16 Bond Yield Curves and Price Senstivities

Bonds, Yield Curves and PriceSensitivitiesThe aim is to explain bonds, yield curves and bond price sensitivities in astraightforward introductory manner. Core mathematical concepts have beenbroken down into longhand constituent elements where possible, enabling ageneral work-through. To achieve a balance towards an approachable introductionseveral concepts will be generalized and others ignored, for simplicity, although itmay be an ongoing article with more detail added over time.

Bonds

Bonds are loans, debt instruments. For example, the US Treasury borrows byissuing bonds, debt, to raise the money needed to operate the FederalGovernment and to pay off maturing obligations. Gilts are UK governmentsecurities issued by HM Treasury for similar purposes. A buyer of bonds is lendingthe issuer money, usually with interest paid in regular instalments and fullrepayment of the loan made on maturity of the bond.

Conceptually

Perhaps the easiest way to conceptualise bond pricing is in reference to presentvalue (PV) and future value (FV). An investment today may be compounded at agiven interest rate to give us a value in the future; FV = PV x interest rate. Equallyone or a series of future cash flows may be discounted at a given interest rate togive us a present value; PV = FV / interest rate. This is known as discounting, ordiscounted cash flows. In calculating PV the higher the numerator (FV) relative tothe interest rate, the higher the PV and vice versa. The higher the denominator inthe expression, the lower the PV and vice versa. This observation underlines thebasics of bond pricing; in that prices (PV) move inversely to yields (interest rates).The higher the yield of a bond the lower the price, and vice versa.

Calculating the Price of a Bond

A bond is just a collection of cash flows paid at varying points in time that may allbe discounted back to the present giving us the price today. Consider a bondwhich consists of n periodic interest payment (coupons) paid by the borrower tothe lender, or buyer, of the bond remaining to be paid, where each coupon is for Ccurrency units per 100 currency units principal amount (known as the ‘facevalue’). If the bond has a yield of Y (the market yield or return if the bond is heldto maturity) the price of the bond is given by:

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Breaking that down with some real numbers, for a two-year par bond paying acoupon rate of 6% and with an annual yield to maturity of 5%. The bond payssemi-annual coupons so cash flows occur every six-months, which gives us 4periodic payments of 3% (6% / 2). Thinking again of that conceptual PVassessment of bond pricing, it’s easiest then to construct the value by interpretingthe two-year bond as 4 separate cash flows, or bonds, with maturities ofsix-months, one year, one and a half years and two years. The first 3 have aredemption value of 3 and the last one of 103 (3 + return of 100 principal at theend of the term). The price of each of the integral bonds is the discounted value oftheir redemption values. So the PV of the series of future cash flows; 1st period =3 / (1 + 0.025) = , 2nd period = 3 / (1 + 0.025)² etc: n = 4, C = 3, Y = 5% / 2 =2.5% (0.025)

The price is equal to the discount of each cash flow with respect to constantyields. Here the coupon rate is greater than the yield. This means the paymentrate is greater than the rate being used to discount to the present value, so thebond will sell at more, or at a premium to the face value of 100. In the case wherethe coupon rate is lower than the yield, the price will be lower than the face valueand is referred to as selling at a discount.

The Yield Curve

Yield curves are diagrams for single dates in time comparing the market yield tothe actual maturity on securities, differing only in terms of the time left to thematurity of the security. Although most bond analysis focuses on yields, it’s worthnoting that price, not yield, is generally the value being established and traded inmarkets. The yield curve estimates the interest rates at which the borrower couldborrow under current market conditions.

The chart below is of a hypothetical yield curve. The yield on the y axis is in basispoints (bp), a market convention for 1/100th of 1%, so 2% would be referred to as200bp. The term to maturity is on the x axis.



At the core of all fixed income markets is the question of how yields on bonds indifferent maturity sectors fluctuate. Changes in yields are imperfectly correlated.Historically, but not always, the shorter end is largely subject to financialinvestment (heavy trading) with higher volatilities. This sector is prone touncertain global economic and political outlooks, and dynamic risk of events forwhich there are no means of prediction or hedging. The intermediate/longermaturities are associated with real investment, such as from pension andinsurance funds.

The principal components theory pioneered by Litterman and Scheinkmanprovides a way to evaluate the fundamental drivers of yield curve (term structure)movements and hedge interest rate risk; represented as independently occurringelasticity characteristics of the yield curve’s dynamic evolution. That is changes inthe level, slope and curvature of the yield curve. The first component represents‘parallel’ shifts in the curve, or a shift in the general level of interest rates,independent of maturity (except very short maturities). The most significant effecton the curve is clearly monetary policy (the interest rate level). The secondrepresents the slope of the term structure, equivalent to the rate of change/delta.It is less representative of underlying monetary issues, possibly due to slowerreaction times, but changes in policy are still a major source of change. The thirdrepresents curvature of the term structure associated with a change in thevolatility structure. These elasticity effects generally result from changing supplyand demand conditions on the bond market reflecting changing habits, quitedifficult to forecast.1) An interesting way to look at the yield curve is as an ‘object’due to strong interaction between the series making up the curve. The cumulativeeffect of these three components usually explains 95% or more of the variation inthe yield curve.



Changes in interest-rate volatility will impact the convexity of the discountingfactors (discounting, as was introduced earlier, enables us to go from yields toprices). This affects the slope of the curve and thus its shape. The concept ofconvexity will be explained later.

Expectations Hypothesis

In general terms, the unbiased expectations hypothesis argues that the yield curveembodies the market’s forecast of future interest rates. Expectations of futureinterest rates affect the shape and position of the curve. A positive slope (longrates higher than short rates) argues for higher rates in the future and an invertedcurve (negative slope) for lower rates in the future. But it is necessary to obtainmore explicit interest rate forecasts, or an idea of the magnitude, of futureinterest rate changes from the yield curve by calculating implied forward ratesfrom current (spot) rates.

Implied Forward rates

Forward-spot relationships (implied forward rates, or future yields) can besynthesized from spot rates. The calculation is based on the assumption that allthe returns over a given period of time are equal, even if the maturities aredifferent (no arbitrage; there should be no difference in the price between twoinvestments with equal cash flow streams). The difference between the currentspot rate and implied forward rate is also referred to that ‘built in to the market’.Extracting implied forward rates therefore gives us a benchmark against whichwe can compare expectations of future short-term. This is calculated usingzero-coupon bond yields, bonds with only one cash flow at maturity. A fullexplanation and analysis of zero coupon bonds is beyond the scope of this article,but this is important to note. We are then able to approximate the yields andcalculate the forward curve. As an example, a five year implied forward rate oneyear from today (or one year forward) where the one year yield is 2% and the sixyear yield is 4% will be 4.405%:

Spot vs. Forward Curve

The Liquidity Premium or Unbiased Expectations (UEH) Hypothesis argues thatinvestors prefer liquid to illiquid securities, and will pay a premium for thatliquidity. Short bonds are more liquid than longer bonds, since they will matureearlier. Investors prefer to preserve their liquidity, and invest funds for shortperiods of time, whereas borrowers prefer to borrow at fixed rates for longperiods of time. This theory argues for a positively sloped curve in all



circumstances in the belief that the forward curve denotes expected (future) spotrates, stemming from the no-arbitrage condition described earlier.

Maturity and Modified Duration

In the analysis of price volatilities we will again start with concepts, and thenmove onto a worked hypothetical example to demonstrate the practicalities ofideas explained.

Duration is a measure of yield change sensitivity. Sensitivity of bond prices toyields is calculated by taking the first derivative of the bond pricing equation, withrespect to yields:

Calculating maturity and modified duration

Duration is an indication of bond responsiveness to the rather unrealisticassumptions of a flat yield curve and parallel shifts in the yield curve (interest ratechanges). Recognition is given to yield changes affecting expected cash flows. Theduration/maturity relationship is not linear because duration places higherweights to first cash flows and less to future ones (i.e. in reality the number ofyears needed to recover an investment is less than years to maturity. This 'reality'measurement is what duration represents by considering when cash flows arereceived). Taking account of the timing of cash flows (Macaulay), reflecting humanbehaviour where we value cash flows coming sooner higher than those cominglater (the time value of money):

Consider a four-year semi-annual (8 periods) paying 10% (y = 10%), trading atpar. If we recall earlier from earlier, trading at a premium refers to > 100,discount < 100, so trading at par means at 100 exactly, where the coupon rate =yield. The duration approximation in half years (cash flows occur every sixmonths) is interpreted as:

(a) Eight zero-coupon bonds with maturities of 6 months to 4 years



(b) The first seven have a redemption value of 5 and the last one of 105. The priceof each of the zero-coupon bonds is the discounted value of their redemptionvalues

(c ) PV of series of future cash flows; 1st = 5 / (1 + 0.05) = 4.761905, 2nd = 5 / (1+ 0.05)² etc., so the weight of each zero-coupon in the portfolio is equal to:

(d) Bond price/value of portfolio (100). Thus, the average weighted time tomaturity of the zero-coupon bonds, calculated in half-yearly periods, is found by:

(e) Multiplying each portfolio weight by its maturity and;

(f) Taking the sum. Equal to the duration (composite measure of bond volatility):

Referring back to the Macaulay Duration expression:

The table below illustrates the (a) to (f) process:

Modified Duration (MD) is an alternative and simpler bond price volatilitymeasure in which cash flows are not assumed to change when interest rates

change: half years.

In years (not half years): Duration = 3.393185 years. Modified Duration =3.231605 years.



Considering a change in yields

If yields on the four-year bond drop by 20bp; the price move using themodified duration equation:

Following on from the MD formula above, the percentage change in the bond

price is (Percentage change in bond price = -MD Change in basis

points/100).

The MD of 3.231605 implies that the bond experiences a 3.231605% change inprice for every 100 basis points (bps) shift in yield. MD is effectively a ‘multiplier’.

MD price move = -3.231605 x (-20 / 100) = 0.646321 (Intuitively = 3.231605 /5)

The actual price move:

The 20bp yield drop equates to 10% - 0.2% = par to yield 9.8%, semi-annually =0.098 / 2 = 4.9%

The bond price = discount of each cash flow with respect to constant yields =100.648942

Actual price move = New price – Old price = 100.648942 - 100 = 0.648942(close, but not equal to that approximated by MD)

Convexity

Estimating the convexity effect

Duration is a useful tool, but an inexact indication of bond responsiveness to theassumption of parallel shifts in a flat yield curve. It suggests there is a linearrelationship between bond yields and prices, so the duration approximation willconsistently underestimate. Trying to manage a bond portfolio by just controllingduration would therefore amount to making it sensitive/not to variations ininterest rates alone.

The example above has demonstrated the inexact nature of the duration measureof bond volatility. If the redemption yield changes by 100bps to 9%, the durationexpected price change is 3.231605 and price is higher 103.231605. If the



redemption yield changes to 11%, duration expected price is lower 96.768395.The greater the yield change the greater the ‘unexplained’ change in price (notestimated by modified duration). This duration price approximation is tangent tothe bond’s price/yield curve on the intersection of the price and the yield of thebond, and will therefore consistently under-estimate. With increasing divergencefrom initial yield levels, modified duration explains increasingly less of a bond'sprice behaviour. For larger yield changes, duration is supplemented with convexityto capture the curvature of the actual price/yield relationship. The tendency ofoption-free bonds is for decreases in yield to have a much greater/faster effectthan increases on the rise and fall in price respectively. Reducing thisrisk-potential is the essence of positive convexity appeal.

This allows for the fact that in reality, the inverse price/yield relationship is notlinear; there is no straight line relationship between the two. Whilst convexity’svalue in price volatility estimation cannot give us a completely accurate picture,the two measures supplemented means we can capture the curvature of the actualprice/yield relationship, and this will be more accurate than duration-alone.

Convexity is a measure of the sensitivity of the bond’s duration to changes in yield(‘a weighted average of the squared difference between the time remaining to afuture payment and the duration of the bond, where the weight is the PV of thefuture payment’). It provides a tool for better approximation of the inverseprice/yield relationship notably given a 25 bps+ yield change; where the lowerestimations obtained through duration alone result in intolerable error levels.Given only a 20bp change in yield the duration measure has logically provided agood price change approximation; in the worked example the small differenceequal to 0.648942 (actual price move) - 0.646231 (MD price move)

= 0.002711, which we can estimate to be (more or less) equal to theconvexity effect.

Estimating the convexity of the bond

To appreciate convexity’s value in price volatility estimation, suppose the currentyield level is y0. Approximating the price change via the first two terms of a Taylorseries [i.e. assuming that the first two approximations to bond price, duration(first derivative) and convexity (second derivative), will estimate price well enoughto not require any further derivatives], bond price at yield level y1 will be:



The table below explains the calculations for the four-year semi-annual 10% parbond:

Convexity in ½ years = 5225.37 / 100 = 52.2537 , Convexity in years = 52.2537 /4 = 13.063 (2 coupons per annum = 2² =4)

Calculation of the true change in price gave us 0.648942 . The price changepredicted by duration was = (-MD)(dy) = -3.231605 x -0.002 x 100 = 0.646321

Price change calculated by convexity = ½(convexity)(dy)² = 6.5315 x(-0.002)² x 100 = 0.0026126 (The convexity inference previously, via divergenceof the duration approximation from true price change was 0.002711).

So the price change predicted by duration and convexity = 0.646321 +0.0026126 = 0.6489336 . The yield change was only 20bp (less than a 25bpduration inaccuracy threshold), but this still demonstrates that a betterapproximation of bond price volatility is obtained by combining both measures.

Greater duration and convexity mean any favourable appreciation in the price ofbonds would be both greater and faster when yields are falling. In addition the



high-convexity analysis also implies that prices would be less sensitive if yieldchanges were detrimental (rising).

The graphical representation below illustrates the linear/non-linear relationshipsin assumptions:

Negative convexity

The implication of negative convexity is that price appreciation will be less/slowerthan the price depreciation for a large change in yield of a given number of basispoints.

Price volatility implications of positive and negative convexity:2)

On the other hand, for an option-free bond exhibiting positive convexity(curvature), the price appreciation will be greater/faster than the pricedepreciation for a large change in yield. In reality, the inference of negativeconvexity is that of embedded option features, making the relationship betweenprice and yield less straightforward than previous examples. By definition, theholder of a callable bond has sold the issuer the right to repurchase thecontractual cash flows prior to the maturity date (i.e. it may be ‘called’), forexample with loans underlying mortgage-backed securities. In the case of aputable bond, the holder has the right to sell the bond back to the issuer at adesignated price and time.

With a callable bond, the price/yield relationship may have negative convexity foryields below a given %, but positive convexity for yields above. This is because asthe bond price exceeds the call price, economic viability dictates an issuerwithdrawal (but for belief that rates will fall further, so bonds will rationally tradea little above call price). Thus, an astute investor will not pay much more than the



call price for the bond due to this call risk. Looking at the simple graphicalrepresentation below, it is evident therefore that below given % yield there is aflattening of the price curve for a callable bond:

3)

The issue for an investor is of risk-assessment. Conventional convexity may beinappropriate given option features in a bond because it does not take intoaccount the effect of yield changes on all cash flows. A fall in rates may changeexpected cash flows for a callable bond, and a rise may change expected cashflows for a putable bond. Embedded options may clearly also impact uponduration, with potential differences between conventional (modified) duration and‘effective’ duration.

Just to introduce one means by which we can begin to evaluate these risks,consider where P- = price if yield is decreased by x bps, P+ = price if yield isincreased by x bps, P0 = initial price (per 100 of par value), Δy = yield changeassumed; thus approximating duration and convexity for any bond employing:4)

The utilisation of both effective-convexity (EC) and effective-duration (ED)measures to evaluate prices given yield changes both up and down should reflectthe expected cash flow changes. The more accurate my price valuation, inassessing any divergence between theoretical and market value, the betterinformed my decision-making and hopefully the greater my ability to maximizeinvestment intentions in evaluation of the fundamental risk/return maxim (subjectto usual option pricing factors).

References

Banque National de Paris, ‘On Deformation of the Yield Curve’ (EconomicResearch Department Eco-Notes No. 1998-5)

Choudhry, M. (2006), The Bond & Money Markets (Strategy, Trading, Analysis)

Hanweck, J & Falkenstein, E., ‘Minimizing Basis Risk from Non-Parallel Shifts inthe Yield Curve. Part II: Principal Components’ (The Journal of Fixed Income Vol 7,Number 1 June 1997)



Bonds

1) Christensen, M. The Pros and Cons of Butterfly Barbells2) Fabozzi,F.J. Investment Management (Second Edition) p.5523) Elton & Gruber (2003), Modern Portfolio Theory and Investment Analysis (SixthEdition), p.5334) Fabozzi, p.555-6

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16 Bond Yield Curves and Price Senstivities

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