on 02-Jul-2014 (Wed)

Coupon Paying bonds are essentially portfolios of Zero Coupon Bond components and the Yield to Maturity of such instruments can be thought of as being a complex blend of the component Zero Coupon bond yields. It therefore can be observed that, for example, in a positively sloped Yield Curve environment and comparing two bonds with the same cash flow dates and maturity, a higher coupon bond will offer a lower YTM than a low coupon bond - assuming the same PV calculated from the same discount curve. (The low coupon bond has cash flows more heavily influenced by a proportionally greater long term component).

Yield to maturity is simply the discount rate at which the sum of all future cash flows from the bond (coupons and principal) is equal to the price of the bond.

• If a bond's coupon rate is less than its YTM, then the bond is selling at a discount.
• If a bond's coupon rate is more than its YTM, then the bond is selling at a premium.
• If a bond's coupon rate is equal to its YTM, then the bond is selling at par.

As some bonds have different characteristics, there are some variants of YTM:

• Yield to call: when a bond is callable (can be repurchased by the issuer before the maturity), the market looks also to the Yield to call, which is the same calculation of the YTM, but assumes that the bond will be called, so the cashflow is shortened.

• Yield to put: same as yield to call, but when the bond holder has the option to sell the bond back to the issuer at a fixed price on specified date.

• Yield to worst: when a bond is callable, puttable, exchangeable, or has other features, the yield to worst is the lowest yield of yield to maturity, yield to call, yield to put, and others.

Yield to Maturity is an estimation of future return, as the rate at which coupon payments can be reinvested at is unknown.

for zero coupon bonds, the yield to maturity and the rate of return are equivalent since there are no coupon payments to reinvest.
it is dubious. YTM is not about reinvesting coupons

yield to maturity is the rate of return that makes the present value (PV) of the cash flow generated by the bond equal to the price.

Face Value, also known as the "par value", is the amount a bond holder will be paid when it matures

The current yield is simply the coupon payment as a percentage of the (current) bond price .

The coupon yield is simply the coupon payment as a percentage of the face value .

TED is an acronym formed from T-Bill and ED, the ticker symbol for the Eurodollar futures contract.

The TED spread is the difference between the interest rates on interbank loans and on short-term U.S. government debt ("T-bills")

For example, if the T-bill rate is 5.10% and ED trades at 5.50%, the TED spread is 40 bps.

The TED spread is an indicator of perceived credit risk in the general economy,^[2] since T-bills are considered risk-free while LIBOR reflects the credit risk of lending to commercial banks

The long term average of the TED has been 30 basis points with a maximum of 50 bps. During 2007, the subprime mortgage crisis ballooned the TED spread to a region of 150–200 bps. On September 17, 2008, the TED spread exceeded 300 bps, breaking the previous record set after the Black Monday crash of 1987.^[4] Some higher readings for the spread were due to inability to obtain accurate LIBOR rates in the absence of a liquid unsecured lending market.^[5] On October 10, 2008, the TED spread reached another new high of 457 basis points. In October 2013, due to worries regarding a potential default on US debt, the 1 month TED went negative for the first time since it started being tracked

A rising TED spread often presages a downturn in the U.S. stock market, as it indicates that liquidity is being withdrawn.

The Interpolated Spread or I-spread or ISPRD is the difference between the yield to maturity of the bond and the linearly interpolated yield to the same maturity on an appropriate reference curve

The Z-spread of a bond is the number of basis points one needs to add to the Treasury spot rates yield curve, so that the NPV of the bond cash flows (using the adjusted yield curve) equals the market price of the bond (after accounting for accrued interest).

Conventionally, the zero rates for calculating Z-spread are determined from the Treasury curve, with semi-annual compounding.

If you calculate the present value of all future cash flows for a bond using prevailing spot rates, you may discover that the price you calculate is greater than the price observed in the market. This difference arises because the market price incorporates additional factors such as liquidity and credit risk. The Z-spread quantifies the impact of these additional factors. It is the spread you need to add to the curve you are discounting with in order to generate a price that matches the market price.

Given a single series of nominal cash flows (like a riskless bond). If these payments are discounted to net present value with a static treasury yield curve the sum of their values will tend to overestimate the market price of the priced instrument. The parallel shift, which, if applied to the yield curve makes the NPV of the anticipated receipts equal to the market price is the Yield curve spread (aka Z-spread).

only the anticipation of falling interest rates will cause an inverted yield curve.

Main article: expectation 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. 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 is the risk premium associated with an 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 guide 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.

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.

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.

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).

at the short end of the curve, where there are few cashflows, the first few elements of P (rate function) may be found by bootstrapping from one to the next.

TED spread is now calculated as the difference between the three-month LIBOR and the three-month T-bill interest rate.

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 is as small a vector as possible

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.

When the yield curve is steep, the bond is predicted to have a large capital gain in the first years (dubious, innit?) before falling in price later.

hen 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.

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 sensitivity of a bond's market price to interest rate (i.e. yield) movements is measured by its duration, and, additionally, by its convexity.

The concept of current yield is closely related to other bond concepts, including yield to maturity, and coupon yield. The relationship between yield to maturity and the coupon rate is as follows:

• When a bond sells at a discount, YTM > current yield > coupon yield.
• When a bond sells at a premium, coupon yield > current yield > YTM.
• When a bond sells at par, YTM = current yield = coupon yield

Bond's duration is approximately equal to the percentage change in price for a given change in yield, and may be thought of as the elasticity of the bond's price with respect to discount rates - it is not elasticity, because yield change is absolute, not relative

market price of a 17-year bond with a duration of 7 would fall about 7% if the market interest rate increased by 1% per annum.

Specifically, duration can be formulated as the first derivative of the price with respect to the interest rate, and convexity as the second derivative

If I lower the price of my product, how much more will I sell?

As the Z-Spread is not dependent upon only one point on the Yield Curve and takes account of all of the relevant term-structure, the distortions of Yield-to-Maturity spreads outlined above are eliminated. It is then a measure of Credit Spread without the distortions of YTM.

Macaulay duration, is the weighted average maturity of cash flows. Consider some set of fixed cash flows. The present value of these cash flows is:

Macaulay duration is defined as:^{[1][2][3] [4]}

(1)

where:

• indexes the cash flows,
• is the present value of the th cash payment from an asset,
• is the time in years until the th payment will be received,
• is the present value of all future cash payments from the asset.

The Eurasian eagle-owl (Bubo bubo) is a species of eagle owl