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73 Cards in this Set
- Front
- Back
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What is a discount factor?
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The discount factor for a particular term gives the present value of one unit of currency to be received at the end of that term.
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What is an arbitrage opportunity?
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An arbitrage opportunity is a trade that generates or that might generate profits without any risk
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How is an arbitrage trade made?
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An arbitrage trade is in which one buys the cheap asset, while simultaneously shorting or selling its replicating portfolio
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What is another name for Zero Coupon Bond and how are they created?
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Zero coupon bonds issued by the U.S. Treasury are called STRIPS (separate trading of registered interest and principal securities). STRIPS are created when someone delivers a particular coupon bond to the Treasury and asks for it to be “stripped” into its principal and coupon components.
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What 2 reasons do investors like zero coupon bonds?
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1. They make it easy to construct any required sequence of cash flows.2. Per dollar invested they have much greater sensitivity to interest rates than coupon bonds (particularly long-term zeros)
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What is the formula for the return on investing x at an annual rate of r compounded semiannually for T years?
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What is the semiannually compounded return from investing x for T years and having w at the end?
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The rate r may be thought of as the semiannually compounded return from investing in a zero coupon bond that matures T years from now.
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What is the spot rate?
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The spot rate is the rate on a spot loan, a loan agreement in which the lender gives money to the borrower at the time of the agreement.
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What is the formula for the value of one unit of currency discounted for t years at the semiannually compounded rate r(t)?
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What is the definition of a forward rate?
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The rate of interest on a forward loan, specified at the time of the agreement as opposed to the time of the loan, is called a forward rate
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What is the relationship between the forward curve and spot curve?
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When the forward rate curve is above the spot rate curve, the spot rate curve is rising or sloping upward.When the forward rate curve is below the spot rate curve, the spot rate curve slopes downward or is falling.tThe term structure of spot rates slopes upward when forward rates are above spot rates.Similarly, the term structure of spot rates slopes downward when forward rates are below spot rates.
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What is the relationship between price, coupon rate and forward rates?
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Price increases with maturity whenever the coupon rate exceeds the forward rate over the period of maturity extension. Price decreases as maturity increases whenever the coupon rate is less than the relevant forward rate.
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When do investors who roll over short-term investments do better than investors in longer-term bonds? When is the opposite true?
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Short term investors do better When the realized short-term rates exceed the forward rates built into bond prices. Investors in bonds do better when the realized short-term rates fall below these forward rates.
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What is the definition of yield-to-maturity?
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Yield-to-maturity, or yield, the single rate that when used to discount a bond’s cash flows produces the bond’s market price.If a bond’s yield-to-maturity remains unchanged over a short time period, that bond’s realized total rate of return equals its yield.
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What does the sum below equal?
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proof: dividing both sides by 1-z gives the equation.
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What happens to the bond price when the coupon rate equals the yield-to-maturity?
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When the coupon rate equals the yield-to-maturity, bond price equals face value, or par. Intuitively, if it is appropriate to discount all of a bond’s cash flows at the rate y, then a bond paying a coupon rate of c is paying the market rate of interest. Investors will not demand to receive more than their initial investment at maturity nor will they accept less than their initial investment at maturity. Hence, the bond will sell for its face value.
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What happens to the bond price when the coupon rate is greater than the yield?
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If the coupon rate exceeds the yield, then the bond sells at a premium to par, that is, for more than face value.
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What happens to the bond price when the coupon rate is less than the yield?
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If the coupon rate is less than the yield, then the bond sells at a discount to par, that is, for less than face value.
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What expression is used for premium bond prices falling and discount bonds increasing as they mature?
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Pull to par
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What is an annuity?
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An annuity is a security that makes a payments for T years but never makes a final “principal” payment.
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What is the relationship between downward sloping, flat and upward sloping spot rates and yields on coupon par bonds?
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spot rates downward sloping : bond yields above spot ratesspot rates flat : bond yields equal spot ratesspot rates upward sloping : bond yields below spot rates
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What is the definition of reinvestment risk?
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Reinvestment risk is the uncertainty of the realized yield relative to the original yield because coupons are invested at uncertain future rates
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What is the definition of Flat Price and Invoice Price?
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Flat Price: Quoted (clean) prices.Invoice Price: When the accrued interest of a bond is non-zero or when the settlement date is not a coupon payment date—the flat and invoice prices of the bond not are equal
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What is the definition of the money market?
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the market to borrow and lend for usually one year or less
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What is a common way to interpolate yield curve data?
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Use a piecewise cubic polynomial, that is, a single function built by joining several cubic polynomials together at some set of knot points.
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What is the relationship between the forward curve and the spot curve?
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The forward rate exceeds the spot rate if and only if spot rates are increasing. Also, the forward rate is less than the spot rate if and only if spot rates are decreasing
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What is the definition of DVO1?
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Dollar value of an ’01 (i.e., .01%) and gives the change in the value of a fixed income security for a one-basis point decline in rates
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What is the definition of duration?
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duration measures the percentage change in the value of a security for a unit change in rates
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Why is using DV01 advantageous of over using duration in an options investment context?
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Duration of options vastly exceeds that of the bond alerts the investor to the far greater risk of investing money in options.
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Why is using DV01 advantageous of over using duration in an options hedging context?
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The DV01 of an option is much less than the DV01 of a bond thus a hedged position must be long much less face amount of bonds than it is short face amount of options.
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What is Convexity?
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Convexity measures how interest rate sensitivity changes with rates.
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What can be said about DV01 when convexity is positive?
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Positive convexity means that DV01 falls as rates increase
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What is a short convexity position?
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This happens when in a hedging context a long option has a P&L that is greater than the long bond position it is hedging.The hedged position loses whether rates rise or fall because the option is more convex than the bond.As a result the trader will need to rehedge the position as rates move. The portfolio will lose if rates move a lot.In this sense the portfolio is also short volatility.
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What is a long convexity position?
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The portfolio will gain if rates move a lot. In this sense the portfolio is also long volatility.
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What is the duration-convexity equation?
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What is the formula for the duration of a portfolio?
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The duration/convexity of a portfolio equals a weighted sum of individual durations/convexities where each security’s weight is its value as a percentage of portfolio value.
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What is the value of a callable bond to an investor?
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The value of a callable bond to an investor equals the value of the underlying non callable bond minus the value of the issuer’s embedded option. For this reason the non callable bond outperforms the callable bond.
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What are the weaknesses of parallel shifts in risk analysis?
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1. They are only valid for securities with fixed cash flows2. will fail against flattening, steepening, or some other twist in the yield curve
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How can DV01 be interpreted?
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DV01 is the sum of the time weighted present values of a bond’s cash flows divided by 10,000 (times 1/[1+y/2] )
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How can Modified Duration be interpreted?
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Modified Duration is the sum of the time weighted present values of a bond’s cash flows divided by price (times 1/[1+y/2] )
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Macaulay duration be interpreted?
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Macaulay duration is the sum of the time weighted present values of a bond’s cash flows divided by price. A convenient property of Macaulay duration is that the Macaulay duration of a T-year zero coupon bond equals T which can be used as a benchmark to judge the sensitivity of other bonds.
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What is the major difference between duration and DV01?
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DV01 measures an absolute change in price while duration measures a percentage change.
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What is the relationship between Duration and Maturity? Duration and Coupon?
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The duration always increases with maturity. For any given maturity, duration falls as coupon increases. Intuitively, higher coupon bonds have a greater fraction or their value paid earlier and thus larger weight on the duration terms of early years relative to later years. Higher coupon bonds are effectively shorter term bonds and therefore have lower durations.
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What is the relationship between DV01 and Maturity?
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DV01 is proportional to Duration and Price. Duration increases with maturity so DV01 increases with maturity (duration effect) except in those cases where prices decrease with increasing maturity (price effect).
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What is the effect of maturity on price of 1) par bonds, 2) premium bonds, 3) discount bonds and 4) zero coupon bonds?
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1) Since the price of par bonds is always 100, the price effect does not come into play2) Extending the maturity of a premium bond increases its price3) Extending the maturity of a discount bond lowers its price.4) Extending the maturity of a discount bond lowers its price to zero eventually
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What is the relationship between DV01 and Coupon?
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DV01 rises with coupon.
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What is the relationship between DV01 and yield?
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DV01 decreases with yield. This is because bonds all non-callable bonds have positive convexity meaning DV01 decreases with yield
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What is meant by positive convexity?
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Price decreases as yield increases but at a slower rate than price increase as rates fallPositive convexity is not a correct mathematical term. positive convexity is just convexity, and negative convexity is just concavity.
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What are 4 properties of positive convexity bonds?
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1. non callable bonds2. perform better when yields change a lot3. positive convexity is paid for at times that yields do not change very much.4. convexity arises and increases with volatility.
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What is the relationship between convexity and maturity of a zero?
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Convexity increases with the square of maturity for zero coupon bonds
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What is the relationship between convexity and maturity of a coupon bond?
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Since a coupon bond is a portfolio of zeros, longer-maturity coupon bonds usually have greater convexity than shorter-maturity coupon bonds.
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What is bucket shift risk analysis, key rate analysis?
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A bucket is a region of some curve. Each bucket shift is a parallel shift of forward rates. Key rates assume the particular local perturbations
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How is a security priced using no arbitrage?
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A security is priced by no-arbitrage by finding and pricing its replicating portfolio.
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How is a security priced using risk neutral pricing?
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1. Find the risk neutral probabilities that equate the prices of the underlying securities with their expected discounted values.2. Price by expected discounted value under these risk-neutral probabilities.
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In arbitrage pricing, what is dynamic replication?
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The replicating portfolio must be adjusted as time passes and as interest rates change.
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What is the advantage of risk neutral pricing to arbitrage pricing?
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Risk-neutral pricing modifies an assumed interest rate process so that any contingent claim can be priced without having to construct and price its replicating portfolio.
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Summarize Black-Scholes pricing?
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Under the assumption that the stock price evolves according to a particular random process and that the short-term interest rate is constant, it is possible to form a portfolio of stocks and short-term bonds that replicates the payoffs of an option.Therefore, by arbitrage arguments, the price of the option must equal the known price of the replicating portfolio.
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Why does Black Scholes not apply to bond pricing?
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1. Bond prices must converge to its face value at maturity while the random process describing the stock price need not be constrained in any similar way.2. The volatility of a bond’s price must eventually get smaller as the bond approaches maturity. The assumption that the stock volatility is constant is not appropriate for bonds.3. It may be relatively harmless to assume that the short term rate is constant. However, simultaneously assuming that the bond price follows some random process and that the short-term interest rate is constant makes little economic sense.
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What is the solution to the inconsistencies of the BS model for bonds?
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Model the random evolution of the entire term structure of interest rates rather than of the bond price.
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What is Jensen's Inequality?
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What is the relationship between bond returns and the economy?
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Bonds with interest rate risk earn a risk premium. This is equivalent to bond returns being positively correlated with the economy or, equivalently, that falling interest rates are associated with good times.Also, interest rates fall when inflation and expected inflation fall and that low inflation is correlated with good times.
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What effect does adding mean reversion to a model have on the term structure of volatility?
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In a model with no mean reversion, rates are determined exclusively by current economic conditions.Shocks to the short-term rate affect all rates equally, giving rise to parallel shifts and a flat term structure of volatility.In a model with mean reversion, shorter-term rates are determined mostly by current economic conditions while longer-term rates are determined mostly by long-term economic conditions.As a result, shocks to the short rate affect shorter-term rates more than longer-term rates and give rise to a downward-sloping term structure of volatility and a downward-sloping factor structure.
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What is the advantage of the CIR and lognormal models?
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The advantage of the CIR and lognormal models in not allowing negative rates.
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What are forward and futures contracts?
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Forward and futures contracts are agreements to trade an asset for delivery on some particular date in the future. The price at which the asset will be traded is fixed today, but the exchange of cash and the asset takes place in the future.
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Why are forwards interesting for fixed income markets?
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1. the common practice of combining a spot position with borrowing or lending creates the equivalent of a forward contract.2. futures contracts are more easily understood3. option contracts may be viewed as derivatives of forward contracts.
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What is the difference between futures and forward contracts?
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Because forward contracts are not marked-to-market, any value, positive or negative, accumulates over time until final settlement at expiration.Futures contracts pay or collect value changes as they occur.As a result, after each day’s mark-to-market a futures contract has zero value.A futures contract is essentially like rolling over one-day forward contracts where each new forward price is that day’s futures settlement price.
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What is the difference between pricing a future an pricing a forward?
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Because of mark-to-market there is an asymmetry relative to rates in futures pricing.When rates rise and the futures position loses, this loss has to be financed at relatively high rates.On the other hand, when rates fall and the futures position gains, this gain is reinvested at relatively low rates.
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What is the futures-forward effect?
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The pure futures-forward effect arises because mark-to-market gains are invested at low rates while mark-to-market losses are financed at high rates. The total futures-forward effect increases with interest rate volatility.
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What is the effect of mark-to-market futures contracts on price?
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While mark-to-market gains can be reinvested and mark-to-market losses must be financed, on average these effects do not cancel out. Rather, on average they make futures contracts less desirable than forward contracts.
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What is the effect of mark-to-market futures contracts on rates?
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The futures rate exceeds the forward rate but may depend on the term structure model.
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What are TED spreads used for?
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TED spreads use implied Eurodollar futures rates to compare securities relative to Eurodollar futures rates or to compare the value of one security relative to another.The name came from the combination of T for Treasury and ED for Eurodollar.
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What do the fixed and float legs of a swap resemble including notional amount?
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The fixed leg resembles a bond which is valued using a swap curve rather than a bond curve.The floating leg resembles a floating rate note.
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What is a floating rate note?
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A floating rate note or floater makes periodic payments that are keyed off some rate index before returning principal at par.
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