Understanding Bond Risk

Farhad Malik
Sep 22, 2018 · 6 min read

In my article “Understanding Bonds”, I explained what bonds are and how their price is calculated. I will provide an overview of risk associated with bonds and how to measure risk so that we can take calculated decisions to mitigate and manage it.

Quick Bond Introduction

Bonds are fixed income products. They provide a regular income for a fixed period of time. Furthermore, bonds are one of the simplest financial instruments and learning about bonds can help us understand how risk management works.

Bond Price And Interest Rate Relationship

When we price bonds, we observe that their price depends on coupon payment, periods and yield rate. There is a unique relationship between bond price and yield rates:

  1. Non linear relationship exists between price of a bond and its yield. To be precise, relationship is quasilinear (like a half-smile).
  2. Market value of a bond decreases with time. Therefore longer the maturity, cheaper the bond.
  3. Yield and bond price are inversely proportional to each other. This is the fundamental relationship in risk management: High yield means low bond price. Therefore increasing the interest rate, decreases bond’s price. As the price of a bond is a sum of its cashflows and yield rate is used to discount future cashflows to present value, if the yield rate is high then bond price will be lower. The relationship is exponential and as a result, bond price never reaches 0 even when the yield rate reaches infinity.

The inverse relationship can be seen in the image below:

Bond Price Plotted On Y Axis, Interest Rate On X Axis

This is the fundamental rule for any asset that pays fixed cash over a period of time.

Interest Rates Importance

  • Assume you want to invest your money and an issuer offers yield rate of 0% on your investment for 10 years.
  • If you invest the money, we won’t earn any interest back. Therefore in 10 years time, you will get the same amount back as what you invest today. However due to inflation, price of other items will increase in the 10 years. This is because the value of money is decreasing over time.
  • As a result, a good investment is one which offers returns based on rate higher than the risk-free rate that government offers.
  • Government offers risk free rate of return. Any issuer who wants to issue a bond has to offer a rate higher than risk-free rate because investors have to be compensated for bearing the risk that the issuer might default.

Bond Risk

There are a number of risks associated with bonds. In this section I will explain the risks along with a wide range of metrics.

  • Reinvestment risk: Cashflows are reinvested at lower than expected rate. Yield to market (YTM) assumes that cashflows will be reinvested at YTM and bond will be held at maturity.
  • We can find price changes over yield rate to get better understanding of risk.
  • Taylor expansion illustrates that the first derivative of a bond is change in price over change in yield rate. This is known as duration of a bond. As the relationship of bond price and interest rate is non linear, we can use taylor series to further estimate risk by calculating convexity. It is the second derivative of a bond and can be computed as:
  • First derivative of bond price * change in yield + 0.5 * second derivative of bond price * change in yield squared.

Bond’s first derivative is known as bond duration. If you compute the first derivative of bond duration then you get bond convexity.

This is the fundamental theory of risk management: we can assess impact of the risk factor (yield) using mathematical derivatives.

Duration And Convexity Effect

We briefly outlined that the change in bond‘s price over change in yield rate is bond’s duration. Additionally, change in duration over change in yield rate is known as bond convexity.

Duration Effect and Convexity Effect can be calculated on bonds to better quantify the risks.

  • Duration Effect = -Duration x Change in Yield x 100
  • Convexity Effect = 0.5 x Convexity x Change in Yield Squared x 100

If convexity is positive and interest rates change then returns will increase.

Notice the negative sign in front of duration to calculate duration effect. This is because bond price is negatively correlated with interest rate.

More On Convexity

Convexity of a bond can help us understand markets better:

  • High convexity means bond is more sensitive to interest rates. Prefer in stable or falling interest rates as price change is larger. As interest rates increase, high convexity means high price changes.

Convexity can be positive or negative:

  • Positive convexity means yield and duration are positively correlated whereas negative convexity implies yield and duration are negative correlated such as in MBS (due to repayments) and bonds with call options (due to premium payment to exercise).
  • Convexity is systemic risk. Convexity is useful when coupons are spread out and coupon rate is low.
  • Effective Convexity is when changes are expected in future cash flows.
  • Modified Convexity is when changes are not expected in future cash flows.

Convexity is positive for bonds with put option. This is because when bond is in the money then if market goes down you can sell the bond and if the market goes up then you can keep all cashflows. Put option gives you the right to sell the bond. On the other hand, if you buy a bond with call options then you would buy the bond if market interest rate decrease and if the market rates go up then you can keep the cashflows. Call option gives you the right to buy the bond.

Bond Risk Metrics

A large number of bond risk measures can be computed to better understand behaviour of bonds. In this section I will outline the most common measures:

  1. Macaulay Duration = Sum (t x Pv(t))/Sum Pv(t)
  2. Modified Duration = Macaulay Duration/(1 + y)

Let’s understand Dollar Duration and DV01:

  • Dollar Duration (DD): It is the negative of first derivative. DD is the change in price of a bond over change in yield rate. It is calculated as modified duration x bond market price (including any accrued interest).
  • Dollar value of basis point (DVBP or DV01): It is the product of dollar duration and change in yield. It is calculated as modified duration * bond market price (incl. any accrued interest) * 1/10000. DV01 is additive across portfolio.

Hedge Ratio

If you want to hedge risk associated with a bond by buying/selling another bond then you can compute hedge ratio and invest in buying the bond with highest hedge ratio. Hedge Ratio is calculated as:

(DV01 of initial bond)/(DV01 of hedging bond)

Bond Trading Strategies

There are two common bond trading strategies:

  1. Barbell Strategy: Investor uses bonds with short and long maturities. Use this strategy if you believe that rates will be volatile.
  2. Bullet Strategy: Investor uses bonds with intermediate maturity rate.

Bond Risk Measures Summary

The table below summarises a number of bond risk measures and how they are calculated:

Portfolio Bond Risk

If we have a number of bonds in a portfolio, we can calculate portfolio duration by summing weighted bonds duration. Each weight of a bond is based on its market value.

  1. Duration of portfolio = Sum Of (weight of each bond x Duration of bond)
  2. Convexity of portfolio = Weighted average of each convexity within the portfolio.

Bond Weight is market value of bond divided by market value of portfolio where market value of portfolio is the sum of market price of all bonds in the portfolio.

Portfolio duration assumes that yield of all bonds are perfectly correlated with each other and does not take diversification effect into account. This very a unlikely assumption.


In this article, we outlined how risk of a bond is calculated. An overview of a large number of bond risk measures were explained which can help us making calculated decisions in our investment strategies.

Let me know if you have any feedback.


This blog aims to bridge the gap between technologists, mathematicians and financial experts and helps them understand how fundamental concepts work within each field. Articles

Farhad Malik

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Explaining complex mathematical, financial and technological concepts in simple terms. Contact: FarhadMalik84@googlemail.com


This blog aims to bridge the gap between technologists, mathematicians and financial experts and helps them understand how fundamental concepts work within each field. Articles