Bonds, Yield Curves and Price Sensitivities

The aim is to explain bonds, yield curves and bond price sensitivities in a straightforward introductory manner. Core mathematical concepts have been broken down into longhand constituent elements where possible, enabling a general work-through. To achieve a balance towards an approachable introduction several concepts will be generalized and others ignored, for simplicity, although it may be an ongoing article with more detail added over time.


Bonds are loans, debt instruments. For example, the US Treasury borrows by issuing bonds, debt, to raise the money needed to operate the Federal Government and to pay off maturing obligations. Gilts are UK government securities issued by HM Treasury for similar purposes. A buyer of bonds is lending the issuer money, usually with interest paid in regular instalments and full repayment of the loan made on maturity of the bond.


Perhaps the easiest way to conceptualise bond pricing is in reference to present value (PV) and future value (FV). An investment today may be compounded at a given interest rate to give us a value in the future; FV = PV x interest rate. Equally one or a series of future cash flows may be discounted at a given interest rate to give us a present value; PV = FV / interest rate. This is known as discounting, or discounted cash flows. In calculating PV the higher the numerator (FV) relative to the interest rate, the higher the PV and vice versa. The higher the denominator in the expression, the lower the PV and vice versa. This observation underlines the basics 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 all be discounted back to the present giving us the price today. Consider a bond which consists of n periodic interest payment (coupons) paid by the borrower to the lender, or buyer, of the bond remaining to be paid, where each coupon is for C currency units per 100 currency units principal amount (known as the ‘face value’). If the bond has a yield of Y (the market yield or return if the bond is held to maturity) the price of the bond is given by:


Breaking that down with some real numbers, for a two-year par bond paying a coupon rate of 6% and with an annual yield to maturity of 5%. The bond pays semi-annual coupons so cash flows occur every six-months, which gives us 4 periodic payments of 3% (6% / 2). Thinking again of that conceptual PV assessment of bond pricing, it’s easiest then to construct the value by interpreting the two-year bond as 4 separate cash flows, or bonds, with maturities of six-months, one year, one and a half years and two years. The first 3 have a redemption value of 3 and the last one of 103 (3 + return of 100 principal at the end of the term). The price of each of the integral bonds is the discounted value of their 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 constant yields. Here the coupon rate is greater than the yield. This means the payment rate is greater than the rate being used to discount to the present value, so the bond will sell at more, or at a premium to the face value of 100. In the case where the coupon rate is lower than the yield, the price will be lower than the face value and is referred to as selling at a discount.

The Yield Curve

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

The chart below is of a hypothetical yield curve. The yield on the y axis is in basis points (bp), a market convention for 1/100th of 1%, so 2% would be referred to as 200bp. 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 in different maturity sectors fluctuate. Changes in yields are imperfectly correlated. Historically, but not always, the shorter end is largely subject to financial investment (heavy trading) with higher volatilities. This sector is prone to uncertain global economic and political outlooks, and dynamic risk of events for which there are no means of prediction or hedging. The intermediate/longer maturities are associated with real investment, such as from pension and insurance funds.

The principal components theory pioneered by Litterman and Scheinkman provides a way to evaluate the fundamental drivers of yield curve (term structure) movements and hedge interest rate risk; represented as independently occurring elasticity characteristics of the yield curve’s dynamic evolution. That is changes in the 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 effect on the curve is clearly monetary policy (the interest rate level). The second represents the slope of the term structure, equivalent to the rate of change/delta. It is less representative of underlying monetary issues, possibly due to slower reaction times, but changes in policy are still a major source of change. The third represents curvature of the term structure associated with a change in the volatility structure. These elasticity effects generally result from changing supply and demand conditions on the bond market reflecting changing habits, quite difficult 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 cumulative effect of these three components usually explains 95% or more of the variation in the yield curve.

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

Expectations Hypothesis

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

Implied Forward rates

Forward-spot relationships (implied forward rates, or future yields) can be synthesized from spot rates. The calculation is based on the assumption that all the returns over a given period of time are equal, even if the maturities are different (no arbitrage; there should be no difference in the price between two investments with equal cash flow streams). The difference between the current spot 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 which we can compare expectations of future short-term. This is calculated using zero-coupon bond yields, bonds with only one cash flow at maturity. A full explanation 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 and calculate the forward curve. As an example, a five year implied forward rate one year from today (or one year forward) where the one year yield is 2% and the six year yield is 4% will be 4.405%:


Spot vs. Forward Curve

The Liquidity Premium or Unbiased Expectations (UEH) Hypothesis argues that investors prefer liquid to illiquid securities, and will pay a premium for that liquidity. Short bonds are more liquid than longer bonds, since they will mature earlier. Investors prefer to preserve their liquidity, and invest funds for short periods of time, whereas borrowers prefer to borrow at fixed rates for long periods of time. This theory argues for a positively sloped curve in all circumstances in the belief that the forward curve denotes expected (future) spot rates, 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 then move onto a worked hypothetical example to demonstrate the practicalities of ideas explained.

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


Calculating maturity and modified duration

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


Consider a four-year semi-annual (8 periods) paying 10% (y = 10%), trading at par. 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 six months) 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 price of each of the zero-coupon bonds is the discounted value of their redemption values

(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 to maturity 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 volatility measure in which cash flows are not assumed to change when interest rates change:bond8.jpg 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 the modified duration equation:

Following on from the MD formula above, the percentage change in the bond price isbond100.jpg(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 in price 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)


Estimating the convexity effect

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

The example above has demonstrated the inexact nature of the duration measure of bond volatility. If the redemption yield changes by 100bps to 9%, the duration expected 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 (not estimated by modified duration). This duration price approximation is tangent to the bond’s price/yield curve on the intersection of the price and the yield of the bond, and will therefore consistently under-estimate. With increasing divergence from initial yield levels, modified duration explains increasingly less of a bond's price behaviour. For larger yield changes, duration is supplemented with convexity to capture the curvature of the actual price/yield relationship. The tendency of option-free bonds is for decreases in yield to have a much greater/faster effect than increases on the rise and fall in price respectively. Reducing this risk-potential is the essence of positive convexity appeal.

This allows for the fact that in reality, the inverse price/yield relationship is not linear; there is no straight line relationship between the two. Whilst convexity’s value in price volatility estimation cannot give us a completely accurate picture, the two measures supplemented means we can capture the curvature of the actual price/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 a future payment and the duration of the bond, where the weight is the PV of the future payment’). It provides a tool for better approximation of the inverse price/yield relationship notably given a 25 bps+ yield change; where the lower estimations obtained through duration alone result in intolerable error levels. Given only a 20bp change in yield the duration measure has logically provided a good price change approximation; in the worked example the small difference equal to 0.648942 (actual price move) - 0.646231 (MD price move)

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

Estimating the convexity of the bond

To appreciate convexity’s value in price volatility estimation, suppose the current yield level is y0. Approximating the price change via the first two terms of a Taylor series [i.e. assuming that the first two approximations to bond price, duration (first derivative) and convexity (second derivative), will estimate price well enough to 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% par bond:


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 change predicted 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 divergence of 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 25bp duration inaccuracy threshold), but this still demonstrates that a better approximation of bond price volatility is obtained by combining both measures.

Greater duration and convexity mean any favourable appreciation in the price of bonds 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 yield changes were detrimental (rising).

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


Negative convexity

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

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 price depreciation for a large change in yield. In reality, the inference of negative convexity is that of embedded option features, making the relationship between price and yield less straightforward than previous examples. By definition, the holder of a callable bond has sold the issuer the right to repurchase the contractual cash flows prior to the maturity date (i.e. it may be ‘called’), for example with loans underlying mortgage-backed securities. In the case of a putable bond, the holder has the right to sell the bond back to the issuer at a designated price and time.

With a callable bond, the price/yield relationship may have negative convexity for yields below a given %, but positive convexity for yields above. This is because as the bond price exceeds the call price, economic viability dictates an issuer withdrawal (but for belief that rates will fall further, so bonds will rationally trade a 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 graphical representation below, it is evident therefore that below given % yield there is a flattening of the price curve for a callable bond:


The issue for an investor is of risk-assessment. Conventional convexity may be inappropriate given option features in a bond because it does not take into account the effect of yield changes on all cash flows. A fall in rates may change expected cash flows for a callable bond, and a rise may change expected cash flows for a putable bond. Embedded options may clearly also impact upon duration, 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 is increased by x bps, P0 = initial price (per 100 of par value), Δy = yield change assumed; 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 reflect the expected cash flow changes. The more accurate my price valuation, in assessing any divergence between theoretical and market value, the better informed my decision-making and hopefully the greater my ability to maximize investment intentions in evaluation of the fundamental risk/return maxim (subject to usual option pricing factors).


Banque National de Paris, ‘On Deformation of the Yield Curve’ (Economic Research 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 in the Yield Curve. Part II: Principal Components’ (The Journal of Fixed Income Vol 7, Number 1 June 1997)


Christensen, M. The Pros and Cons of Butterfly Barbells
Fabozzi,F.J. Investment Management (Second Edition) p.552
Elton & Gruber (2003), Modern Portfolio Theory and Investment Analysis (Sixth Edition), p.533
Fabozzi, p.555-6

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