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MACAULAY DURATION

Macaulay duration provides a measure of the risk of a bond. It is the weighted-average term to maturity of the cash flows from a bond, expressed in number of years.

Cpn: Amount of paid coupon
Rbt: Redemption value
YTM: Yield to maturity
Freq: Number of coupons paid per year
N: Total number of coupons to be received
a: Fraction of year covered by the accrued coupon

The result of this calculation can be interpreted in two ways :

Effective Life
The Macaulay duration shows a bond’s effective life, that is, the time a bond must be held in order to recover one’s investment. An investor wishing to recover an investment will prefer to invest in a bond with the shortest possible effective life, that is, a bond with a short duration .

Approximation of Price Sensitivity to Interest Rate Variations
The Macaulay duration is the first derivative of the function between price and yield at a certain point (the slope of the tangent). If duration is an essential concept for professionals, it is because it allows the price variation of a bond to be determined by small variations in yield. In other words, an approximation of the function is made along the slope of its tangent. The bond with the longer duration will have a larger price variation, and therefore will be the riskiest .

Although the formula used to calculate the Macaulay duration gives a positive number, the inverse relationship between price and yield must always be kept in mind .

Percentage variation in price = (- duration) x (percentage variation in yield)

The P(y) curve represents the evolution of the bond’s price according to the evolution of interest rates. This relationship is convex, not linear.

The limit for using the duration is determined by the mathematical tool used. The use of the duration tends to overestimate this variation in case of rising interest rates and underestimate it in times of declining rates. This phenomenon is due to the fact that the price of a bond is a decreasing and convex function with regard to the rate, and not a linear function as one might suppose by using the duration. Consequently, the results are valid only for slight variations in interest rates. Therefore, the use of this concept would not be advisable for abrupt rate movements .

Duration Properties

Duration and Life Span
With regard to the effective life, it is intuitively understood that the duration cannot be longer than the bond’s life, and that the duration will be shorter than maturity as soon as a coupon is paid. Only zero-coupon bonds have a duration matching maturity. Consequently, it can also be understood that all other things being equal, a bond with a longer maturity will have a longer duration .

Duration and Coupon
The higher the coupon, the shorter the duration. It is easy to understand this inverse relationship between the coupon level and duration by referring to the effective life of the bond. If the coupon is high, the investor recovers the investment more quickly .

Duration and Interest Rate Levels
It is common knowledge that the lower the interest rates, the higher the present value of a payment to be received in the future. Therefore, by referring to effective life, the investor will have a shorter wait to recoup the initial investment by updating the cash flows at high yields. Consequently, a decline in interest rates increases the duration.

Example : What will the price of the bond shown below be if rates vary + or – 10 bps ?

A yield variation equal to +1% will result in a price variation of –17.027%.
A yield variation equal to +0.10% will result in a price variation of -1.7027
Y = 3.76% => estimated price = (119.93 – 1.7027%(119.93)) = 121.96% as opposed to the actual 121.93%.
Y = 3.96% => estimated price = (119.93 + 1.7027%(119.93)) = 117.88% as opposed to the actual 117.95%.

MODIFIED DURATION

Modified duration is very close to Macaulay duration in that it also provides an estimation of the range of price variation for a given yield variation .

The result obtained is a percentage rather than a duration in years .

Keeping in mind that the ratio between price and yield is negative, and consequently that a lower figure indicates a flatter slope, the use of modified duration better takes into account the convex nature of the ratio between price and yield when rates are rising .

As long as yield is a positive number, the modified duration is always shorter than the Macaulay duration.

In case of rising interest rates, the price variation calculated with the modified duration is closer to the actual situation. In comparison with the Macaulay duration, the estimated price reduces the overestimation of the loss.

Example : What will the price of the bond shown below be if rates vary + or – 10 bps ?

Modified duration (P = 119.93%) = 17.027 / (1 + 0.0386) = 16.394

Rdt = 3.76% => estimated price = (119.93 – 1.6394%(119.93)) = 121.88%
as opposed to 121.96% estimated with Macaulay duration and 121.93% in reality.

Rdt = 3.96% => estimated price = (119.93 + 1.6394%(119.93)) = 117.95%
as opposed to 117.88% estimated with Macaulay duration and 117.95% in fact.

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