In this material, the concepts of duration and convexity are presented in an accessible way, but with a responsible degree of mathematical rigor. It is assumed that the reader has basic knowledge of the bond market and differential calculus.
BONDS
A bond is understood as a set of future cash flows through which a debtor pays a creditor the principal plus interest for financing received.
Cash flows are generally called coupons in addition to the fact that the face or face value is included in the last payment. Sometimes, under a traditional amortization scheme or variations thereof, part of the nominal value is paid in each flow. There can also be a prepayment of the principal or a pause in the payment of the flows.
There are several types of bonds that are observed in the financial markets. One classification involves bonds issued by companies (corporate) and those issued by the government. Another classification is given based on the risk of default through the investment grade category (BBB-). Due to the place of issue, there are Eurobonds that are issued in a foreign capital market, such as Samurai bonds. In globalized financial markets a bond can be issued in any country and can be placed in some hard currency or emerging market. No less important, carbon credits are a great opportunity for emerging economies to steer towards sustainable development.
For explanatory purposes only, the classic bond consisting of coupons and payment of principal at maturity is treated. However, the concepts of duration and convexity can be applied to any fixed income instrument. In light of the above, the following notation is available:
- P: Bond price VN: Face or face value of bond n: Number of coupons paid by bond C t: Coupon paid in period 1 ≤ t ≤ n R: Market rate at maturity S: Days of a payment period Coupon D: Macauly Duration D *: Modified Duration C: Convexity
The price P = P (R) of a bond is the aggregate of the present values of the future cash flows C t plus the nominal value VN.
1+ R t 1
This price is assumed to depend solely on R, the market rate at maturity, so there is a flat yield curve. This means that the discount rate is the same for all cash flows. Also, between each coupon payment there are S days.
Illustration 1. Inverse relationship between the price of a bond and the market rate it pays.
From the function P (R) and indicated by Illustration 1 an inverse relationship between the interest rate and the price of the bond can be seen. The economic importance of this relationship emerges when you think that investment increases when you have low interest rates, among other factors. Regarding investment in foreign markets, if the bonds of a country offer better yields than foreign bonds, an appreciation of the currency occurs. If a country's currency is appreciated then foreign investors will find juicy dollar returns when investing in bonds.
DURATION AND CONVEXITY
Why study the concepts of duration and convexity?
It is important to study these concepts because the variability of interest rates modifies the value of a position in fixed income. For example, when yields increase, bondholders suffer losses.
The duration and convexity serve to estimate the variations in the values of the bond portfolios and, therefore, they are a valuable tool in managing interest rate risk.
In financial slang, changes in interest rates are measured in basis points, each of which is one-hundredth of a percentage point. In other words, 100 base points equals 1%.
Illustration 2. In April 2004, the so-called Mexican market bond crack occurred.
Changes in the value of assets and liabilities sensitive to movements in the interest rate are relevant for banking institutions. In this case, an element known as mismatch risk is added in which the duration intervenes. A bank is barefoot when the duration of its liabilities is less than the duration of assets.
From Illustration 2 it can be seen that the variation in the rates can be from a few to several tens of base points. In April 2004, some treasuries in Mexico suffered from a variation in interest rates in what was interpreted as a bond crash.
Two concepts of duration
The duration indicates the sensitivity of the relative changes in the price of a fixed income instrument to changes in the market interest rate. This degree of sensitivity is a consequence of the application of Taylor's theorem.
Bonus under study has:
⇒ DP dR R 360 S
In this way, the definition of the duration of Macaulay can be given as the indicator of the average time in which the holder of the bond obtains its benefits.
In subsequent paragraphs it will be proved that the modified duration of a strip is equivalent to the time to expiration of the same. If an investor buys a strip that matures in 5 years, then to receive the principal payment, he must wait five years. Instead, with a couponed bond the investor receives cash flows. In this situation, the duration of Macaulay contributes to estimating the time in which the principal payment is received.
Properties of duration
The duration has properties that sometimes depend on the characteristics of the bond as shown in the following list:
- The duration is less than or equal to the term to maturity of the bond. Equality occurs when it comes to a strip when it is the length of Macaulay. Ceteris paribus the longer it expires, the longer it lasts. Ceteris paribus the shorter the coupon, the longer the duration. 4. Ceteris paribus at a higher market rate has a shorter duration.
Convexity
When interest rates vary by too many basis points, duration is no longer a good measure of sensitivity and convexity is used. Figure 3 shows that the estimation of the variation in the price of a bond through duration is actually a linear approximation. When there are sudden fluctuations in the interest rate, the result that it gives lasts loses effectiveness, so the alternative is to estimate the value of the bond with a parable.
Figure 3. Convexity adjusts the estimate obtained by duration.
Actually what you do is approximate the price of the bond with the second order Taylor polynomial as shown.
Coupon rate: 8%
Market rate: 10%
Days between coupon payment (S): 180
Age: 5
With these data, the price of the bond amounts to 92.27 monetary units (CU). This result is logical because it is a low par bond. Next, the durations and convexity are obtained and then applied by means of questions that are answered.
Determination of values
- Modified Duration
92.27 360 t = 1 360 360
- Macaulay duration
4.- Once again find the variation of the market price without using the concept of duration.
Reply
1+ R
Thus the price of a perpetual bond with these conditions is P = B. The first derivative R turns out to be negative dP dR = - R B 2 ⇒ - P 1 dP dR = P 1 R B 2 but remember
that the price of the bond is P = B, so its duration is D * = 1. The
convexity is obtained in a similar way. d dR 2 P = 2 B 3 ⇒ C = P 1 d dR 2 P 2 = B R 2 R B 3 = R 2 2 .
2 R
7.- Inquire the values of the duration and convexity of a zero coupon bond.
Reply
It is very easy to solve this question because the formulas obtained with C t = 0 are simply applied.
To estimate the VaR of a bond, it is necessary to use the modified duration.
It is known that dP P = - D *. To calculate the volatility of the relative prices of the bond, the right member of this equality must be multiplied and divided by R and then the variance calculated.
var æ ç dP = var æ - D * R dR = (- D * R) 2 var æ ç dR these matches can be
P R R
volatility σ P of the price of the bond from the volatility of changes in interest rates σ R.
σ P = D * Rσ R
To estimate the value at risk, the volatility of the bond price is multiplied by a factor appropriate to the desired level of confidence.
Where
Example: Suppose a bond with a market price of CU100. whose modified duration is 3.98 with quarterly volatility of interest rates of 3% and R 0 = 10%. Estimate the VaR at a 99% confidence level.
The maximum loss at a 99% confidence level is expected to be CU70.
DURATION AND CONVEXITY OF A PORTFOLIO OF BONDS
The concept of duration can be extended in the participation of more than one bond. Taylor's theorem is again the vehicle for achieving the duration of a bond portfolio.
Consider a portfolio made up of K bonds where P i (R i) and m i are the price and the number of units of the i-th bond. The value of the portfolio P = P (R 1, R 2,…, R K) is the sum of the amounts invested in the bonds that compose it. With these assumptions we have the following theorems:
Theorem 1. The duration D p of a portfolio of K bonds is the weighted average of the particular durations. The weighting w i of the i-Øsima duration is the percentage of portfolio that represents the i-Øsimo bond.
Demonstration.
By construction P = ∑ m i P i (R i). The total differential of P (R 1, R 2,…, R K) is
If we multiply and divide by i by the i-th sum of dP, then we have that dP = ∑ iK = 1 m i P i D i * dR i. Defining w i = m P i P i and dividing dP by P gives the desired result.
case i = j leads to dP = ∑ i = 1 ∂ R ii + 2 ∑ i = 1 mi ∂∂ R 2 P i 2 (dR i) 2. One more time
multiply and divide by P i by the i-th sum of each sum and divide by P through dP to obtain the expected result.
The values obtained for the duration and convexity of the portfolio are used in the same way as in the case of an individual bond.
SUMMARY
After reading this presentation, the reader should bear in mind the following:
- The duration indicates the degree of sensitivity of the value of a portfolio of fixed income instruments to changes in the interest rate. The duration of Macaulay indicates the average time in which the investor receives the payment of the principal. Before large variations in the interest rate, the convexity of the bond is used to adjust the estimate obtained through the duration. The duration (convexity) of a portfolio is the weighted average of the durations (convexities) of each bond.
REFERENCES
- Fabozzi, Frank (2000). Bond Markets Analysis and Strategies. Prentice Hall. Lara Haro (2005), Alfonso de. Control and measurement of financial risks. Limusa.
Recently, regional banks in Germany and the Austrian government have placed bonds in Mexican pesos giving rise to the Europeso.
There is also a risk of currency mismatch.
Prepaid bonds have negative convexity but require somewhat different treatment.
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