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Call Options on Bonds

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images Call Options on Bonds
Many bonds include a call feature that allows the issuer to redeem or “call” all or part of the issue before the maturity date.

The call option has value to the issuer for several reasons. In the future the borrower may wish to remove restrictions placed on him by the bond indenture. For a corporation these might be restrictions on merger, the sale of assets, or the payment of dividends. Without the call provision, the bondholders by hard bargaining might be able to utilize their monopoly position to extract a large premium from the issuer before selling back the bonds or agreeing to change these clauses.

A second source of value is that the borrower may find that he wants to decrease the amount of his borrowing before the bonds mature. Essentially the same effect as retiring his own bonds could be obtained by buying on the market similar bonds issued by someone else. However, because of the transaction costs involved in making interest payments, the cost of bonds to the issuer will always be somewhat greater than their value on the market. There will therefore be some saving to the issuer in retiring his own bonds rather than buying someone else’s.

The third and probably most significant source of value of the option to the issuer is the ability it gives him to refinance the issuer in the future if interest rates should fall. This implies three risks from the investor:
(a) The cash flow pattern becomes uncertain;
(b) The investor becomes exposed to reinvestment risk because the issuer will call the bond when interest rates drop;
(c) The capital appreciation potential of a bond will be reduced, because the price of a callable bond may not rise much above the price at which the issuer will call the bond.

Option writing for bonds is used both for risk and hedge purposes. Bond call options provide liquidity with little cash requirement. Bond options also have a finite life and can be expensive to purchase and hold until maturity.

Mechanics of Bond Option Writing

The seller of the contract must have on account securities or funds equal to the amount of the potential sale until the contract expires. The seller is also called the writer of the contract. The incentive for writing a bond call option is the premium payment reflecting the risk of loss of the bonds. The premium payment is in addition to the strike price. The strike price is the market value the bond call option must reach in order to be exercised, or called.

Trading on an Exchange

For a stated par, or maturity value, the bond call writer initiates a contract through his broker on a nationally recognized exchange. The exchange provides a uniform bond option contract detailing the responsibilities of both the bond call option buyer and seller, including the length of the call and what bonds constitute good value for delivery. Rarely are bonds ever delivered. Instead contracts are offset by cash payments at maturity. The contract also denotes an exercise price. The fee received, the premium, is determined by market conditions. This includes the current level of interest rates, the price volatility of the bonds and the number of business days the bond option seller is at risk.

Leverage

Rather than incur the price volatility and cash payment for bonds, traders often use bond call options. Traders know from the beginning of the trade the absolute amount of loss they may incur while freeing up cash for other trading purposes. The cost of a bond option varies depending on the premium charged, but a one month bond call option may cost as little as 2 percent of the face value of the bonds. This implies that a 1 percent increase in bond values on a $1 million call would increase bond option values by nearly $10,000.

Hedging

Bond options are popular as hedging vehicles to reduce risk. Traders can buy options and use them to hedge, or offset, an equal amount of futures contracts, bonds and other options, such as bond put options. Bond call options are also used like stock options in a variety of mathematical strategies. By buying and selling different bond call and bond put options, income can be derived from the time to maturity, the difference in coupon or even bond prices. Bond call options, with their liquidity, and known maximum loss, provide traders many opportunities to employ profitable, leveraged, strategic portfolio outcomes.

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How to Measure Interest Rate Risk

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Financial Risk Management How to Measure Interest Rate Risk

Bond prices change inversely with interest rates, and, hence, there is interest rate risk with bonds. One method of measuring interest rate risk is by the full valuation approach, which simply calculates what bond prices will be if the interest rate rose by specific amounts. The full valuation approach is based on the fact that the price of a bond is equal to sum of the present value of each coupon payment plus the present value of the principal payment.

Bond Value = Present Value of Coupon Payments + Present Value of Par Value

Another method, which is less computationally intensive, is by calculating the duration of a bond, which is the weighted average of the present value of the bond’s payments, and can be viewed as the average, or effective, maturity of a bond. Graphically, the duration of a bond can be envisioned as a seesaw where the fulcrum is placed so as to balance the weights of the present values of the payments and the principal payment. The longer the duration, the longer is the average maturity, and, therefore, the greater the sensitivity to interest rate changes.

Although the effective duration is measured in years, it is more useful to interpret duration as a means of comparing the interest rate risks of different securities. Securities with the same duration have the same interest rate risk exposure. For instance, since zero-coupon bonds only pay the face value at maturity, the duration of a zero is equal to its maturity. It also follows that any bond of a certain duration will have an interest rate sensitivity equal to a zero-coupon bond with a maturity equal to the bond’s duration.

Duration is also often interpreted as the percentage change in a bond’s price for a 1% change in its yield to maturity (YTM). So, for instance, the price of a bond with a 10-year duration would change by 10%.

Duration can be estimated by the following equation:

Duration Approximation Formula

duration estimation How to Measure Interest Rate Risk

P0 = bond price
P- = bond price when interest rate is incremented
P+ = bond price when interest rate is decremented
∆y = change in interest rate in decimal form

The interest rate is shocked up and down by a specific amount to obtain the new bond prices. Note that even if the interest rates are shocked by an amount different from 1%, duration is still interpreted as the percentage change in bond price for a 1% change in the YTM.

Macaulay Duration

It was Frederick Macaulay who developed the concept of duration, equating it to the average time to maturity or the time required to receive half of the present value, discounted by the bond’s yield to maturity, of the bond’s cash flow. The Macaulay duration is calculated by 1st calculating the weighted average of each cash flow at time t by the following formula:

weighted average cash flow How to Measure Interest Rate Risk

wt = weighted average of cash flow at time t
CFt = cash flow at time t
y = yield to maturity

Then these weighted averages are summed:

Macaulay Duration Formula

macaulay duration formula How to Measure Interest Rate Risk

T = number of cash flow periods

Hence, the Macaulay duration measures the effective maturity of a bond, and can also be used to calculate the average maturity of a portfolio of fixed-income securities.

Modified Duration

Modified duration is a modification of the Macaulay duration to estimate interest rate risk, calculating the change in a bond’s price to a change in its yield to maturity by the following formula:

Modified Duration Formula

modified duration formula How to Measure Interest Rate Risk

Dm = Modified Duration
DMac = Macaulay Duration
y = yield to maturity
k = number of payments per year

The modified duration formula is valid only when the change in yield will not alter the cash flow of the bond, such as may occur, for instance, if the price change for a callable bond increases the likelihood that it will be called.

It is also only valid for small changes in yield, because duration itself changes as the yield changes. It is a 1st derivative of the price-yield curve, which is a line tangent to the curve at the current price-yield point.

Duration and Modified Duration Formulas for Bonds Using Microsoft Excel
Duration = DURATION(settlement,maturity,coupon,yield,frequency,basis)

Modified Duration = MDURATION(settlement,maturity,coupon,yield,frequency,basis)

  • Settlement = Date in quotes of settlement.
  • Maturity = Date in quotes when bond matures.
  • Coupon = Nominal annual coupon interest rate.
  • Yield = Annual yield to maturity.
  • Frequency = Number of coupon payments per year.
    • 1 = Annual
    • 2 = Semiannual
    • 4 = Quarterly
  • Basis = Day count basis.
    • 0 = 30/360 (U.S. NASD basis). This is the default if the basis is omitted.
    • 1 = actual/actual (actual number of days in month/year).
    • 2 = actual/360
    • 3 = actual/365
    • 4 = European 30/360
1. Example—Calculating Modified Duration using Microsoft Excel

Calculate the duration and modified duration of a 10-year bond paying a coupon rate of 6%, a yield to maturity of 8%, and with a settlement date of 1/1/2008 and maturity date of 12/31/2017.

Duration = DURATION(“1/1/2008″,”12/31/2017″,0.06,0.08,2) = 7.45

Modified duration = MDURATION (“1/1/2008″,”12/31/2017″,0.06,0.08,2) = 7.16

Modified duration is always slightly less than duration, since the modified duration is the duration divided by 1 plus the yield per payment period.

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Convexity

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Convexity means a measure of the curvature in the relationship between bond prices and bond yields that demonstrates how the duration of a bond changes as the interest rate changes. Convexity is used as a risk-management tool, and helps to measure and manage the amount of market risk to which a portfolio of bonds is exposed.

convexity22 ConvexityIn the example above, Bond A has a higher convexity than Bond B, which means that all else being equal, Bond A will always have a higher price than Bond B as interest rates rise or fall.

As convexity increases, the systemic risk to which the portfolio is exposed increases. As convexity decreases, the exposure to market interest rates decreases and the bond portfolio can be considered hedged. In general, the higher the coupon rate, the lower the convexity (or market risk) of a bond. This is because market rates would have to increase greatly to surpass the coupon on the bond, meaning there is less risk to the investor.

Convexity is the rate that the duration changes along the price-yield curve, and, thus, is the 1st derivative to the equation for the duration and the 2nd derivative to the equation for the price-yield function, and is calculated by the following equation:

Convexity Formula

convexity formula.png1 Convexity

P = bond price

y = yield to maturity in decimal form

T = maturity in years

CFt= cash flow at time t

The equation for duration can be improved by adding the convexity term:

Calculating the Change in Bond Prices with Interest Rates Using Duration + Convexity Adjustment

duration convexity formula Convexity

∆y = yield change

∆P = bond price change

Convexity can also be estimated with a simpler formula, similar to the approximation formula for duration:

Convexity Approximation Formula

Convexity = P+ + P- – 2P0
─────────────
2 x P0(Δy)2

P0 = bond price
P- = bond price when interest rate is incremented
P+ = bond price when interest rate is decremented
∆y = change in interest rate in decimal form

However, that this convexity approximation formula must be used with this convexity adjustment formula, then added to the duration adjustment:

Convexity Adjustment Formula

Convexity Adjustment = Convexity x 100 x (Δy)2

∆y = change in interest rate in decimal form

Hence:

Bond Price Change Formula

Bond Price Change = Duration x Yield Change + Convexity Adjustment

Convexity is usually a positive term regardless of whether the yield is rising or falling, hence, it is positive convexity. However, sometimes the convexity term is negative, such as occurs when a callable bond is nearing its call price. Below the call price, the price-yield curve follows the same positive convexity as an option-free bond, but as the yield falls and the bond price rises to near the call price, the positive convexity becomes negative convexity, where the bond price is limited at the top by the call price. Hence, similar to the terms for modified and effective duration, there is also modified convexity , which is the measured convexity when there is no expected change in future cash flows, and effective convexity , which is the convexity measure for a bond for which future cash flows are expected to change.

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CMO’s: Plain Vanilla Bonds

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3d wallpaper collection 300x225 CMOs: Plain Vanilla BondsA plain vanilla bond is a bond with no unusual features, paying a fixed rate of interest and redeemable in full on maturity. The term derives from vanilla or ‘plain’ flavoured ice-cream.

Out of all high yield bond securities, plain vanilla securities are the most widely used. They are exactly as they sound; they pay a cash interest payments at a fixed rate. These securities typically have a term to maturity of 7 to 12 years and have a callable option built in within the first 3 to 5 years to allow for a company to pay off expensive debt as their revenues and credit ratings improve. The corporation can then issue more lower yield debt as a substitute due to their improved position. Plain vanilla bonds have prepayment and extension risk.

Plain-vanilla are collateralized mortgage obligations.

Collateralized Mortgage Obligations or CMO’s are a series of bonds backed by an agency and their mortgage backed securities. These investments are AAA rated and pay monthly principal and interest.

Various Types of CMOs

The most basic CMO structure has tranches that pay in a strict sequence. Each tranche receives regular interest payments, but the principal payments received are made to the first tranche alone, until it is completely retired. Once the first tranche is retired, principal payments are applied to the second tranche until it is fully retired, and the process continues until the last tranche is retired. The first tranche of the offering may have an average life of 2-3 years, the second tranche 5-7 years, the third tranche 10-12 years, and so forth. This type of CMO is known as a “sequential pay,” “clean,” or “plain vanilla” offering.

The CMO structure allows the issuer to meet different maturity requirements and to distribute the impact of prepayment variability among tranches in a deliberate and sometimes uneven manner. This flexibility has led to increasingly varied and complex CMO structures. CMOs may have 50 or more tranches, each with unique characteristics that may be interdependent with other tranches in the offering. The types of CMO tranches include:

Planned Amortization Class (PAC) Tranches

PAC tranches use a mechanism similar to a “sinking fund” to establish a fixed principal payment schedule that directs cash-flow irregularities caused by faster- or slower-than-expected prepayments away from the PAC tranche and toward another “companion” or “support” tranche. With a PAC tranche, the yield, average life, and lockout periods estimated at the time of investment are more likely to remain stable over the life of the security.

PAC payment schedules are protected by priorities which assure that PAC payments are met first out of principal payments from the underlying mortgage loans. Principal payments in excess of the scheduled payments are diverted to non-PAC tranches in the CMO structure called companion or support tranches because they support the PAC schedules. In other words, at least two bond tranches are active at the same time, a PAC and a companion tranche. When prepayments are minimal, the PAC payments are met first and the companion may have to wait. When prepayments are heavy, the PAC pays only the scheduled amount, and the companion class absorbs the excess.

“Type I PAC” tranches maintain their schedules over the widest range of actual prepayment speeds—say, from 100% to 300% PSA. “Type II” and “Type III PAC” tranches can also be created with lower priority for principal payments from the underlying loans than the primary or Type I tranches. They function as support tranches to higher-priority PAC tranches and maintain their schedules under increasingly narrower ranges of prepayments.

PAC tranches are now the most common type of CMO tranche. Because they offer a high degree of investor cash-flow certainty, PAC tranches are usually offered at lower yields.

Targeted Amortization Class (TAC) Tranches

TAC tranches also provide more cash-flow certainty and a fixed principal payment schedule, based on a mechanism similar to a sinking fund, but this certainty applies at only one prepayment rate rather than a range. If prepayments are higher or lower than the defined rate, TAC bondholders may receive more or less principal than the scheduled payment. TAC tranches’ actual performance depends on their priority in the CMO structure and whether or not PAC tranches are also present. If PACs are also present, the TAC tranche will have less cash-flow certainty. If no PACs are present, the TAC provides the investor with some protection against accelerated prepayment speeds and early return of principal. The yields on TAC bonds are typically higher than yields on PAC tranches but lower than yields on companion tranches.

Companion Tranches

Every CMO that has PAC or TAC tranches in it will also have companion tranches (sometimes called support bonds), which absorb the prepayment variability that is removed from the PAC and TAC tranches. Once the principal is paid to the active PAC and TAC tranches according to the schedule, the remaining excess or shortfall is reflected in payments to the active companion tranche. The average life of a companion tranche may vary widely, increasing when interest rates rise and decreasing when rates fall. To compensate for this variability, companion tranches offer the potential for higher expected yields when prepayments remain close to the rate assumed at purchase. Similar to Type II and Type III PACs, TAC tranches can serve as companion tranches for PAC tranches. These lower-priority PAC and TAC tranches will in turn have companion tranches further down in the principal payment priority. Companion tranches are often offered for sale to retail investors who want higher income and are willing to take more risk of having their principal returned sooner or later than expected.

Z-Tranches (also known as Accretion Bonds or Accrual Bonds)

Z-tranches are structured so that they pay no interest until the lockout period ends and they begin to pay principal. Instead, a Z-tranche is credited “accrued interest” and the face amount of the bond is increased at the stated coupon rate on each payment date. During the accrual period the principal amount outstanding increases at a compounded rate and the investor does not face the risk of reinvesting at lower rates if market yields decline. Typical Z-tranches are structured as the last tranche in a series of sequential or PAC and companion tranches and have average lives of 18 to 22 years. However, Z-tranches can be structured with intermediate-term average lives as well. After the earlier bonds in the series have been retired, the Z-tranche holders start receiving cash payments that include both principal and interest.

While the presence of a Z-tranche can stabilize the cash flow in other tranches, the market value of Z-tranches can fluctuate widely, and their average lives depend on other aspects of the offering. Because the interest on these securities is taxable when it is credited, even though the investor receives no interest payment, Z-tranches are often suggested as investments for tax-deferred retirement accounts.

Principal-Only (PO) Securities

Some mortgage securities are created so that investors receive only principal payments generated by the underlying collateral. These Principal-Only (PO) securities may be created directly from mortgage pass-through securities, or they may be tranches in a CMO. In purchasing a PO security, investors pay a price deeply discounted from the face value and ultimately receive the entire face value through scheduled payments and prepayments.

The market values of POs are extremely sensitive to prepayment rates and therefore interest rates. If interest rates are falling and prepayments accelerate, the value of the PO will increase. On the other hand, if rates rise and prepayments slow, the value of the PO will drop. A companion tranche structured as a PO is called a “Super PO.”

Interest-Only (IO) Securities

Separating principal payments to create PO mortgage securities necessarily involves the creation of Interest-Only (IO) securities. CMOs that have PO tranches will therefore also have IO tranches. IO securities are sold at a deep discount to their “notional” principal amount, namely the principal balance used to calculate the amount of interest due. They have no face or par value. As the notional principal amortizes and prepays, the IO cash flow declines.

Unlike POs, IOs increase in value when interest rates rise and prepayment rates slow; consequently, they are often used to “hedge” portfolios against interest rate risk. IO investors should be mindful that if prepayment rates are high, they may actually receive less cash back than they initially invested.

The structure of IOs and POs exaggerates the effect of prepayments on cash flows and market value. The heightened risk associated with these securities makes them unsuitable for certain investors.

Floating-Rate Tranches

First offered in 1986, “floating-rate CMO” tranches carry interest rates that are tied in a fixed relationship to an interest rate index, such as the London Interbank Offered Rate (LIBOR), the Constant Maturity Treasury (CMT), or the Cost of Funds Index (COFI), subject to an upper limit, or “cap,” and sometimes to a lower limit, or “floor.” The performance of these investments also depends on the way interest rate movements affect prepayment rates and average lives.

Sometimes the interest rates on these tranches are stated in terms of a formula based on the designated index, meaning they move up or down by more than one “basis point” (1/100 of one percent) for each basis point increase or decrease in the index. These so-called “superfloaters” offer leverage when rates rise. The interest rates on “inverse floaters” move in a direction opposite to the changes in the designated index and offer leverage to investors who believe rates may move down. The potential for high coupon income in a rally can be rapidly eroded when prepayments speed up in response to falling interest rates. All types of floating-rate tranches may be structured as PAC, TAC, companion, or sequential tranches, and are often used to hedge interest rate risks in portfolios.

Residuals

CMOs also contain a “residual” interest tranche, which collects any cash flow remaining from the collateral after the obligations to the other tranches have been met. Residuals are not classified as regular interest and may be structured as sequential, PAC, floating-rate, or inverse-floater tranches, and differ from regular tranches primarily in their tax characteristics, which can be more complex than other CMO tranches. CMOs issued as non-REMICs also have residuals which are sold as a separate security such as a trust certificate or a partnership interest.

Collateralized Mortgage Obligations are generally meant for institutional investors or wealthy bond investors. The money invested, while earning monthly income – can take a while if interest rates rise. When interest rates rise, these bonds will pay slower. The refinancing that normally can happen with mortgage pools will slow down or stop when interest rates or bond yields rise.

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