Group Members:

Sumit Arya, 106Amit Bansal, 107

Sourav Gupta, 123Firoz RV, 145

Prashant Shukla, 155Ankur Agrawal, 205Gaurav Kumar, 230

1. Interest Rate Risk2. Ways of measuring Interest Rate Risk3. Full valuation approach

Example4. Duration/Convexity Approach

Duration What is duration? Calculation Properties Application Limitations

Convexity Introduction Predicting percentage price change using convexity Examples Convexity for callable and putable bonds Applications

Changes in Bond prices due to interest rates fluctuations

Formula of Bond Value calculation.

Where Po=value of bond c1,c2,…,cn=cash flows expected from bond

over ‘n’ periods r=discount rate

Full Valuation Approach

Duration - Convexity Approach

The full valuation approach to measuring the interest rate risk involves using a pricing model to value individual bonds and can be used to find the price impact of any scenario of interest rate/yield curve changes.

Advantages of this approach:Precision ComplexFlexibility

The Full Valuation ApproachScenario Yield ∆ Bond X

(in millions)Bond Y

(in millions)Portfolio Portfolio

Value ∆%

Current +0 bp ($108.42) ($81.78) ($190.21) 1 +50 bp ($106.23) ($77.93) ($184.17) -3.18%2 +100 bp ($104.10) ($74.32) ($178.42) -6.20%

-2.02%

-3.99%

-4.71%

-9.12%

Bond X Bond Y

Coupon 8% 5%

Maturity (yr) 5 15

FV($) 100 100

YTM 6.00% 7.00%

This approach provides an approximation of the actual interest rate sensitivity of a bond or a bond portfolio.

Its main advantage is itsSimplicityLesser Time for calculation

1. Duration is the slope of the price-yield at the current YTM.

2. Weighted average of time until each cash flow would be received. Weights are proportions of total bond value that each cash flow presents

3. Approximate percentage change in price for 1% change in yield

Example: 20yr, 8% coupon PV=908 If yield decreases by 50 bps, price=952.3 If yield increases by 50 bps, price=866.8

1. Coupon Rate: Lower Coupon rate means Higher duration

2. Maturity: Longer Maturity means Higher duration

The Full Valuation ApproachScenario Yield ∆ Bond X

(in millions)Bond Y

(in millions)Portfolio Portfolio

Value ∆%

Current +0 bp ($108.42) ($81.78) ($190.21) 1 +50 bp ($106.23) ($77.93) ($184.17) -3.18%2 +100 bp ($104.10) ($74.32) ($178.42) -6.20%

-2.02%

-3.99%

-4.71%

-9.12%

3. Higher market yield means lower duration.

Price Yield Curve for an option free bond 8% coupon rate, 20 year bond

It is a good measure of sensitivity of a portfolio, and can be used to reduce or increase the exposure to a particular term interest rate risk

1) Large changes in interest rates

2) This approach is applicable for a portfolio of bonds with only parallel yield curve shifts.

Measures how much a bond’s price-yield curve deviates from a straight line

Second derivative of price with respect to yield divided by bond price

Allows us to improve the duration approximation for bond price changes

Duration underestimates actual prices Previous Example: 20yr, 8% coupon PV=908

Duration : 9.42

Recall approximation using only duration:

The predicted percentage price change accounting for convexity is:

PP

100 Dm* y100

PP

100 Dm* y100

1

2Convexity (y)2 100

Comparison of bonds: A bond with greater convexity is less

affected by interest ratesA bond with greater convexity will have

higher prices than bonds with lower convexity, regardless of interest rate rise or fall

FIN 509Class session 2 20

Consider a 20-year 9% coupon bond selling at $134.6722 to yield 6%. Coupon payments are made semiannually.

Dm= 10.66

The convexity of the bond is 164.106.

FIN 509Class session 2 21

If yields increase instantaneously from 6% to 8%, the percentage price change of this bond is given by: First approximation (Duration):

–10.66 .02 100 = –21.32

Second approximation (Convexity)

0.5 164.106 (.02)2 100 = +3.28

Total predicted % price change: –21.32 + 3.28 = –18.04%

(Actual price change = –18.40%.)

FIN 509Class session 2 22

What if yields fall by 2%? If yields decrease instantaneously from 6% to 4%,

the percentage price change of this bond is given by: First approximation (Duration):

–10.66 –.02 100 = 21.32

Second approximation (Convexity)

0.5 164.106 (–.02)2 100 = +3.28

Total predicted price change: 21.32 + 3.28 = 24.60%

Note that predicted change is NOT SYMMETRIC.

With a callable or prepayable debt, the upside price appreciation in response to decreasing yields is limited.

When the price begins to rise at a decreasing rate in response to further decreases in yield, the price yield curve bends over to the left and exhibits negative convexity.

Compared to an option-free bond, a putable bond will have less price volatility at higher yields.

Convexity is a interest rate risk management tool, which is used in managing bond portfolios

If the combined convexity and duration of a trading book is high, so is the risk. However, if the combined convexity and duration are low, the book is hedged, and little money will be lost even if fairly substantial interest movements occur.

Fixed Income, Derivatives, and Alternative Investments – CFA L 1, Kaplan Schweser

Valuation of Financial Assets – A.S.Ramasastri

Presentation on the session “Duration and Convexity” by Prof. Jonathan Karpoff

www.investopedia.com

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