Question:

# How much does Curves cost for a month?

## The monthly fees for Curves vary widely, depending on the area, country, cost of doing business, and whether you are signing a year's contract or just going month-to-month. On average it costs around $35 a month. ### More Info: In economics, fixed costs are business expenses that are not dependent on the level of goods or services produced by the business. They tend to be time-related, such as salaries or rents being paid per month, and are often referred to as overhead costs. This is in contrast to variable costs, which are volume-related (and are paid per quantity produced). In management accounting, fixed costs are defined as expenses that do not change as a function of the activity of a business, within the relevant period. For example, a retailer must pay rent and utility bills irrespective of sales. In marketing, it is necessary to know how costs divide between variable and fixed. This distinction is crucial in forecasting the earnings generated by various changes in unit sales and thus the financial impact of proposed marketing campaigns. In a survey of nearly 200 senior marketing managers, 60 percent responded that they found the "variable and fixed costs" metric very useful. Fixed costs are not permanently fixed; they will change over time, but are fixed in relation to the quantity of production for the relevant period. For example, a company may have unexpected and unpredictable expenses unrelated to production; and warehouse costs and the like are fixed only over the time period of the lease. By definition, there are fixed costs in the long run. Investments in facilities, equipment, and the basic organization that can't be significantly reduced in a short period of time are referred to as committed fixed costs. Discretionary fixed costs usually arise from annual decisions by management to spend on certain fixed cost items. Examples of discretionary costs are advertising, machine maintenance, and research & development expenditures. Discretionary fixed costs can be expensive. In business planning and management accounting, usage of the terms fixed costs, variable costs and others will often differ from usage in economics, and may depend on the intended use. Some cost accounting practices such as activity-based costing will allocate fixed costs to business activities for profitability measures. This can simplify decision-making, but can be confusing and controversial. In accounting terminology, fixed costs will broadly include almost all costs (expenses) which are not included in cost of goods sold, and variable costs are those captured in costs of goods sold. The implicit assumption required to make the equivalence between the accounting and economics terminology is that the accounting period is equal to the period in which fixed costs do not vary in relation to production. In practice, this equivalence does not always hold, and depending on the period under consideration by management, some overhead expenses (e.g., sales, general and administrative expenses) can be adjusted by management, and the specific allocation of each expense to each category will be decided under cost accounting. In economics, a cost curve is a graph of the costs of production as a function of total quantity produced. In a free market economy, productively efficient firms use these curves to find the optimal point of production (minimizing cost), and profit maximizing firms can use them to decide output quantities to achieve those aims. There are various types of cost curves, all related to each other, including total and average cost curves, and marginal ("for each additional unit") cost curves, which are equal to the differential of the total cost curves. Some are applicable to the short run, others to the long run. Average variable cost (which is a short-run concept) is the variable cost (typically labor cost) per unit of output: SRAVC = wL / Q where w is the wage rate, L is the quantity of labor used, and Q is the quantity of output produced. The SRAVC curve plots the short-run average variable cost against the level of output, and is typically drawn as U-shaped. The average total cost curve is constructed to capture the relation between cost per unit of output and the level of output, ceteris paribus. A perfectly competitive and productively efficient firm organizes its factors of production in such a way that the average cost of production is at the lowest point. In the short run, when at least one factor of production is fixed, this occurs at the output level where it has enjoyed all possible average cost gains from increasing production. This is at the minimum point in the diagram on the right. Short-run total cost is given by where PK is the unit price of using physical capital per unit time, PL is the unit price of labor per unit time (the wage rate), K is the quantity of physical capital used, and L is the quantity of labor used. From this we obtain short-run average cost, denoted either SATC or SAC, as STC / Q: where APK = Q/K is the average product of capital and APL = Q/L is the average product of labor.:191 Short run average cost equals average fixed costs plus average variable costs. Average fixed cost continuously falls as production increases in the short run, because K is fixed in the short run. The shape of the average variable cost curve is directly determined by increasing and then diminishing marginal returns to the variable input (conventionally labor).:210 The long-run average cost curve depicts the cost per unit of output in the long run—that is, when all productive inputs' usage levels can be varied. All points on the line represent least-cost factor combinations; points above the line are attainable but unwise, while points below are unattainable given present factors of production. The behavioral assumption underlying the curve is that the producer will select the combination of inputs that will produce a given output at the lowest possible cost. Given that LRAC is an average quantity, one must not confuse it with the long-run marginal cost curve, which is the cost of one more unit.:232 The LRAC curve is created as an envelope of an infinite number of short-run average total cost curves, each based on a particular fixed level of capital usage.:235 The typical LRAC curve is U-shaped, reflecting increasing returns of scale where negatively-sloped, constant returns to scale where horizontal and decreasing returns (due to increases in factor prices) where positively sloped.:234 Contrary to Viner][, the envelope is not created by the minimum point of each short-run average cost curve.:235 This mistake is recognized as Viner's Error.][ In a long-run perfectly competitive environment, the equilibrium level of output corresponds to the minimum efficient scale, marked as Q2 in the diagram. This is due to the zero-profit requirement of a perfectly competitive equilibrium. This result, which implies production is at a level corresponding to the lowest possible average cost,:259 does not imply that production levels other than that at the minimum point are not efficient. All points along the LRAC are productively efficient, by definition, but not all are equilibrium points in a long-run perfectly competitive environment. In some industries, the bottom of the LRAC curve is large in comparison to market size (that is to say, for all intents and purposes, it is always declining and economies of scale exist indefinitely). This means that the largest firm tends to have a cost advantage, and the industry tends naturally to become a monopoly, and hence is called a natural monopoly. Natural monopolies tend to exist in industries with high capital costs in relation to variable costs, such as water supply and electricity supply.:312 A short-run marginal cost curve graphically represents the relation between marginal (i.e., incremental) cost incurred by a firm in the short-run production of a good or service and the quantity of output produced. This curve is constructed to capture the relation between marginal cost and the level of output, holding other variables, like technology and resource prices, constant. The marginal cost curve is U-shaped. Marginal cost is relatively high at small quantities of output; then as production increases, marginal cost declines, reaches a minimum value, then rises. The marginal cost is shown in relation to marginal revenue (MR), the incremental amount of sales revenue that an additional unit of the product or service will bring to the firm. This shape of the marginal cost curve is directly attributable to increasing, then decreasing marginal returns (and the law of diminishing marginal returns). Marginal cost equals w/MPL.:191 For most production processes the marginal product of labor initially rises, reaches a maximum value and then continuously falls as production increases. Thus marginal cost initially falls, reaches a minimum value and then increases.:209 The marginal cost curve intersects both the average variable cost curve and (short-run) average total cost curve at their minimum points. When the marginal cost curve is above an average cost curve the average curve is rising. When the marginal costs curve is below an average curve the average curve is falling. This relation holds regardless of whether the marginal curve is rising or falling.:226 The long-run marginal cost curve shows for each unit of output the added total cost incurred in the long run, that is, the conceptual period when all factors of production are variable so as minimize long-run average total cost. Stated otherwise, LRMC is the minimum increase in total cost associated with an increase of one unit of output when all inputs are variable. The long-run marginal cost curve is shaped by return to scale, a long-run concept, rather than the law of diminishing marginal returns, which is a short-run concept. The long-run marginal cost curve tends to be flatter than its short-run counterpart due to increased input flexibility as to cost minimization. The long-run marginal cost curve intersects the long-run average cost curve at the minimum point of the latter.:208 When long-run marginal costs are below long-run average costs, long-run average costs are falling (as to additional units of output).:207 When long-run marginal costs are above long run average costs, average costs are rising. Long-run marginal cost equals short run marginal-cost at the least-long-run-average-cost level of production. LRMC is the slope of the LR total-cost function. Cost curves can be combined to provide information about firms. In this diagram for example, firms are assumed to be in a perfectly competitive market. In a perfectly competitive market the price that firms are faced with would be the price at which the marginal cost curve cuts the average cost curve. Assuming that factor prices are constant, the production function determines all cost functions. The variable cost curve is the inverted short-run production function or total product curve and its behavior and properties are determined by the production function.:209 Because the production function determines the variable cost function it necessarily determines the shape and properties of marginal cost curve and the average cost curves. If the firm is a perfect competitor in all input markets, and thus the per-unit prices of all its inputs are unaffected by how much of the inputs the firm purchases, then it can be shown that at a particular level of output, the firm has economies of scale (i.e., is operating in a downward sloping region of the long-run average cost curve) if and only if it has increasing returns to scale. Likewise, it has diseconomies of scale (is operating in an upward sloping region of the long-run average cost curve) if and only if it has decreasing returns to scale, and has neither economies nor diseconomies of scale if it has constant returns to scale. In this case, with perfect competition in the output market the long-run market equilibrium will involve all firms operating at the minimum point of their long-run average cost curves (i.e., at the borderline between economies and diseconomies of scale). If, however, the firm is not a perfect competitor in the input markets, then the above conclusions are modified. For example, if there are increasing returns to scale in some range of output levels, but the firm is so big in one or more input markets that increasing its purchases of an input drives up the input's per-unit cost, then the firm could have diseconomies of scale in that range of output levels. Conversely, if the firm is able to get bulk discounts of an input, then it could have economies of scale in some range of output levels even if it has decreasing returns in production in that output range. Basic: For each quantity of output there is one cost minimizing level of capital and a unique short run average cost curve associated with producing the given quantity. These statements assume that the firm is using the optimal level of capital for the quantity produced. If not, then the SRAC curve would lie "wholly above" the LRAC and would not be tangent at any point. Both the SRAC and LRAC curves are typically expressed as U-shaped.:211; 226 :182;187-188 However, the shapes of the curves are not due to the same factors. For the short run curve the initial downward slope is largely due to declining average fixed costs.:227 Increasing returns to the variable input at low levels of production also play a role, while the upward slope is due to diminishing marginal returns to the variable input.:227 With the long run curve the shape by definition reflects economies and diseconomies of scale.:186 At low levels of production long run production functions generally exhibit increasing returns to scale, which, for firms that are perfect competitors in input markets, means that the long run average cost is falling;:227 the upward slope of the long run average cost function at higher levels of output is due to decreasing returns to scale at those output levels.:227 In finance, the yield curve is a curve showing several yields or interest rates across different contract lengths (2 month, 2 year, 20 year, etc...) for a similar debt contract. The curve shows the relation between the (level of) interest rate (or cost of borrowing) and the time to maturity, known as the "term", of the debt for a given borrower in a given currency. For example, the U.S. dollar interest rates paid on U.S. Treasury securities for various maturities are closely watched by many traders, and are commonly plotted on a graph such as the one on the right which is informally called "the yield curve." More formal mathematical descriptions of this relation are often called the term structure of interest rates. The shape of the yield curve indicates the cumulative priorities of all lenders relative to a particular borrower (such as the US Treasury or the Treasury of Japan). Usually, lenders are concerned about a potential default (or rising rates of inflation), so they offer long-term loans for higher interest rates than they offer for shorter-term loans. Occasionally, when lenders are seeking long-term debt contracts more aggressively than short-term debt contracts, the yield curve "inverts," with interest rates (yields) being lower for the longer periods of repayment so that lenders can attract long-term borrowing. The yield of a debt instrument is the overall rate of return available on the investment. In general the percentage per year that can be earned is dependent on the length of time that the money is invested. For example, a bank may offer a "savings rate" higher than the normal checking account rate if the customer is prepared to leave money untouched for five years. Investing for a period of time t gives a yield Y(t). This function Y is called the yield curve, and it is often, but not always, an increasing function of t. Yield curves are used by fixed income analysts, who analyze bonds and related securities, to understand conditions in financial markets and to seek trading opportunities. Economists use the curves to understand economic conditions. 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). 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 depressions. 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). 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. 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. 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 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. 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. An inverted yield curve occurs when long-term yields fall below short-term yields. See. Under unusual circumstances, long-term investors will settle for lower yields now if they think the economy will slow or even decline in the future. 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 6 out of 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. Since the financial crisis of 2007–2008, the U. S. Federal Reserve has maintained a zero interest-rate policy (ZIRP) for an "extended period of time", where the short term interest rate is practically set to zero. Under this circumstance the nominal yield curve is unlikely to be inverted even in the event of a recession. Thus the absence of an inverted yield curve is not a sufficient condition to preclude recessions. 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. 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. Using this, future rates, along with the assumption that arbitrage opportunities will be minimal in future markets, and that future rates are unbiased estimates of forthcoming spot rates, is 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 2-year interest rate can be calculated as the compounding of this year's interest rate by next year's interest rate. More generally, rates on a long-term instrument are equal to the geometric mean of the yield on a series of short-term instruments. This theory perfectly explains 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: Neglects the risks inherent in investing in bonds (because forward rates are not perfect predictors of future rates). 1) Interest rate risk 2) Reinvestment rate risk 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: Where $rp_n$ is the risk premium associated with an ${n}$ year bond. 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). The preferred habitat theory is another guise 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. 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. 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. 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 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 The significant difficulty in defining a yield curve therefore is to determine the function P(t). P is called the discount factor function. 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, 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 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: 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. 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. 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. 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 rates and the federal funds rate) is included in the Index of Leading Economic Indicators. An inverted yield curve is often a harbinger of recession. A positively sloped yield curve is often a harbinger of inflationary growth. Work by Dr. Arturo Estrella & Dr. 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 (he uses 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 that a rise in unemployment usually occurs. All of the recessions in the US since 1970 (up through 2011) 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. Dr. Estrella has postulated that the yield curve affects the business cycle via the balance sheet of banks. When the yield curve is inverted banks are often caught paying more on short-term deposits than they are making on 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 long-term loans so they are eager to supply credit to borrowers. Markup is the difference between the cost of a good or service and its selling price. A markup is added on to the total cost incurred by the producer of a good or service in order to create a profit. The total cost reflects the total amount of both fixed and variable expenses to produce and distribute a product. Markup can be expressed as a fixed amount or as a percentage of the total cost or selling price. Retail markup is commonly calculated as the difference between wholesale price and retail price, as a percentage of wholesale. Other methods are also used. P = (1+μ) W. Where μ is the markup over costs. This is the pricing equation. W = F(u,z) Pe . This is the wage setting relation. u is unemployment which negatively affects wages and z the catch all variable positively affects wages. P = Pe(1+μ) F(u,z). This is the aggregate supply curve. Where the price is determined by expected price, unemployment and z the catch all variable. In production, research, retail, and accounting, a cost is the value of money that has been used up to produce something, and hence is not available for use anymore. In business, the cost may be one of acquisition, in which case the amount of money expended to acquire it is counted as cost. In this case, money is the input that is gone in order to acquire the thing. This acquisition cost may be the sum of the cost of production as incurred by the original producer, and further costs of transaction as incurred by the acquirer over and above the price paid to the producer. Usually, the price also includes a mark-up for profit over the cost of production. More generalized in the field of economics, cost is a metric that is totaling up as a result of a process or as a differential for the result of a decision. Hence cost is the metric used in the standard modeling paradigm applied to economic processes. Costs (pl.) are often further described based on their timing or their applicability. In accounting, costs are the monetary value of expenditures for supplies, services, labor, products, equipment and other items purchased for use by a business or other accounting entity. It is the amount denoted on invoices as the price and recorded in bookkeeping records as an expense or asset cost basis. Opportunity cost, also referred to as economic cost is the value of the best alternative that was not chosen in order to pursue the current endeavor—i.e., what could have been accomplished with the resources expended in the undertaking. It represents opportunities forgone. In theoretical economics, cost used without qualification often means opportunity cost.][ When a transaction takes place, it typically involves both private costs and external costs. Private costs are the costs that the buyer of a good or service pays the seller. This can also be described as the costs internal to the firm's production function. External costs (also called externalities), in contrast, are the costs that people other than the buyer are forced to pay as a result of the transaction. The bearers of such costs can be either particular individuals or society at large. Note that external costs are often both non-monetary and problematic to quantify for comparison with monetary values. They include things like pollution, things that society will likely have to pay for in some way or at some time in the future, but that are not included in transaction prices. Social costs are the sum of private costs and external costs. For example, the manufacturing cost of a car (i.e., the costs of buying inputs, land tax rates for the car plant, overhead costs of running the plant and labor costs) reflects the private cost for the manufacturer (in some ways, normal profit can also be seen as a cost of production; see, e.g., Ison and Wall, 2007, p. 181). The polluted waters or polluted air also created as part of the process of producing the car is an external cost borne by those who are affected by the pollution or who value unpolluted air or water. Because the manufacturer does not pay for this external cost (the cost of emitting undesirable waste into the commons), and does not include this cost in the price of the car (a Kaldor-Hicks compensation), they are said to be external to the market pricing mechanism. The air pollution from driving the car is also an externality produced by the car user in the process of using his good. The driver does not compensate for the environmental damage caused by using the car. A psychic cost is a subset of social costs that specifically represent the costs of added stress or losses to quality of life. When developing a business plan for a new or existing company, product, or project, planners typically make cost estimates in order to assess whether revenues/benefits will cover costs (see cost-benefit analysis). This is done in both business and government. Costs are often underestimated, resulting in cost overrun during execution. Cost-plus pricing, is where the price equals cost plus a percentage of overhead or profit margin. Manufacturing Costs are those costs that are directly involved in manufacturing of products. Examples of manufacturing costs include raw materials costs and charges related workers. Manufacturing cost is divided into three broad categories: Non-manufacturing Costs are those costs that are not directly incurred to manufacture a product. Examples of such costs are salary of sales personnel and advertising expenses. Generally non-manufacturing costs are further classified into two categories: A defensive cost is an environmental expenditure to eliminate or prevent environmental damage. They form part of the genuine progress indicator (GPI) calculations. Labour on costs would include travel time, holiday pay, training costs, working clothes. Path cost is a term in networking to define the worthiness of a path, see Routing. In microeconomic theory, the opportunity cost of a choice is the value of the best alternative forgone, in a situation in which a choice needs to be made between several mutually exclusive alternatives given limited resources. Assuming the best choice is made, it is the "cost" incurred by not enjoying the benefit that would be had by taking the second best choice available. The New Oxford American Dictionary defines it as "the loss of potential gain from other alternatives when one alternative is chosen". Opportunity cost is a key concept in economics, and has been described as expressing "the basic relationship between scarcity and choice". The notion of opportunity cost plays a crucial part in ensuring that scarce resources are used efficiently. Thus, opportunity costs are not restricted to monetary or financial costs: the real cost of output forgone, lost time, pleasure or any other benefit that provides utility should also be considered opportunity costs. The term was coined in 1914 by Austrian economist Friedrich von Wieser in his book . It was first described in 1848 by French classical economist Frédéric Bastiat in his essay "What Is Seen and What Is Not Seen". Opportunity cost may be expressed in terms of anything which is of value. For example, an individual might decide to use a period of vacation time for travel rather than to do household repairs. The opportunity cost of the trip could be said to be the forgone home renovation.][ Opportunity costs may be assessed in the decision-making process of production. If the workers on a farm can produce either one million pounds of wheat or two million pounds of barley, then the opportunity cost of producing one pound of wheat is the two pounds of barley forgone (assuming the production possibilities frontier is linear). Firms would make rational decisions by weighing the sacrifices involved. Explicit costs are opportunity costs that involve direct monetary payment by producers. The opportunity cost of the factors of production not already owned by a producer is the price that the producer has to pay for them. For instance, a firm spends$100 on electrical power consumed, their opportunity cost is $100. The firm has sacrificed$100, which could have been spent on other factors of production. Implicit costs are the opportunity costs in factors of production that a producer already owns. They are equivalent to what the factors could earn for the firm in alternative uses, either operated within the firm or rent out to other firms. For example, a firm pays $300 a month all year for rent on a warehouse that only holds product for six months each year. The firm could rent the warehouse out for the unused six months, at any price (assuming a year-long lease requirement), and that would be the cost that could be spent on other factors of production. Opportunity costs are not always monetary units or being able to produce one good over another. The opportunity cost can also be unknown, or spawn a series of infinite sub opportunity costs. For instance, an individual could choose not to ask a girl out on a date, in an attempt to make her more interested ("playing hard to get"), but the opportunity cost could be that they get ignored - which could result in other opportunities being lost. Note that opportunity cost is not the sum of the available alternatives when those alternatives are, in turn, mutually exclusive to each other – it is the value of the next best use. The opportunity cost of a city's decision to build the hospital on its vacant land is the loss of the land for a sporting center, or the inability to use the land for a parking lot, or the money which could have been made from selling the land. Use for any one of those purposes would preclude the possibility to implement any of the other. Cost accounting is a process of collecting, analyzing, summarizing and evaluating various alternative courses of action. Its goal is to advise the management on the most appropriate course of action based on the cost efficiency and capability. Cost accounting provides the detailed cost information that management needs to control current operations and plan for the future. Since managers are making decisions only for their own organization, there is no need for the information to be comparable to similar information from other organizations. Instead, information must be relevant for a particular environment. Cost accounting information is commonly used in financial accounting information, but first we are concentrating on its use by managers to make decisions. Unlike the accounting systems that help in the preparation of financial reports periodically, the cost accounting systems and reports are not subject to rules and standards like the Generally Accepted Accounting Principles. As a result, there is wide variety in the cost accounting systems of the different companies and sometimes even in different parts of the same company or organization. All types of businesses, whether service, manufacturing or trading, require cost accounting to track their activities. Cost accounting has long been used to help managers understand the costs of running a business. Modern cost accounting originated during the industrial revolution, when the complexities of running a large scale business led to the development of systems for recording and tracking costs to help business owners and managers make decisions. In the early industrial age, most of the costs incurred by a business were what modern accountants call "variable costs" because they varied directly with the amount of production. Money was spent on labor, raw materials, power to run a factory, etc. in direct proportion to production. Managers could simply total the variable costs for a product and use this as a rough guide for decision-making processes. Some costs tend to remain the same even during busy periods, unlike variable costs, which rise and fall with volume of work. Over time, these "fixed costs" have become more important to managers. Examples of fixed costs include the depreciation of plant and equipment, and the cost of departments such as maintenance, tooling, production control, purchasing, quality control, storage and handling, plant supervision and engineering. In the early nineteenth century, these costs were of little importance to most businesses. However, with the growth of railroads, steel and large scale manufacturing, by the late nineteenth century these costs were often more important than the variable cost of a product, and allocating them to a broad range of products lead to bad decision making. Managers must understand fixed costs in order to make decisions about products and pricing. For example: A company produced railway coaches and had only one product. To make each coach, the company needed to purchase$60 of raw materials and components, and pay 6 laborers $40 each. Therefore, total variable cost for each coach was$300. Knowing that making a coach required spending $300, managers knew they couldn't sell below that price without losing money on each coach. Any price above$300 became a contribution to the fixed costs of the company. If the fixed costs were, say, $1000 per month for rent, insurance and owner's salary, the company could therefore sell 5 coaches per month for a total of$3000 (priced at $600 each), or 10 coaches for a total of$4500 (priced at $450 each), and make a profit of$500 in both cases. The following are different cost accounting approaches: Basic cost elements are: (In some companies, machine cost is segregated from overhead and reported as a separate element) Classification of cost means, the grouping of costs according to their common characteristics. The important ways of classification of costs are: In modern cost accounting, the concept of recording historical costs was taken further, by allocating the company's fixed costs over a given period of time to the items produced during that period, and recording the result as the total cost of production. This allowed the full cost of products that were not sold in the period they were produced to be recorded in inventory using a variety of complex accounting methods, which was consistent with the principles of GAAP (Generally Accepted Accounting Principles). It also essentially enabled managers to ignore the fixed costs, and look at the results of each period in relation to the "standard cost" for any given product. This method tended to slightly distort the resulting unit cost, but in mass-production industries that made one product line, and where the fixed costs were relatively low, the distortion was very minor. An important part of standard cost accounting is a variance analysis, which breaks down the variation between actual cost and standard costs into various components (volume variation, material cost variation, labor cost variation, etc.) so managers can understand why costs were different from what was planned and take appropriate action to correct the situation. As business became more complex and began producing a greater variety of products, the use of cost accounting to make decisions to maximize profitability came into question. Management circles became increasingly aware of the Theory of Constraints in the 1980s, and began to understand that "every production process has a limiting factor" somewhere in the chain of production. As business management learned to identify the constraints, they increasingly adopted throughput accounting to manage them and "maximize the throughput dollars" (or other currency) from each unit of constrained resource. Throughput accounting aims to make the best use of scarce resources(bottle neck) in a JIT environment. Activity-based costing (ABC) is a system for assigning costs to products based on the activities they require. In this case, activities are those regular actions performed inside a company. "Talking with customer regarding invoice questions" is an example of an activity inside most companies. Companies may be moved to adopt ABC by a need to improve costing accuracy, that is, understand better the true costs and profitability of individual products, services, or initiatives. ABC gets closer to true costs in these areas by turning many costs that standard cost accounting views as indirect costs essentially into direct costs. By contrast, standard cost accounting typically determines so-called indirect and overhead costs simply as a percentage of certain direct costs, which may or may not reflect actual resource usage for individual items. Under ABC, accountants assign 100% of each employee's time to the different activities performed inside a company (many will use surveys to have the workers themselves assign their time to the different activities). The accountant then can determine the total cost spent on each activity by summing up the percentage of each worker's salary spent on that activity. A company can use the resulting activity cost data to determine where to focus their operational improvements. For example, a job-based manufacturer may find that a high percentage of its workers are spending their time trying to figure out a hastily written customer order. Via ABC, the accountants now have a currency amount pegged to the activity of "Researching Customer Work Order Specifications". Senior management can now decide how much focus or money to budget for resolving this process deficiency. Activity-based management includes (but is not restricted to) the use of activity-based costing to manage a business. While ABC may be able to pinpoint the cost of each activity and resources into the ultimate product, the process could be tedious, costly and subject to errors. As it is a tool for a more accurate way of allocating fixed costs into product, these fixed costs do not vary according to each month's production volume. For example, an elimination of one product would not eliminate the overhead or even direct labor cost assigned to it. ABC better identifies product costing in the long run, but may not be too helpful in day-to-day decision-making. Recently, Mocciaro Li Destri, Picone & Minà (2012). proposed a performance and cost measurement system that integrates the Economic Value Added criteria with Process Based Costing (PBC). The EVA-PBC methodology allows us to implement the EVA management logic not only at the firm level, but also at lower levels of the organization. EVA-PBC methodology plays an interesting role in bringing strategy back into financial performance measures. Lean accounting has developed in recent years to provide the accounting, control, and measurement methods supporting lean manufacturing and other applications of lean thinking such as healthcare, construction, insurance, banking, education, government, and other industries. There are two main thrusts for Lean Accounting. The first is the application of lean methods to the company's accounting, control, and measurement processes. This is not different from applying lean methods to any other processes. The objective is to eliminate waste, free up capacity, speed up the process, eliminate errors & defects, and make the process clear and understandable. The second (and more important) thrust of Lean Accounting is to fundamentally change the accounting, control, and measurement processes so they motivate lean change & improvement, provide information that is suitable for control and decision-making, provide an understanding of customer value, correctly assess the financial impact of lean improvement, and are themselves simple, visual, and low-waste. Lean Accounting does not require the traditional management accounting methods like standard costing, activity-based costing, variance reporting, cost-plus pricing, complex transactional control systems, and untimely & confusing financial reports. These are replaced by: As an organization becomes more mature with lean thinking and methods, they recognize that the combined methods of lean accounting in fact creates a lean management system (LMS) designed to provide the planning, the operational and financial reporting, and the motivation for change required to prosper the company's on-going lean transformation. The cost-volume-profit analysis is the systematic examination of the relationship between selling prices, sales, production volumes, costs, expenses and profits. This analysis provides very useful information for decision-making in the management of a company. For example, the analysis can be used in establishing sales prices, in the product mix selection to sell, in the decision to choose marketing strategies, and in the analysis of the impact on profits by changes in costs. In the current environment of business, a business administration must act and take decisions in a fast and accurate manner. As a result, the importance of cost-volume-profit is still increasing as time passes. CONTRIBUTION MARGIN A relationship between the cost, volume and profit is the contribution margin. The contribution margin is the revenue excess from sales over variable costs. The concept of contribution margin is particularly useful in the planning of business because it gives an insight into the potential profits that can generate a business. The following chart shows the income statement of a company X, which has been prepared to show its contribution margin: CONTRIBUTION MARGIN RATIO The margin contribution can also be expressed as a percentage. The contribution margin ratio, which is sometimes called the profit-volume ratio, indicates the percentage of each sales dollar available to cover fixed costs and to provide operating revenue. For the company Fusion, Inc. the contribution margin ratio is 40%, which is computed as follows: The contribution margin ratio measures the effect on operating income of an increase or a decrease in sales volume. For example, assume that the management of Fusion, Inc. is studying the effect of adding $80,000 in sales orders. Multiplying the contribution margin ratio (40%) by the change in sales volume ($80,000) indicates that operating income will increase $32,000 if additional orders are obtained. To validate this analysis the table below shows the income statement of the company including additional orders: Variable costs as a percentage of sales are equal to 100% minus the contribution margin ratio. Thus, in the above income statement, the variable costs are 60% (100% - 40%) of sales, or$648,000 ($1'080,000 X 60%). The total contribution margin$432,000, can also be computed directly by multiplying the sales by the contribution margin ratio (\$1'080,000 X 40%).
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Mathematical finance is a field of applied mathematics, concerned with financial markets. Generally, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock (see: Valuation of options; Financial modeling). The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black–Scholes equation and formula are amongst the key results.

Mathematical finance also overlaps heavily with the field of computational finance (as well as financial engineering). The latter focuses on application, while the former focuses on modeling and derivation (see: Quantitative analyst), often by help of stochastic asset models. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk- and portfolio management on the other.

Personal finance refers to the financial management of which an individual or a family unit is required to make to obtain, budget, save, and spend monetary resources over time, taking into account various financial risks and future life events. When planning personal finances the individual would consider the suitability to his or her needs of a range of banking products (checking, savings accounts, credit cards and consumer loans) or investment (stock market, bonds, mutual funds) and insurance (life insurance, health insurance, disability insurance) products or participation and monitoring of individual- or employer-sponsored retirement plans, social security benefits, and income tax management.

The key component of personal finance is financial planning, which is a dynamic process that requires regular monitoring and reevaluation. In general, it involves five steps:

The term annual percentage rate of charge (APR), corresponding sometimes to a nominal APR and sometimes to an effective APR (or EAPR), describes the interest rate for a whole year (annualized), rather than just a monthly fee/rate, as applied on a loan, mortgage loan, credit card, etc. It is a finance charge expressed as an annual rate. Those terms have formal, legal definitions in some countries or legal jurisdictions, but in general:

In some areas, the annual percentage rate (APR) is the simplified counterpart to the effective interest rate that the borrower will pay on a loan. In many countries and jurisdictions, lenders (such as banks) are required to disclose the "cost" of borrowing in some standardized way as a form of consumer protection. APR is intended to make it easier to compare lenders and loan options.

Fee Pricing

An interest rate is the rate at which interest is paid by a borrower (debtor) for the use of money that they borrow from a lender (creditor). Specifically, the interest rate (I/m) is a percent of principal (P) paid a certain amount of times (m) per period (usually quoted per annum). For example, a small company borrows capital from a bank to buy new assets for its business, and in return the lender receives interest at a predetermined interest rate for deferring the use of funds and instead lending it to the borrower. Interest rates are normally expressed as a percentage of the principal for a period of one year.

Interest-rate targets are a vital tool of monetary policy and are taken into account when dealing with variables like investment, inflation, and unemployment. The central banks of countries generally tend to reduce interest rates when they wish to increase investment and consumption in the country's economy. However, a low interest rate as a macro-economic policy can be risky and may lead to the creation of an economic bubble, in which large amounts of investments are poured into the real-estate market and stock market. This happened in Japan in the late 1980s and early 1990s, resulting in the large unpaid debts to the Japanese banks and the bankruptcy of these banks and causing stagflation in the Japanese economy (Japan being the world's second largest economy at the time), with exports becoming the last pillar for the growth of the Japanese economy throughout the rest of 1990s and early 2000s. The same scenario resulted from the United States' lowering of interest rate since late 1990s to the present (see 2007–2012 global financial crisis) substantially by the decision of the Federal Reserve System. Under Margaret Thatcher, the United Kingdom's economy maintained stable growth by not allowing the Bank of England to reduce interest rates. In developed economies, interest-rate adjustments are thus made to keep inflation within a target range for the health of economic activities or cap the interest rate concurrently with economic growth to safeguard economic momentum.