to donating $1 million to Feeding America, a charity founded by John van Hengel, House for Hunger has helped fund the efforts of The Feed Foundation, started by Laura Bush, allowing it to distribute over two million school meals throughout Africa.

Discography

บทความหลัก: Avicii discography

Studio albums

• True (2013)
• Stories (2015)

Concert tours

• True Tour (2014)
• Stories World Tour (2015)

Awards and nominations

American Music Awards

The American Music Awards is an annual American music awards show created by Dick Clark in 1973, which airs on ABC. Avicii received two nominations and won one.

ปี	รางวัลที่ได้เข้าชิง	รางวัล/สาขา	ผล
2013	Avicii	Favorite Electronic Dance Music Artist	ชนะ
2014		Favorite Electronic Dance Music Artist	เสนอชื่อเข้าชิง

Billboard Music Awards

The Billboard Music Awards are an annual awards show from Billboard Magazine, Avicii received six nominations.

ปี	รางวัลที่ได้เข้าชิง	รางวัล/สาขา	ผล
2014	Avicii	Top Dance/Electronic Artist	เสนอชื่อเข้าชิง
	True	Top Dance/Electronic Album	เสนอชื่อเข้าชิง
	"Wake Me Up"	Top Radio Song	เสนอชื่อเข้าชิง
		Top Dance/Electronic Song	ชนะ
2015	Avicii	Top Dance/Electronic Artist	เสนอชื่อเข้าชิง
	True	Top Dance/Electronic Album	เสนอชื่อเข้าชิง

Echo Music Awards

The Echo Music Awards is an annual music awards ceremony presented by Deutsche Phono-Akademie in Germany. Avicii won two awards.

ปี	รางวัลที่ได้เข้าชิง	รางวัล/สาขา	ผล
2014	Avicii	Electronic Dance Music National/International	ชนะ
		Hit des Jahres

Grammy Awards

The Grammy Awards are awarded by the National Academy of Recording Arts and Sciences of the United States. Avicii received one nomination in 2012 and one nomination in 2013.

ปี	รางวัลที่ได้เข้าชิง	รางวัล/สาขา	ผล
2012	"Sunshine" (David Guetta & Avicii)	Best Dance Recording	เสนอชื่อเข้าชิง
2013	"Levels"	Best Dance Recording	เสนอชื่อเข้าชิง

iHeartRadio Music Awards

The iHeartRadio Music Awards is an awards show created by iHeartRadio. Avicii received two nominations and won an award.

ปี	รางวัลที่ได้เข้าชิง	รางวัล/สาขา	ผล
2014	"Wake Me Up"	Best Lyrics	เสนอชื่อเข้าชิง
		EDM Song of the Year	ชนะ

International Dance Music Awards

The International Dance Music Awards are held annually as part of the Winter Music Conference. Avicii has received four nominations and won one.

ปี	รางวัลที่ได้เข้าชิง	รางวัล/สาขา	ผล
2013	Avicii	Best European DJ	เสนอชื่อเข้าชิง
		Best Remixer
	Levels Podcast	Best Podcast
	Avicii	Best Artist (Solo)	ชนะ

MTV Europe Music Awards

The MTV Europe Music Awards were established in 1994 by MTV Networks Europe, Avicii received five nominations and won two.

ปี	รางวัลที่ได้เข้าชิง	รางวัล/สาขา	ผล
2012	Avicii	Best Electronic	เสนอชื่อเข้าชิง
		Best Swedish Act	เสนอชื่อเข้าชิง
2013		Best Electronic	ชนะ
		Best Swedish Act	ชนะ
		Best Northern European Act	เสนอชื่อเข้าชิง
2015		Best Electronic	เสนอชื่อเข้าชิง
		Best Swedish Act	เสนอชื่อเข้าชิง

MTV Video Music Awards

The MTV Video Music Awards are presented annually by MTV and honor accomplishments in the music video medium. Avicii has received two nominations.

ปี	รางวัลที่ได้เข้าชิง	รางวัล/สาขา	ผล
2014	"Wake Me Up"	Best Pop Video	เสนอชื่อเข้าชิง
	"Hey Brother"	Best Video with a Social Message

Teen Choice Awards

The Teen Choice Awards is an annual award ceremony to honour the year's biggest achievements in entertainment and is voted by teen viewers. Avicii has won an award.

ปี	รางวัลที่ได้เข้าชิง	รางวัล/สาขา	ผล
2014	"Wake Me Up"	Choice EDM Song	ชนะ

World Music Awards

The World Music Awards is an annual awards ceremony that honors best selling artists in the record industry. Avicii received eight nominations.

ปี	รางวัลที่ได้เข้าชิง	รางวัล/สาขา	ผล
2014	"Wake Me Up"	World's Best Song	เสนอชื่อเข้าชิง
		World's Best Video	เสนอชื่อเข้าชิง
	True	World's Best Album	เสนอชื่อเข้าชิง
	Avicii	World's Best Male	เสนอชื่อเข้าชิง
		World's Best Live Act	เสนอชื่อเข้าชิง
		World's Best Electronic Dance Artist	ชนะ
		World's Best Entertainer of the Year	เสนอชื่อเข้าชิง
		World's Best Swedish Act	ชนะ

See also

• Swedish popular music

External links

• Avicii X You Project
• Avicii on Best4Ears.com
• Salacious Sound Interviews Avicii

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สถานีวัดมังกร

U3 ชั้นร้านค้า (เฉพาะทางออก 2)	-	ศูนย์การค้าเมโทรมอลล์
U2 ชั้นร้านค้า (เฉพาะทางออก 2)	-	ศูนย์การค้าเมโทรมอลล์
G ระดับถนน	-	ป้ายรถประจำทาง, วัดมังกรกมลาวาส, ย่านเยาวราช
B1 ทางเดินระหว่างชั้นระดับถนน กับชั้นขายตั๋วโดยสาร	ชั้น Subway	-
B2 ชั้นขายบัตรโดยสาร	ชั้นขายบัตรโดยสาร	ทางออก 1-2, ห้องขายบัตรโดยสาร, เครื่องขายบัตรโดยสาร
B4 ชานชาลา	ชานชาลา 2	แม่แบบ:BTS Lines มุ่งหน้า สถานีหัวลำโพง - สถานีบางซื่อ - สถานีท่าพระ
	ชานชาลาด้านข้าง, ประตูรถจะเปิดทางด้านขวา
B5 ชานชาลา	ชานชาลา 1	แม่แบบ:BTS Lines มุ่งหน้า สถานีหลักสอง
	ชานชาลาด้านข้าง, ประตูรถจะเปิดทางด้านซ้าย