imported>Doug Williamson |
imported>Administrator |
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| (EAR). | | (MC). Broadly speaking, modified convexity measures the curvature of an instrument’s price function, as yields change - from a given starting point - by a small amount. |
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| | More strictly, it is the rate of change of modified duration with respect to yield - at the given starting yield. |
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| | Modified convexity can be calculated from Convexity as follows: |
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| A quoting convention under which interest at the quoted effective annual rate is calculated and added to the principal annually.
| | '''Modified Convexity = C<sub>MOD</sub> = Convexity / (1+r)<sup>2</sup>''' |
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| EAR is the most usual conventional quotation basis for instruments with maturities of greater than one year.
| | The estimation of price change for a given small change in yield can then be calculated as follows: |
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| | Price change estimation using Modified Duration (MD) only: |
| | = - Price x MD x Change in yield |
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| 2. | | Price change estimation using Modified Convexity (C<sub>MOD</sub>): |
| | = - [Price x MD x (Change in yield)] + ½ x [Price x C<sub>MOD</sub> x (Change in yield)<sup>2</sup>] |
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| A conventional measure which usefully expresses the returns on different instruments on a comparable basis.
| | Because the value v yield relationship is a curve and not a straight line (values do not change linearly as yields change) the estimate of change in value using only modified duration will generally underestimate the new value (because the curve lies above its tangent). Therefore the modified convexity adjustment is always positive - it always adds to the estimate of the new price whether yields increase or decrease. |
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| The EAR basis of comparison is the ''equivalent'' rate of interest paid and compounded annually, which would give the same all-in rate of return as the instrument under review.
| | It is also possible to estimate the MD and the C<sub>MOD</sub> from given observations of Price and Yield, by rearranging them to solve for MD and C<sub>MOD</sub> - effectively running the price change estimation formulae in the other direction. |
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| For this reason, 'EAR' is sometimes expressed as <u>equivalent</u> annual rate.
| | == See also == |
| | | * [[Convexity]] |
| | | * [[Matching]] |
| | | * [[Modified duration]] |
| ==Conversion formulae==
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| ====Nominal annual rate to periodic rate====
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| r = R / n
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| ''Where:''
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| r = periodic interest rate or yield
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| R = nominal annual rate
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| n = number of times the period fits into a conventional year (for example, 360 or 365 days)
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| ====Periodic interest rate or yield to Effective annual rate====
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| EAR = (1 + r)<sup>n</sup> - 1
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| ''Where:''
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| EAR = effective annual rate or yield
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| r = periodic interest rate or yield, as before
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| n = number of times the interest calculation period fits into a calendar year of 365 days (or 366 days in a leap year)
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| ==Calculating EAR from overnight quotes==
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| <span style="color:#4B0082">'''Example 1: EAR from overnight quote'''</span>
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| GBP overnight interest is conventionally quoted on a simple interest basis for a 365-day fixed year.
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| So GBP overnight interest quoted at R = 5.11% means:
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| Interest of:
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| r = R / n
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| r = 5.11% / 365
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| r = 0.014% (= 0.00014) is paid per day.
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| (ii)
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| The ''equivalent'' effective annual rate is calculated from (1 + r).
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| 1 + r = 1 + 0.00014 = 1.00014
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| EAR = (1 + r)<sup>n</sup> - 1
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| EAR = 1.00014<sup>365</sup> - 1
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| EAR = '''5.2424%'''.
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| <span style="color:#4B0082">'''Example 2: EAR from 360-day overnight quote'''</span>
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| USD short term interest is conventionally quoted on a simple interest basis for a 360-day year.
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| So USD overnight interest quoted at R = 5.11% means:
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| (i)
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| Interest of:
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| r = R / n
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| r = 5.11% / 360
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| r = 0.01419444% (= 0.0001419444) is paid per day.
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| (ii)
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| The ''equivalent'' effective annual rate is calculated from (1 + r).
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| 1 + r = 1 + 0.0001419444 = 1.0001419444
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| EAR = (1 + r)<sup>n</sup> - 1
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| EAR = 1.0001419444<sup>365</sup> - 1
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| EAR = '''5.3171%'''.
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| <span style="color:#4B0082">'''Example 3: EAR in a leap year'''</span>
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| The strict calculation of the effective annual rate is based on the prevailing calendar year, which is 365 days in a normal year, and 366 days in a leap year.
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| For the same periodic rate of interest (r), the effective annual rate is greater in a leap year.
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| For example, where (r) = 0.00014 overnight (as in Example 1).
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| The number of times (n) that the one-day period fits into the calendar year in a leap year = 366.
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| EAR = (1 + r)<sup>n</sup> - 1
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| EAR = 1.00014<sup>366</sup> - 1
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| EAR = '''5.2572%'''.
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| == See also ==
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| * [[AER]]
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| * [[ACT/365 fixed]]
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| * [[Annual effective rate]]
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| * [[Annual effective yield]]
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| * [[Annual percentage rate]]
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| * [[Calculating effective annual rates]]
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| * [[Capital market]]
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| * [[Certificate in Treasury Fundamentals]]
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| * [[Certificate in Treasury]]
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| * [[Continuously compounded rate of return]]
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| * [[Effective annual yield]]
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| * [[Equivalent Annual Rate]]
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| * [[Leap year]]
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| * [[LIBOR]]
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| * [[Nominal annual rate]]
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| * [[Periodic discount rate]]
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| * [[Periodic rate of interest]]
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| * [[Periodic yield]]
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| * [[Rate of return]]
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| * [[Real]]
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| * [[Return]]
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| * [[Semi-annual rate]]
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(MC). Broadly speaking, modified convexity measures the curvature of an instrument’s price function, as yields change - from a given starting point - by a small amount.
More strictly, it is the rate of change of modified duration with respect to yield - at the given starting yield.
Modified convexity can be calculated from Convexity as follows:
Modified Convexity = CMOD = Convexity / (1+r)2
The estimation of price change for a given small change in yield can then be calculated as follows:
Price change estimation using Modified Duration (MD) only:
= - Price x MD x Change in yield
Price change estimation using Modified Convexity (CMOD):
= - [Price x MD x (Change in yield)] + ½ x [Price x CMOD x (Change in yield)2]
Because the value v yield relationship is a curve and not a straight line (values do not change linearly as yields change) the estimate of change in value using only modified duration will generally underestimate the new value (because the curve lies above its tangent). Therefore the modified convexity adjustment is always positive - it always adds to the estimate of the new price whether yields increase or decrease.
It is also possible to estimate the MD and the CMOD from given observations of Price and Yield, by rearranging them to solve for MD and CMOD - effectively running the price change estimation formulae in the other direction.
See also
