Interest Rate Markets and Futures

8/9/2019 Interest Rate Markets and Futures

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Interest Rate MarketsInterest Rate Markets

Prof Mahesh KumarAmity Business School

[email protected]


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Types of RatesTreasury Rates: They are the interest rates applicable toborrowing by a government in its own currency. It isusually assumed that there is no chance that agovernment will default on an obligation denominated in

its own currency. For this reason, Treasury rates areoften termed risk-free rates.LIBOR Rates: LIBOR is abbreviated for London InterbankOffer Rate. It is the rate at which one large internationalbank is willing to lend money to another largeinternational bank. LIBOR rates are determined in

trading between banks and change as economicconditions change. LIBOR rates are generally higher thanthe corresponding Treasury rates because they are notrisk free rates.


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Types of RatesRepo Rates: Repo or repurchase agreementsare contracts where an investment dealer whoowns securities agrees to sell them to another

company now and buy them back later at aslightly higher price.Zero Rates: The n-zero rate (zero couponrate) is the rate of interest earned on an

investment that starts today and lasts for nyears. There are no intermediate payments.The n-year zero rate is sometimes also referredto as the n-year spot rate.


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Important ConceptsBond Pricing: The theoretical price of a bondis calculated as the present value of all thecash flows that will be received by the owner

of the bond using the appropriate zero ratesas discount rates.Bond Yield: The yield on a coupon bearingbond is the discount rate that equates thecash flows on the bond to its market value.Par Yield: The par yield for a certain maturityis the coupon rate that causes the bond priceto equal its face value.


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Important ConceptsForward Rates: They are the rates which are specifiednow for a loan that will occur at a specified future date.Forward interest rates are the rates of interest implied bycurrent zero rates for period of time in the future.Example

Year N Zero rate for an n-yr inv Forward rate for the nthyr (%p.a.)

1 10

2 10.5 11.0

3 10.8 11.4

4 11.0 11.6

5 11.1 11.5


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Important ConceptsWhen interest rates are continuously compounded andrates in successive time periods are combined, theoverall equivalent rate is simply the average rate duringthe whole period (In our example, 10% for the first year& 11% for the second year average to 10.5% for twoyears).To generalize, if R 1 & R 2 are the zero rates for maturitiesT1 & T2 respectively and R F is the forward interest ratefor the period of time between T1 & T2 then

RF=R 2 T2-R 1 T1 / T2- T1This equation can be rewritten as

RF=[R 2 +(R 2-R 1)]*{ T1 / T2- T1 }From the above curve it implies that if zero curve is upwardsloping between T1 & T2 so that R 2 >R 1 , then R F>R 2 .Similarly if the zero curve is downward sloping with R 2


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Day Count ConventionThe interest rate earned between two days is:No of days between dates * Int earned in ref pdNo of days in referred pdThe day count conventions commonly used are:

1. Actual/ Actual (in period)2. 30/3603. Actual/360


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QuotationsThe quoted price is not the same as the cash price thatis paid by the purchaser. In general,Cash price= Quoted price+ Accrued int since last coupondate


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Forward V/s Future Interest RatesFor short maturities future interest rate can be assumedto be same as the corresponding forward interest rate.For long dated contracts, the difference between futuresand forward rates is adjusted by parameter known asconvexity adjustment and the relationship is depicted asForward rate=Futures rate-1/2 W2 t 1 t 2where t 1 is the time to maturity of the futures contractt 2 is the time to the maturity of the rate underlying thefutures contractW is the standard deviation of the change in the shortterm interest rate in one year and typically its value istaken as 0.012 (or 1.2%)


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Interest Rate SensitivityBond prices are inversely related: as yieldincrease, bond prices fall; as yield fall, bond pricerise.An increase in a bonds yield to maturity results in

a smaller price change than a decrease in yield of equal magnitude (due to convexity of yield curvein the bonds).Prices of long term bond tend to be more sensitiveto interest rate changes than the prices of shortterm bonds. The reason being if rates increase, for

example, the bond is less valuable as its cashflows are discounted now at a higher rate. Theimpact of the higher discounted rate will be greateras that rate is applied to more distant cash flows.


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Interest Rate SensitivityThe sensitivity of bond prices to changes inyields increases at a decreasing rate asmaturity increases.Interest rate risk is inversely related to thebonds coupon rate. Prices of high couponbonds are less sensitive to changes in interestrate than prices of low-coupon bonds.The sensitivity of a bonds price to a change inits yield is inversely related to the yield to

maturity at which the bond is currently selling.


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DurationFredrick Macaulay termed the effective maturity concept asthe duration of the bond.Macaulays duration is computed as the weighted averageof times to each coupon or principal payment made by the

bond.The weight associated with each payment is related to theimportance of that payment to the value of the bond andshould be the proportion of the total value of the bondaccounted for by that payment. This proportion is thepresent value of the payment divided by the bond price.

The coupon payments made prior to maturity makeeffective (i.e. weighted average) maturity less than itsactual time to maturity.


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DurationTo calculate the weighted average directly, we definethe weight w t , associated with the cash flow made attime t (denoted by CF t) as:

wt= CF

t /(1+y) t

Bond priceWhere y is the bonds yield to maturity.Macaulays duration formula is:

T

D = 7 t*w tt=1


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DurationDuration has a three fold importance:

1. It gives effective maturity of the portfolio.2. It is an essential tool in immunizing portfolios

from interest rate risk.3. Duration is the interest rate sensitivity of a

portfolio.


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D ifference Between Duration &

Modified DurationMacaulay Duration The only duration that can accuratelybe quoted as length of time. D iscovered in 1938, byFrederic Macaulay, this duration is calculated as the

present-value-weighted time to receipt of cash flows, which is why it is quoted in years. Each cash flow time is

multiplied by the present value of the associated cash flowsand then the sum of all of these terms is divided by thesum of the present values.Modified Duration In 1966, Larry Fisher presented a proof of the relationship between duration and bond pricechanges. Modified duration is a measurement of the changein value of an instrument in response to a change in

interest basis (payment frequency). This "modifies"Macaulay duration. The relationship of duration and pricevolatility can be expressed as follows:Percentage price change = -Modified duration x Yieldchange x 100


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DurationWhen the interest rate change, the proportional changein a bonds price can be related to the change in itsYTM, y according to the rule,( P/P = - D x [ ( (1+y)/(1+y)]

The proportional price change equals the proportionalchange in 1 plus the bonds yield times the bondsduration.Modified Duration D* = D /(1+y) and ( y)= ( y,therefore above equation can be rewritten as ,( P/P = - D* ( yi.e. The percentage change in bond price is the productof modified duration and the change in the bonds yieldto maturity.


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Bond Pricing & DurationBonds price changes in market interest rate isinfluenced by three factors:

1. Time to maturity2. Coupon rate3. Yield to Maturity.

Rule #1 : The duration of a zero coupon bondequals its time to maturity.A coupon bond has a lower duration than a zerowith equal maturity because coupons early in

bonds life lower the bonds weighted average timeuntil payments.Rule #2 : Holding maturity constant, a bondsduration is higher when the coupon rate is lower.


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Bond Pricing & DurationRule #3 : Holding the coupon rate constant, abonds duration generally increases with itstime to maturity. Duration always increaseswith maturity for bonds selling at par or apremium to par.Rule #4 : Holding maturity constant, a bondsduration is higher when the coupon rate islower.Rule #5 : Duration of perpetuity =(1+y)/y

e.g. at a 10% yield, the duration of perpetuitythat Rs.100 forever is 1.10/0.10=11 years butat 8% yield it is 1.08/0.08=13.5 years.


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Bond Pricing & DurationRule #6 : The duration of a level annuity is equal to

Duration of annuity =(1+y)/y- T /[(1+y) T-1]where T is the number of payments and y is theannuitys yield per payment period.

Rule #7 : The duration of coupon bonds equals:Duration of coupon bond=(1+y)/y-[(1+y)+t(c-y)]/{c[(1+y) T-1]+y}

where c is the coupon rate per payment period, T is thenumber of payment periods and y is the bonds yieldper payment period.Rule #8 : For coupon bonds selling at par value

Duration=(1+y)/y*{1-1/(1+y) T}


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ConvexityPrice Yield relationship are said to be convexand the curvature of the price-yield curve iscalled the convexity of the bond.

Price sensitivity of the bond when theconvexity is also accounted for is:( P/P = - D* ( y+1/2 x Convexity x ( ( y) 2

Convexity is considered to be desirable trait.Bonds with greater curvature gain more inprice when yields fall than they lose whenyields rise. Thus if interest rates are volatile,this is an attractive asymmetry that increasesthe expected return on the bond.


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ImmunizationIncrease in interest rates causes capital lossesbut at the same time it increases the rate atwhich reinvested income will grow. If theportfolio duration is chosen appropriately,these two effects will cancel out exactly.When the portfolio duration is set equal to theinvestors horizon date, the accumulated valueof the investment fund at the horizon date will

be unaffected by the interest rate fluctuations.For a horizon equal to the portfolios duration,price risk and reinvestment risk exactly cancelout.


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Duration Based Hedging StrategiesFC: Contract price for the interest rate futures contractDF: Duration of the asset underling the futures contract at the

maturity of the futures contractP: Forward value of the portfolio being hedged at the maturity

of the hedgeDP: Duration of the portfolio at the maturity of the hedge.

( P = -P x DP x ( yTo reasonable approximation

( FC = -F C x DF x ( yThe number of contracts required to hedge against uncertain ( y

is, thereforeN=P DP

FCDFThis is duration based hedge ratio and is also at times referred

as price sensitivity hedge ratio.


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Strategies Using Fixed Income Futures1. Changing the duration of the portfolio

Investors with strong expectations about thedirection of the future course of interest rates will

adjust the duration of their portfolio to capitalizeon their expectations.If they expect interest rates to increase they willshorten the duration of the portfolio; if they expectthe interest rates to decrease, they will lengthen

the duration of the portfolio.Future contracts offer quick and less expensivemethod for changing the duration of the portfolio.


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Strategies Using Fixed Income FuturesBy buying futures contracts on Treasury bonds or notesthey can increase the duration of the portfolio.Conversely they can shorten the duration of the portfolioby selling futures contracts on Treasury bonds or notes.Approximate no. of contracts= ( DT- D I)P I / DFPFDT= target modified duration for the portfolio.D I= initial modified duration for the portfolio.DF= modified duration for the futures contractsPI= initial market value of the portfolio.PF= market value of the futures contract.

If the investor wishes to increase the duration, thenD

T willbe greater than D I and the above equation will have + signmeaning thereby that future contracts will be purchased.The opposite is true if the objective is to shorten theduration of the portfolio.


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Strategies Using Fixed Income Futures2. Yield Enhancement

Consider an investor who own a 20 year T-bill and sellsTreasury futures that call for the delivery of that particularbond 3-mths from now.Although the maturity of bond owned is 20 years but theinvestor has effectively shortened the maturity of the bondto 3-mths.Consequently a long position in the 20 year bond and theshort position are equivalent to a long position in a 3 mthriskless security. It is riskless because the investor islocking the price that he will get 3 mths from now- thefuture price.If the yield on the synthetic 3-mth Treasury bill is greaterthan the yield on the cash market t-bill, the investor willrealize an enhanced yield by creating the synthetic shortterm security.


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Strategies Using Fixed Income Futures3. Hedging

Hedging with futures involves employing a futuresposition as a temporary substitute for transactionsto be made in the cash market at a later date.

If the cash market and future prices movetogether, any loss realized by the hedger from oneposition (whether cash or futures) will be offset bya profit on the other position.When the profit and loss from each position areequal, the hedge is referred to as a perfect hedge.The difference between the cash price and thefuture price is called the basis and the risk that thebasis will change in way that adversely impacts thehedge is called the basis risk.


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Strategies Using Fixed Income FuturesWhen a bond to be hedged is not identical to the bondunderlying the futures contract it is referred as cross hedging.An unhedged position is exposed to price risk and a hedgedposition to basis risk.A short (or sell) hedge is used to protect against a decline in

the cash price of a fixed income security resulting from anincrease in interest rates. To execute a short hedge , futurecontracts are sold. Thus, the hedger has fixed the future cashprice and transferred the price risk of ownership to the buyerof futures of contract.A long (or buy) hedge is undertaken to protect against anincrease in cash price of fixed income security resulting fromdecline in interest rates. In this hedger buys a future contractto lock in a purchase price. This strategy is used whensustained cash contributions are expected and interest ratesexpected to fall. It can also be used by investors who knowsthat bonds are maturing in the near future and expectsinterest rates to fall.

Interest Rate Markets and Futures

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