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# Convexity formula

#### mathman

##### New Member
Hi,

There are 2 formulas for convexity that I know:

1. C = [ BV(2) + BV(1) - 2*BV(0) ] / BV(0) * (delta y)^2

2. C = Summation of ( year number)^2 * Present Value of Cash Flow / Current Bond Value

Now I did a question using both methods and got 2 different answers.

Here is the question: What is the convexity of a 5% 2 year Semi Annual Bond when the market rate is 4.5%.

Soln1: C= 4.56 and Soln2: C = 3.8 (different solutions!)

My complete solutions can be found on the image I uploaded. Please let me know if there is something wrong.

Thanks.

#### ShaktiRathore

##### Well-Known Member
Subscriber
Try the formula as:
C= Summation of (( year number)^2+( year number)) * Present Value of Cash Flow / Current Bond Value*(1+y)^2

this is as given in john hull.

#### mathman

##### New Member
Still doesnt give me 4.556

When using this formula: C = [ 1/ {B * (1 + y)^2 } ] * Sum of [ { t^2 + t } * PVCF ] what should be the value of y when the yield is semi annual? So a 4.5% semi annual rate would come out to be 1.045 squared or should be 1.0225 to the power 4? Either way the answer comes to 5.22 to 5.24

#### ShaktiRathore

##### Well-Known Member
Subscriber
Convexity calculation
coupon 5%
y 9.20% you need to convert it into annual yield
BondFaceValue 100

t | CFt | |PV of CFt | PV of CFt*(t*(t+1))
0.5| 5 | 4.784689 | 3.588517
1 | 5 | 4.57865 | 9.1573
1.5 | 5 | 4.381483 | 16.43056
2 | 105 | 88.04894 | 528.2936
sum | 101.7938 | 557.47

BVo=summation of PV of CFt =101.7938
summation[CFt/(1+y)^t]*(t*(t+1))=557.47
Convexity=(1/1+y)^2*(1/Bvo)*summation[CFt/(1+y)^t]*(t*(t+1))
=(1/1+9.2%)^2*(1/101.7938)*557.47
= 4.592352

See the above one it comes nearly close to yours 4.55. ask if any doubt in above.

thanks

#### mathman

##### New Member
Hi, Thanks for looking into it. But the coupons are 2.5 each, coz its a 5% semi annual bond. yielding to give 4.5% for 2 years.

And yes, the formula I initially had was incorrect. The one you gave was listed on many sites where you find sum of t(t+1) PVCF as the numerator and in the denominator have bond value times (1+y/2) squared................or should it be (1 +y/2) to the power 4?

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hey mathman,
coupons are given 5 . 5 is 6 month coupon rate because coupons are always given every six months.
5% 2 year Semi Annual Bond=>The rate here implies that coupons are paid every six months at rate 5% of the face value at semi annual yield which is 4.5%.This is how we read.
No dont change the yield take it as it is which can be yearly or semi annual yield which u used to calculate the PV's of CFs.
thanks

#### mathman

##### New Member
Hmm. Okay. Since I designed the question: suppose that the coupons are 2.5, 2.5, 2.5, 102.5. The market yield is 4.5%

#### ShaktiRathore

##### Well-Known Member
Subscriber
In this case nothing will change except that coupon rate now is 2.5% down from 5%. You just need to replace 5% with 2.5% as coupon rate for bond or convexity calculations.Now Bond is paying 2.5% on FV of bond every six months . In that case bond values will change and consecutevely the convexity also.

thanks

#### mathman

##### New Member
Here is the calculation:

t(t+1) * PVCF = 1.84, 4.78, 8.77, 562.62 = 578.01

denominator: B ( 1 + y )^2 .... Now B = 100.95 and taking y = 4.5% we get this = 110.24

so its: 5.24

#### ShaktiRathore

##### Well-Known Member
Subscriber
see you are taking t as .5,1,1.5 and also taking semi annual yields for calculation so make t as 1,2,3,4 above and see the results.Since your rates are semi annual its necessary that you take period wise PVs instead of yearwise. We always calculate the PVs looking into periods and yield over that period.
I consider that you have not understood it. Please refer to any book for further clarification.
thanks

#### mathman

##### New Member
Taking higher t values will only increase the convexity value. still wont get 4.55

Thanks.

#### mathman

##### New Member
Can you please calculate the value according to the data provided, and give me a value. Its a very very simple calculation.

#### mathman

##### New Member
i have sent u a request to be able to view that.

#### mathman

##### New Member
The coupons are 2.5 every 6 months. Why do you keep taking 5? I designed the question. The market rate is 4.5% compounded semi annually.

#### mathman

##### New Member
I love you man. Thanks for helping. But I am now telling you forget 5. Take 2.5 Why not answer the question as it is? Why?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
this is the same as i just answered here, yes? http://www.bionicturtle.com/forum/threads/convexity-calculation.6257/
i think the mistake above is the multiplier in the third column, it is not semi-annual, the third column, under s.a., should be PV(cash flow)*T*(T+0.5), then the sum of that column (480.6822) is multiplied by 1/[price*(1+4.5%/2)^2] = 4.554508.