While analyzing bonds, it is important to apply the concepts of duration and convexity. Duration has been an excellent tool to forecast the approximate price change of a bond or portfolio of bonds. However, with larger yield changes and securities with embedded options, duration becomes less accurate at estimating price changes. Duration can be combined with convexity to arrive at a closer approximation of price for a given change in yield. This article discusses the relationship between duration, convexity and price-yield curve. Community bankers should be aware of this relationship, as well as how duration and convexity can be used together to estimate approximate price changes.
<h3>Duration</h3>
For both regulatory and performance reasons, bankers need a way to manage interest rate risk while realizing the current and future effects on their investment portfolio. Duration can be used to predict the market value change of a security or an entire portfolio given a rate shock on interest rates. Secondly, duration is a method for comparing the interest rate risk of securities with different coupons, maturities, cash flow and pricing characteristics. Lastly, since a specific duration value can be placed on a portfolio or each individual security, this becomes an excellent way to set interest-rate risk limits on an institution’s asset structure and capital adequacy.
Duration can be defined as the weighted average maturity of an instruments total cash flows (interest, principal and prepaid principal) in present value terms. When adjusted for yields, modified duration measures the volatility of an instrument’s market value in response to an increase or decrease in interest rates. Consider an 8% 10-year note at a price of 100 and a modified duration of 6.80. if the yield increases by 10%, duration estimates the price change as follows:
Price change = -[Duration] x [Yield Change]
Price change = -[6.80] x [2]
Price change = -13.60
The price change is negative to reflect the fact that prices decline as yields rise. The duration estimated price is 86.40 (100 – 13.60). Similarly, for a 2% decrease in yields, the estimated price is 113.60 (100 + 13.60).
The actual prices should be 87.54 and 114.88, and therefore, duration tends to underestimate the price during both rising and falling rates.
Figure 1 shows the actual price-yield relationship of the 8% 10-Year note and the straight line representing the duration estimated prices.
<img class="aligncenter size-full wp-image-814" title="duration_estimate" src="https://financetrain.com/wp-content/uploads/2010/08/duration_estimate.gif" alt="" width="546" height="423" />
When examining the price-yield curve, it is important to remember three properties:
<ul>
<li>The duration estimated dollar prices are on the tangent line drawn from the price of 100.</li>
</ul>
<ul>
<li>The accuracy of the duration estimate depends on the magnitude of the yield change. For small yield changes, the tangent line is close to the curve and duration does a good job of estimating the price change. For larger changes, the divergence of the price-yield curve and the tangent line increases, and the approximation worsens.</li>
</ul>
<ul>
<li>Regardless of whether yields increase or decrease, the duration estimated price will always be lower than the actual new price. In other words, when yields rise, the actual drop in price is smaller than the duration estimated drop in price, and when yields fall, the actual rise in the price is larger than the duration estimated rise in price. Because of this, duration is a conservative estimate of price changes.</li>
</ul>


<div class="wp-block-button aligncenter download-common"><a class="wp-block-button__link" href="https://financetrain.com/spreadsheet-duration-convexity-and-price-yield-curve/">Download the Duration and Convexity Excel Spreadsheet</a></div>



View Part 2 of this post - <a href="https://financetrain.com/bond-duration-and-convexity-simplified-part-2-of-2/">Bond Duration and Convexity Simplified – Part 2 of 2</a>

While analyzing bonds, it is important to apply the concepts of duration and convexity. Duration has been an excellent tool to forecast the approximate price change of a bond or portfolio of bonds. However, with larger yield changes and securities with embedded options, duration becomes less accurate at estimating price changes. Duration can be combined with convexity to arrive at a closer approximation of price for a given change in yield. This lesson discusses the relationship between duration, convexity and price-yield curve. Community bankers should be aware of this relationship, as well as how duration and convexity can be used together to estimate approximate price changes.

Bond Duration and Convexity Simplified – Part 1 of 2

As we learnt in <a href="https://financetrain.com/bond-duration-and-convexity-simplified-part-1-of-2/">part 1</a>, the duration, as measured by the slope of the curve, changes as yields change. The slope of the curve is steeper (and the duration is larger) at low yield levels, and becomes flatter (and the duration becomes smaller) at high yield levels.
This property is called convexity. Convexity is the rate of change of duration, as yields change. Graphically, convexity measures the relative “bend” of a bond’s price-yield curve, whereas duration measures its relative “slope”. A bond is positively convex when the price increases from falling yields is greater than the price decrease caused by rising yields. Bonds without any embedded options, such as call features, will always have positive convexity. Callable bonds at par, including most mortgage-backed securities, have negative convexity. The following figure compares positive and negative convexity and the duration line (Convexity=0).
With large shifts in interest rates, bonds with positive convexity generally perform better than bonds with negative convexity. Consider a 5-year Agency, callable in 1 year and priced at par. If rates rise, the agency is priced to maturity and has approximately the same depreciation as a 5-year Treasury. If rates fall however, the security is called away and the investor realizes the appreciation of a 1-year. In other words, the callable agency priced at par has the downside risk of a 5-year with the upside potential of a 1-year. This is a perfect example of a negatively convex bond.
<img class="aligncenter size-full wp-image-822" title="Positive and Negative Convexity" src="https://financetrain.com/wp-content/uploads/2010/08/convexity1.gif" alt="" width="530" height="414">
Although callable bonds priced at par exhibit negative convexity, deep discount or high premium callable bonds often perform similar to non-callable bonds and may have slight positive convexity. This is because, the deeper the discount of a callable bond, the less likely it will be called, and the more likely it will trade to maturity like a non-callable bond.
<h3>Combining Duration and Convexity</h3>
In estimating price changes, duration treats the price-yield function as though it were along the line tangent to the actual price-yield curve. Because duration provides a linear or first-order approximation of price changes, is often referred to as the first derivative of the price-yield function. Since the actual price-yield relationship is not linear, we can get a better price approximation if we use a higher-order derivative. Convexity represents the second derivative of the price-yield function.
By including convexity in our price change formula. We can get a better approximation of the new price as follows:
Price Change = (- Duration x Price Yield) + (0.5 x Convexity x (Yield Change)^2))
Using our previous example, if the 8% 10-year note has a 0.60 convexity, the new estimated price change is calculated as follows:
Price Change = (-6.80 x 2) + (0.5 x 0.60 x 4)
Price Change = -12.40
The estimated price using convexity is now 87.60 (100 – 12.40). The convexity estimate of 87.60 is much closer to the actual price of 87.54 than the duration estimate. The new estimated price-yield curve using both duration and convexity is shown below.
<img class="aligncenter size-full wp-image-823" title="Price estimation using duration and convexity" src="https://financetrain.com/wp-content/uploads/2010/08/convexity2.gif" alt="" width="528" height="413">
Notice that the convexity estimates still underestimates the price increase when yields fall, but it also underestimates the price decline when yields rise. For this reason, duration is a more conservative, although less accurate, estimation of price declines in a rising rate scenario.
Due to the increased volatility in the bond market and the heightened regulatory scrutiny, managing the interest rate risk of a bank’s investment portfolio has become a top priority. Duration has proven to be an effective tool for measuring interest rate risk and setting portfolio limits. Analysts should be aware that duration is only an estimate, and convexity can be used in conjunction with duration to arrive at a more accurate estimation of a bond’s future price.

Bond Duration and Convexity Simplified – Part 2 of 2

Bonds are a lucrative investment class for investors and portfolio managers. However, just like any other investment, investing in bonds also has many risks associated with it. This article lists the key risks:

Interest Rate Risk: Bond prices are inversely related to interest rates. When interest rates rise, bond prices fall and vice versa.

Call and Prepayment Risk: Bonds can have embedded options such as an embedded call in which case the bond may be called back before maturity. Bonds may have <a title="Cash Flows and Prepayment Risk" href="https://financetrain.com/cash-flows-and-prepayment-risk/">prepayment risk</a>, in cases where the underlying cash flow is backed by a loan pool. 

Yield Curve Risk: The term structure of interest rates, or yield curve, may change causing the value of bond portfolio to change.

Reinvestment Risk: The <a title="YTM and Reinvestment Risk" href="https://financetrain.com/ytm-and-reinvestment-risk/">yield-to-maturity (YTM)</a> assumes that you will be able to reinvest the intermediate cash flows at the same yield. This may not always be possible and the investor may not realize the YTM.

Credit Risk: The investor is also exposed to credit risk in the form of default risk, credit spread risk, and downgrade risk.

Liquidity Risk: This is the risk that the investor may have to sell the bond at a price lower than the expected price. This is measured using bid-ask spread.

Exchange-rate Risk: Bonds with cash flows in foreign currency will add foreign exchange risk to the bond portfolio.

Volatility Risk: The higher is the expected yield volatility, the higher is the price of the bond.

Inflation Risk: The increase in inflation may wipe out the profits from the bond.

Event Risk: At times, the issuer may not be able to make the interest and principal payments on time due to events such as natural disasters, merger/acquisition, or a change in regulation. Such events collectively refer to event risk.

Sovereign Risk: Bonds issued by foreign entities also pose sovereign risk. The risk is that some action of the foreign government affects the bonds price or cash flows.

In the following articles, we will look at all these risks faced by a bond investor in detail.

Key Risks Associated with Investing in Bonds

The investors in bonds face interest rate risk because the price of the bond is inversely proportional to the changes in interest rates. So, if interest rates rise, the bond’s price will fall and if interest rates fall, bond’s price will rise.

But why this inverse relationship? Let’s understand this with the help of an example. Let’s say that an investor purchases a $100 8% coupon 5 year maturity bond at par. Since the bond is selling at par, the yield from the bond is the same as its coupon, i.e., 8%.

Let’s say the interest rates rise to 8.5%. At the same time, the investor decides to sell the bond. Any new investor would expect a yield of 8.5% and if he has to purchase a par bond, he will expect the bond to pay a coupon of 8.5%. However, in our bond, the coupon is fixed at 8%. The investor cannot change that. So, how will he convince the new investor who wants 8.5% yield from this bond. The only way to do so is to adjust the price downwards so that the new investor can make up for the loss of 0.5% in the coupon. Using <a title="How to Price a Bond Using Spot Rates (Zero Curve)" href="https://financetrain.com/how-to-price-a-bond-using-spot-rates-zero-curve/">bond valuation</a>, the price at which the new investor would be willing to buy this bond at $98.03 which will then provide him a yield of 8.5%.

Similarly, if the interest rates had fallen to 7.5% (instead of rising), then the new investor will be very happy to buy this bond at par as it is providing higher yield. However, the current investor holding the bond knows that the market is offering only 7.5%, not 8%. Since his bond is providing a higher yield, he will increase the price of the bond, so that effectively provides a yield of 7.5%. In this case, he will increase the price to $102.02.

This inverse relationship can be summarized as below:
<ul>
	<li>The bond trades at par when its coupon rate is equal to the required yield.</li>
	<li>If required yield is greater than the coupon rate, then the bond price will be below par (sell at a discount)</li>
	<li>If required yield is lower than the coupon rate, then the bond price will be above par (sell at premium)</li>
</ul>

Understanding Inverse Price/Yield Relationship in Bonds

We know that a <a title="Understanding Inverse Price/Yield Relationship in Bonds" href="https://financetrain.com/understanding-inverse-priceyield-relationship-in-bonds/">bond’s price is inversely related to the yield</a>. How sensitive a bond price is to yield depends on the various features of the bond such as its maturity, coupon rate, and any embedded options in the bond. Let’s look at how these factors influence the impact of interest rate changes on a bond’s price.

Maturity

In general, the longer the maturity, the higher the interest rate sensitivity. This means that for a given change in interest rates, everything else remaining the same, the price of a bond with higher maturity will change more compared to another similar bond with lower maturity. This happens because a higher number of cash flows need to be discounted.

Coupon Rate

All other factors remaining the same, the lower the coupon, the higher is the interest rate sensitivity. This happens because when the bond has lower coupon, its value is more dependent on the par amount to be received at maturity. In other words, it takes longer for a bondholder to get back its capital when the interest rates are low. On the other hand, a bond with high coupon rate has higher cash flows in the beginning which reduces its dependency on the maturity value. In this sense, zero-coupon bonds have highest interest rate sensitivity compared to a similar coupon paying bond.

Embedded Bonds

Some <a title="Common Options Embedded in a Bond Issue" href="https://financetrain.com/common-options-embedded-in-a-bond-issue/">bonds can have embedded options</a> such as a call option attached to it. For a callable bond, the bond can be called back by the issuer. Such an option also affects the interest rate sensitivity of the bond. All other factors remaining the same, a bond with embedded call option will be less sensitive to interest rate changes. This happens because the price of a callable bond is lower than a similar non-callable bond by an amount equal to the value of the option.

Price of Callable Bond = Price of Non-callable Bond – Price of Embedded Option

The price is reduced because the call option is a benefit for the issuer. The opposite will be true for a puttable bond which provides the right to the investor to return the bond to the issuer.

When interest rates rise, the price of the embedded call option declines. Therefore, the overall effect on the decline in the price of the bond is less.

Bond Features Affecting Interest Rate Risk

In the previous article we learned about how maturity, coupon, and embedded options impact interest rate risk in bonds. Apart from this, the price sensitivity of a bond to interest rate change is also affected by the level of yield (interest rate). Note that other factors remaining the same, the different bonds trade at different yield because of the credit risk premium.

The rule is as follows:
<blockquote>The higher is the yield, the lower is the interest rate sensitivity. That is, the price of the bond will change less for a one basis point change in yield, compared to a bond with lower yield.</blockquote>
Consider two 5-year bonds, both paying 7% coupon, one selling at an initial yield of 7% (trading at par) and the other selling at a yield of 9% (selling at a discount). Now, if interest rates increase by 1%, let’s see how the bonds’ prices will change.

For the first bond, the initial price is $100 (as it is selling at par). When the yield increases by 1%, the price drops to $94, a drop of $4 or 4%.

For the second bond, the initial price is $92.22. When the interest rates rise by 1%, the price drops to 88.62, a drop of $3.59, or 3.89%.

When we compare the two bonds, we can conclude that bond with lower yield is more volatile and vice versa. This happened because a 1% increase in interest rates reflects a higher % increase in interest rates for a lower initial yield. A shift from 7% to 8% is an increase of 14.28% in the yield, while a shift from 9% to 10% is an increase of only 11.11% in the yield.

We can also say that bond prices are more volatile in a low interest rate environment and less volatile in a high interest rate environment.

Impact of Yield Level on Bond’s Price Sensitivity

The debt securities issued in the market can have many features; one such feature is an embedded option such as a call option. The call option gives the issuer the right to but the bond back from the investor at prespecified terms after a certain period. Similarly, there could be a put feature in the bond which will give the bondholder the right to return the bond to the issuer.

Such embedded options can influence the price of the bond depending on in whose favour the embedded option works. As we can see, the call option in a callable bond is a clear benefit to the issuer as it can call the bond back under unfavourable circumstances and issue a new bond. Therefore, the price that an investor would be willing to pay for this bond will be lower than an option-free bond.

The general equation is as follows:
<blockquote>Price of Callable Bond = Price of Option-free Bond – Price of Call Option</blockquote>
Due to the lower price, the callable bonds are also less sensitive to interest rate changes.

For puttable Bonds, the pricing is opposite, i.e., the price of the put option is added to the price of option-free bond, since it’s an additional benefit for the investor allowing him to sell the bond back to the issuer.

Here are some more interesting points about callable bonds:
<ul>
	<li>The issuer compensates the investor for the call option in the form of higher coupon.</li>
	<li>Government-sponsored entities are the largest issuers of callable bonds.</li>
	<li>Callable bonds provide natural hedge to the issuers; if the interest rates fall, they can call their bonds back and reissue at lower rates.</li>
	<li>Since callable bonds have lower price than option-free bonds, they also provide higher yield.</li>
	<li>Unlike a warrant, the call option is not detachable from the bond.</li>
	<li>The value of the call option converges to zero at maturity. It’s wiser to exercise as early as possible after the call protection period.</li>
	<li>Once the rates decline, the call should be exercised immediately before the rates rise again and the refinancing opportunity extinguishes.</li>
</ul>

Price of a Callable Bond

In case of fixed-rate bonds, the coupon is set as a reference rate plus a margin. Since the reference rates changes periodically, the coupon rate for the bond is also reset periodically, such as monthly, quarterly, or every six months. Every time the coupon is reset, the bond’s price also resets to par. However, the price of a floating-rate bond is not always the same and does very over time. Let’s look at the three key factors that could affect the price of such a bond.

Length of Time Between Coupon Reset Dates

Since coupon rates are not set every day, on a particular day the market interest rate could differ from the recently set coupon rate. Say for example, the coupon is reset every six months, and just after the day the coupon is reset, the market interest rate changes. So, for the next 179 days, the coupon rate will be different from the market rate, which will affect the price of the bond. If the market rate is higher than the coupon rate, the bond price will decline and vice versa. A bond with shorter reset dates will be less price sensitive compared to a bond with longer reset dates.

Change in Margin Required by the Investor

As we mentioned above, the coupon of a floating-rate bond is set as reference rate + a margin. The margin reflects risk taken by the investor compared to a risk-free bond, such as credit risk and liquidity risk. It could be so that the market conditions have changed and the investor now demands a higher margin for the same bond. This will also increase the expected yield of the investor and inversely affect the price.

Cap Rate in the Bond

The floating rate of the bond may be capped, and if the market interest rates rise above this rate, the bond’s coupon rate will be capped to this rate. In this scenario again, the market interest rate will become higher than the coupon rate and lower the price of the bond.

To conclude, while measuring the interest rate risk in a floating-rate bond, all the above factors should be considered.

Interest Rate Risk of Floating-rate Bonds

We know that the bond prices are sensitive to changes in interest rates. When interest rates increase, the bond prices fall and vice versa. A bond’s maturity also influences its price sensitivity to interest rate changes.

When we are looking at a single bond, measuring the interest rate risk (sensitivity) is relatively easier because we are talking about one bond with one maturity. However, in case of a portfolio, things become a bit complex. A bond portfolio will usually have bonds with different maturities. For example, there could be two bonds with 2-years of maturity, 2 bonds with 5-year maturity, and 2 more bonds with 10-year maturity.

While measuring the interest rate risk in such a portfolio, we can make a simple assumption such as, “Change in price of the portfolio if the interest rates increase by 50 basis points”. This will do the job, but there is one problem. In the real world, the interest rates or yields are different for different maturities. There is a structural relationship between the yield and the maturity, and the graph that plots maturity on x-axis and yield on y-axis is called the yield curve. If we simply say that the interest rates have increased by 50 basis points, we are assuming that interest rates have risen by 50 bps across the yield curve for all maturities. This is also called a parallel shift in the yield curve. This may not always be the case. It could be so that yields for different maturities change by different percentage. For example, the 5 year rate rises by 50 bps while the 10-year rate rises by just 25 basis points. This is called a non-parallel shift in yield curve and is a more commonly seen phenomenon.

Due to this reason the bond portfolios will have different exposures to how yield curve shifts, and this risk is called the yield curve risk.

This also brings us to one problem with <a title="Bond Duration and Convexity Simplified – Part 1 of 2" href="https://financetrain.com/bond-duration-and-convexity-simplified-part-1-of-2/">calculating duration</a>. Duration doesn’t account for yield curve risk because it assumes a single change in yield across all maturities on the yield curve. As per the definition of duration, it measures the approx. change in the price of a bond or a portfolio, for a 1% change in the yield. Due to this drawback, we need to look at other measures of interest rate risk that could capture the exposure even in case of a non-parallel shift in yield.  One such measure is ‘rate duration’. Rate duration measures the change in the value of a portfolio considering the change in yield of only a particular maturity, the yield of all other maturities remaining constant. For example, 10-year rate duration of 3% means that the portfolio value will change by 3% for a 1% change in the 10-year yield.

Yield Curve Risk

A bond may have a call provision that allow the issuer to call the bond back before its maturity date. Since a call feature is a benefit to the issuer, its value is lower than a similar plain vanilla bond. The call provision brings three key disadvantages to the investor.
<ol>
	<li>The investor cannot be certain about the cash flow pattern of its investment, as he doesn’t know when the bond could be called back.</li>
	<li>There is higher reinvestment risk. This is because the changes of a bond call back increases if interest rates decline, and the investor will have to reinvest the funds at a lower rate.</li>
	<li>In a declining interest rate environment, the price appreciation of a callable bond is limited compared to an option-free bond. This phenomenon is also called price compression.</li>
</ol>
Put together all these form the call risk for the investors. Similar to call risk, there is <a title="Cash Flows and Prepayment Risk" href="https://financetrain.com/cash-flows-and-prepayment-risk/">prepayment risk</a> in <a title="Introduction to Mortgage-backed Securities – Part 1 of 3" href="https://financetrain.com/introduction-to-mortgage-backed-securities/">mortgage-backed securities</a>. The mortgage-backed securities are backed by a pool of thousands of loans. The cash flow generated from this mortgage pool is used to pay the cash flow to the MBS investors. The borrowers of the loans underlying these MBS can prepay their loans in a falling interest rate environment, which will affect the cash flows of the securities.

In mortgage-backed securities, the investors, along with the scheduled amortized payments, also receive unscheduled payment arising due to prepayments from the borrowers. Due to these unscheduled payments, it becomes difficult to assess the exact cash flow and maturity of the MBS.

Call and Prepayment Risk

Investors in fixed income securities, such as bonds, face reinvestment risk. The risk arises from the fact that the investor may have to invest the interim cash flows from the bonds at a lower interest rate than what he earns from the security. For example, if the security has a <a title="YTM and Reinvestment Risk" href="https://financetrain.com/ytm-and-reinvestment-risk/">yield-to-maturity</a> (YTM) of 10%, this assumes that the investor will be able to reinvest the coupon payments also at the same rate of 10%. Only then will the investor truly realize the YTM on his investment. However, this may not always be possible and investor may have to settle for a lower interest rate for these cash flows, and hence the reinvestment risk. Let’s say it’s a 10-year bond, and the first coupon is paid after 6 months. Then investor will have to ensure that this coupon payment continues to earn 10% for the next 9.5 years till the maturity of the bond.

The case of reinvestment risk can also be seen in callable bonds. The issuer will typically call back the bond in a falling interest rate environment as he would be able to come out with a new issue of bonds at lower interest rates. The investors (whose bonds are called back) will receive their principal earlier and will have to find new avenues for investment. Since it’s a falling interest rate environment, the investors will not be able to match their earlier proceeds.

Apart from regular bonds, the amortizable fixed income securities face even more reinvestment risk. In an amortizing security, the interest and a portion of principal is paid every month. If the interest rates in the market are falling, the principal may be prepaid prior to the scheduled payment dates, and the investor in such securities will have to reinvest at a lower interest rates. This is also cumbersome because the payments are received monthly and every month the investor will have to ensure the right investment to keep up the desired yield. In a rising interest rate environment, monthly proceeds may be advantageous because the investor can investment faster instead of waiting for 6 months. However, in a falling interest rate environment, this is a disadvantage to the investor.

Due to the presence of reinvestment risk in these securities, many investors prefer zero-coupon bonds because in zero-coupon bonds there are no coupon payments and hence no reinvestment risk. However, no coupon also means higher interest rate risk.

Reinvestment Risk in Bonds

The investors in a bond issue also face credit risk as they are actually lending money to the issuer. The credit risk can arise in three forms, namely, default risk, credit spread risk, and downgrade risk. Let’s look at each of them in detail.

Default Risk

This is the simplest form of credit risk and is the risk that the issuer will be unable to meet its payment obligations or keep up with the terms of payments. Such an event is called a default. Since each issuer is different we don’t have a direct measure of the possibility of default. However, in various studies, people have looked at thousands of bond issues and measured the probabilities of default of these issues. This probability of default is called default rate. Default rate is used as a proxy while measuring the default risk in an issue. When a default occurs the investor does not lose all its money. It will typically recover some money with is referred to as the recovery rate. Using the default rate and recovery rate we can estimate the loss given default in a bond issue.

Credit Spread Risk                                                               

Apart from default risk, there is also the risk that the price of the bond will decline due to increase in the yield from the bond. Let’s try to understand how this happens. The yield of a non-treasury bond is made up of two components: The interest rate on a risk-free security, plus a risk premium for the risk associated with the bond.

The risk premium acts as the compensation for all risks associated with the bond issue, and one of its components is the credit spread which is for default risk. If the default risk of a bond issue increases, its credit spread will increase (widen), in turn increasing the expected yield of the bond. This will lead to a fall in the price of the bond.

Downgrade Risk

Since the credit spread of a bond issue increases based on the increase in default risk, how does one measure the changes in default risk or the credit quality of a bond? This is generally done using credit ratings. Credit rating agencies such as Moody’s Standard &amp; Poor’s, and Fitch Ratings, assess the default risk of every debt issue and assign a credit rating to it, which reflects the credit quality of the issue. For example, a bond issue rated AAA reflects high credit quality (low credit risk) compared to another issue rated CCC which reflects poor credit quality (high credit risk). The symbol D represents default. Once an initial rating is assigned the rating agencies continuously monitors the issue and its credit quality. Depending on the change in circumstances, the rating agency may upgrade or downgrade the rating of the debt issue. For example, a debt issue rating AAA may be downgraded to AA on account of poor financial performance of the firm. A downgrade will widen the credit spread leading to a decrease in bond price. This is called downgrade risk.

The portfolio managers also using <a title="Defaults and Ratings Changes" href="https://financetrain.com/defaults-and-ratings-changes/">rating transition matrix</a> to measure downgrade risk.

Credit Risk in Bonds

Investors in bonds also face liquidity risk. This is the risk that the investor may have to sell the bond at a price lower than the expected price. Based on the market conditions and also on a review of recent market transactions, the investor can get an idea of the indicative price at which he will be able to sell his security. But when he approaches a dealer/broker and gets a lower rate than what he expects, it’s due to lower liquidity in the particular instrument.

Let’s review some important points about liquidity risk in bonds:
<ul>
	<li>Liquidity risk is measured using the <a title="Bid-Ask Spread of Bonds" href="https://financetrain.com/bid-ask-spread-of-bonds/">bid-ask spread</a>. Bid price is he price at which a dealer will buy the security from the investor, and ask price is the price at which the dealer will sell the security to the investor. For instruments having high levels of liquidity, the bid-ask spread will be very small and will not be affected by the size of the transaction. However, as liquidity deteriorates, the bid-ask spread widens.</li>
	<li>Since it’s a dealers market and every dealer offers its own bid-ask quote, different people may interpret liquidity differently. A good way is to compare quotes from different dealers and select the best bid price (lowest) and best ask price (highest). This is referred to as the market bid-ask spread.</li>
	<li>For investors holding the bond till maturity but not marking their position to market, liquidity risk is not very important.</li>
	<li>For investors who are required to mark their position to market (such as mutual funds), liquidity risk is of concern, as the bid price will change. To mark a position to market the investor may take quotes from many dealers and then use the best bid price as the new price for the position.</li>
	<li>For instruments that are very illiquid, the investor may not be able to rely on a quote from the dealer. Instead he will use models to estimate the fair value of the instrument.</li>
	<li>As market conditions change, bid-ask spread may change over time. Changing market conditions may also change impact liquidity and widen bid-ask spread as investors wait before taking new positions.</li>
	<li>The market for new products may be less liquid initially but the liquidity may improve as more brokers/dealers take interest in these products.</li>
</ul>

Liquidity Risk in Bonds

Some bonds and other fixed income products also expose the investor to exchange rate risk (also known as currency risk). This happens when the investor purchases bonds that have cash flows in a foreign currency instead of his domestic currency. In such a case, when the investor receives the foreign currency-denominated cash flow, he will have to convert the same into his domestic currency at the prevailing exchange rate exposing him to currency risk.

Consider a US-based portfolio manager who purchases a bond that has its cash flows in Euro. Since the managers domestic currency is US dollars and the bond’s cash flows are in euros, he faces exchange rate risk, in case the foreign currency (euro) depreciates. Let’s say each coupon payment is EUR 10,000 and the initial exchange rate is 1 EUR = 1.5 USD. In that case, he expects to receive a cash flow of USD 15,000. However, if euro depreciates and the new exchange rate is 1 EUR = 1.4 USD, then the actual payment he will receive will be USD 14,000 lower than his expectations.

To conclude, exchange rate risk in bonds is the risk that the portfolio manager’s domestic cash flow will reduce because of a change in the exchange rate.

Exchange Rate Risk in Bonds

Almost all bonds expose an investor to inflation risk, also known as purchasing power risk. It is a risk that the increase in inflation may wipe out the profits from the bond. I’ll take a simple example to explain this.

Let’s say you buy a 1-year $100 bond that pays 8% coupon. At the end of one year you will receive a $8 coupon and $100 of your principal. Your investment has grown to $108. That’s like a 8% return on investment. But has your purchasing power also increased that much?

Let’s look at what you could do with your $100. One year back, you could buy 1 kg of apples. That represents the <a title="Purchasing Power Parity (PPP)" href="https://financetrain.com/purchasing-power-parity-ppp/">purchasing power</a> of $100. Fast forward one year, and given an inflation rate of 5%, now the same 1kg of apples will cost you $105. If you went to the shop keeper with a $100 bill, he will give you only 0.95 kg of apples.

If we compare the above with your investment, then we can say that even though your money grew by $8, out of that $5 is wiped out by inflation, and your real wealth grew by only $3.

Since most bonds pay a fixed coupon for the life of the issue, an investor is exposed to inflation risk. There are now inflation-protected or <a title="Inflation-Indexed Bonds" href="https://financetrain.com/inflation-indexed-bonds/">inflation-indexed bonds</a> in which the interest and principal payments are indexed to the inflation rate. This way the inflation-indexed bonds protect the investors from any rise in the inflation rate, and thereby allow investors to maintain their purchasing power.

Inflation Risk in Bonds

<a title="Common Options Embedded in a Bond Issue" href="https://financetrain.com/common-options-embedded-in-a-bond-issue/">Bonds with embedded options</a> such as call options and put options also have volatility risk. This happens because any factor that affects the value of the embedded option will also impact the value of the bond. We earlier learned that interest rates affect embedded options. When interest rates rise, the price of the embedded call option declines. Therefore, the overall effect on the decline in the price of the bond is less. Apart from interest rates, there are other factors that affect the price of the embedded options. One such factor is expected volatility. For bonds this is called expected yield volatility.

In general, the higher the expected yield volatility, the greater is the value of the option.

Let’s review how the price of a call option is affected by the embedded option.
<blockquote>Price of Callable Bond = Price of Option-free Bond – Price of Call Option</blockquote>
So, how will the bond price change when volatility increases? When volatility increases, the price of the call option increases. This will result in a decrease in the price of callable bond. In callable bonds, therefore, the volatility risk arises from an increase in volatility.

The opposite is true for puttable bonds.
<blockquote>Price of Puttable Bond =  Price of Option-free Bond + Price of Put Option</blockquote>
When volatility increases, the price of put option increases, and hence the price of puttable bond will also increase. In puttable bonds, therefore, the volatility risk arises from a decrease in volatility.

Volatility Risk in Bonds with Embedded Options

Event Risk

The investors in bonds may also face event risk, i.e., the risk that the issuer will not be able to meet its interest and principal payment obligations due to a major event that impacts its financial condition. There are three broad categories of event risk:
<ol>
	<li>A natural disaster such as an earthquake, a flood or fire in the factory. As an example, imagine an earthquake destroys lots of property in a region. The bonds issued by insurers in that area will be hit because these insurers will have to make huge settlements for the policyholders.</li>
	<li>Merger and Acquisition or Corporate Restructuring: In such an event the company may face ratings downgrade exposing the company to downgrade risk. The company’s overall debt burden may increase due to either issuing new debt or debt that comes as a part of takeover.</li>
	<li>Regulatory changes such as imposition of a new tax or government’s decision to divest from certain securities can also impair an issuer’s ability to meet its obligations. Regulatory risk can come in a variety of forms and affects almost all companies.</li>
</ol>
Sovereign Risk

Sovereign risk occurs in bonds held by a portfolio manager that are issued by foreign entities, for example, an American investor holding a bond issued by Japanese Government bond. The risk occurs due to the fact that the foreign government may default on its obligation, or takes some action that restricts the issuer to meet its bond obligations. Sovereign risk can take primarily two forms: 1) the foreign government is just unwilling to pay its debt (repudiation). 2) the government is unable to pay due to unfavourable economic conditions.

Event Risk and Sovereign Risk in Bonds

Risks of Investing in Bonds

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Risks of Investing in Bonds

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