Convexity And Duration

Combined Effects: Duration and Convexity

Duration and Convexity Negative Duration Positive and Negative Convexity Positioning :

Callable Bond Price = Noncallable Bond Price less Call Option Price

Effective Convexity Calculated using cash flows Positive Convexity vs Negative Convexity

Convex vs Concave Effective Convexity :

P + P - 2P +

_

0

----------------------------------------

2 2P0 ( ∆Ί)

Sample Problem Due to the change of interest

movement, from 8% to 7% in the market rate, the bond price resulted to 99.0768 and 97.2691 from its original value of 98.166. What would be its computed effective convexity? What would be its estimated price change?

Solution:  97.2691 + 99.0768 – 2(98.166)

-----------------------------------2 2(98.166)(1%)

= .7079

2 - .9108 (-1%)*100 + .7079 * (-1%)

x 100 = .2029

Combined Effects Given : 18 year bond, 12% coupon, 9% YTM Price : 126.50 Modified Duration : 8.38 Convexity : 107.70 Interest change : 100 bps

Calculate the combined effects of duration and convexity.

Solution Change in Yield : -100BP Duration change : -8.38 x (-100/100) =+ 8.38%

8.38% x 126.50

= $10.60

2

Convexity Change: ½ x (126.50) x 107.70 x (.01)

6,812.03 x .0001

=$ .68

Combined Effect : 126.50 +10.60 +.68 = $137.78

Immunization matches the durations of assets and liabilities thereby minimizing the

impact of interest rates on the net worth ensures that a change in interest rates will not affect the value of a security (Frank Redington) Ways to Immunize: -matching the duration, cash flow, volatility and convexity  trading thru derivatives (forward, futures and options) If the immunization is incomplete, the strategies are usually hedging while If immunization is complete, the strategies are usually arbitrage.

Passive and Active Strategies in Bonds Benchmarking

benchmark interest rate = “base interest rate” - minimum interest rate investors will accept for investing in a non-Treasury security ; either an estimated yield curve or an estimated spot rate curve. It is the yield that is earned on the most recent “on the run” Treasury security. Relative Value Analysis : identifies securities being overpriced (“rich”) or underpriced (“cheap”) or fairly priced thru benchmark interest rates *spread measure depends on the benchmark used Option Adjusted Spread (OAS) (spread of the forward rates in the Interest rate tree)

Interest Rate Tree Constructed using a process similar to “bootstrapping” with interest

rates that will produce a value for the “on the run” issues The illustration may explain: 9.60% 8.48% 7.50%

1. Backward induction 7.86%

6.94% 6.43%

Entrepreneurship perspective Finance perspective

2. arbitrage free 3. Interest rate shock

Valuing Bonds with Embedded Options Valuation models can give different values for the

same bond depending on the assumptions: 1. volatility assumption 2. benchmark interest rates 3. call rule However, 1. the OAS is constant even when interest rates change 2. an OAS of zero means that the issue is fairly priced

Callable Bonds – Conversion Conversion ratio = Par Value / Conversion Price  Conversion value = (market price of stock) (conversion ratio)  Market Conversion Price = Market Price of Convertible Bond Conversion ratio

 Conversion Premium per Share = MCP - MPS  Conversion Premium Ratio = CP per share / MPS  Premium Over Straight Value = MPCB / Straight value – 1  Income Differential : Coupon interest from bond-(conversion ratio x dividend/share) conversion ratio

 Premium Payback Period = Market Conversion Premium/ID

Sample Problem A convertible bond is priced at $900 with coupon

interest of $85 while the common stock is worth $25 which earns a dividend of $1 per share. If the conversion ratio is at 30 and straight value of the bond is estimated to be at $700, calculate the following: 1.1 1.2 1.3 1.4 1.5 1.6 1.7

conversion value market conversion price conversion premium per share conversion premium ratio premium over straight value income differential premium payback

Solution: 1.1 1.2 1.3 1.4 1.5 1.6 1.7

CV = $25 x 30 = $750 MCP = $900 / 30 = $30 CP/share = $30 - $25 = $5 CP Ratio = $5/$25 = 20% P/SV = $900/$700 – 1 = 28.6% ID = $85- (30 x $1) / 30 =$1.833 PB = $5/$1.833 = 2.73 years

Bond Strategies Horizon Matching and Immunization

Investment Horizon = Duration

Immunized

YTM

Flat Interest Rates : wealth position is compounded e.g $1,000,000 @ 8% semi annual coupon rate for 20yrs 20 $1,000,000 x (1.04) = $2,191,123

Bond Strategies Risk for Not having Flat Interest Structure

Interest Rate Risk 1. Price Risk : Duration > Investment Horizon 2. Reinvestment Risk Duration < Investment Horizon

Illustrations: Interest Risks on Bonds Bond A : a 10 yr bond, paying a 9% annual coupon, YTM of 10% and sell it in 3 years at 8% (prevailing market rate)

Initial Purchase = 10 90/(1+.10) + 1,000/(1+.10)

= $938.55 7 Ending Purchase = 1,000/(1+.08) =$1,052.06 (P) 2 dd =90(1+.08) +90(1.08) + 90 = $292.18 (C) = $1,344.24 3 RY = √ 1,344.24/938.55 – 1 = 12.72%

Illustrations: Interest Risks on Bonds Bond B: 3 consecutive one-year “pure discount” bond, YTM of 10%, market rate declined to 8% for 2nd & 3rd yr. with initial purchase of 1,000 (roll over strategy)

Year 1 : (1,000)(1+.10) = $1,100.00 Year 2 : (1,100)(1.08) = 1,188.00 Year 3 : (1,188) (1.08) = $1,283.04

RY =

3 √ 1,283.04 / 1,000 - 1 = 8.66%

Illustrations: Interest Risks on Bonds Bond C : three-year pure discount bond, YTM of

10%, 8% market rate after 3 years (straight) Initial Purchase : 3 1,000 / (1+.10) = $751.31 3 RY = √ 1,000/751.31 – 1 = 10.00%

Sample Problem Assume that you have these bonds with YTM as 12%,

but the market rate flanked to 9% after 3 years and projected to be steady by this rate for the next 3 more years. Under this scenario, would you recommend to sell on the 4th year? What interest rate risks are these bonds exposed of? How will you immunize? Bond X = 5 years; annual coupon of 6% Bond Y = 4 year consecutive one year pure discount bond Bond Z = 4 year pure discount bond