Properties concerning the price volatility of an option free bond:
· Price is inversely related to yield.
· Price sensitivity depends on maturity and coupon.
· For small changes in yield, the price change is same whether yield moves up or down.
· For large changes in yield, the price increase for a fall in yield is greater than the price decrease for a similar rise in yield, i.e. the price yield curve is convex.
· The lower the coupon, the longer the term to maturity, the lower the initial yield -> the greater the bond price volatility
· Yield/price relationship for option free bonds is not linear but convex, referred to as positive Convexity.
Positive convexity:
· The decrease in an option-free bond’s price due to a rise in yield is lower than the increase for an equal fall in yield. This results a positive adjustment to the duration-based estimate of price change whether yield goes up or down.
Price-Volatility Characteristics of Callable and Prepayable Securities
· Embedded options can dramatically change the price-yield profile.
· At high yields the profiles of these bonds are similar, but at low yields price of callable/prepayable securities grows slower with a fall in yield than the price of option-free bonds {price compression}.
Negative convexity:
· For callable and prepayable securities, at low levels of yield, the price increase for a given fall in yield is less than the price decrease for an equal rise in yield. ie, Prices rise at a decreasing rate. The point where the curve starts to flatten is at ( or near a yield level of y’
Price Volatility Characteristics of Putable Bonds
· The advantage of these bonds to an investor is that if market yields rise and the value of the bond falls below the put price, the investor can exercise the put option and stem his losses to the put price.
· The price of a puttable bond will react same way as an option-free bond at low yield levels. As rates rise, the puttable bond’s price will decrease at the same rate as an option-free bond, but the decline will be lessened because of the value of the put option.
Value of puttable bond = value of option free bond + the option.
Showing posts with label Measurement of interest rate risk. Show all posts
Showing posts with label Measurement of interest rate risk. Show all posts
Thursday, January 1
Convexity
Convexity is a measure of the degree of curvature of the price/yield relationship, to indicate the error in the estimated change in bond's price based on duration.
Effective convexity = (V_ + V+ - 2 x Vo) / (2 x Vo x dy2).
It is the 2nd derivative of price function wrt yield. For callable bond, V is capped at call price.
Contribution of convexity = Convexity x dy2.
For noncallable bond, convexity effect is always +ve no matter which direction interest rate move but it can be –ve if the bond has embedded options.
Thus, approximate bond price change (using duration and convexity), i.e.
Approx. price change = -1 x Duration x dy + Convexity x dy2
Modified Convexity vs. Effective Convexity
With modified convexity the cash flows do not change due to a change in interest rates.
Effective Convexity, on the other hand, assumes that cash flow does change due to a change in interest rates.
When bonds have options, it is best to use effective convexity just like you should use effective duration. For option-free bonds, either convexity measure will be a positive value, whereas when it comes to bonds with options, the effective convexity could be negative even if the modified convexity is positive.
Effective convexity = (V_ + V+ - 2 x Vo) / (2 x Vo x dy2).
It is the 2nd derivative of price function wrt yield. For callable bond, V is capped at call price.
Contribution of convexity = Convexity x dy2.
For noncallable bond, convexity effect is always +ve no matter which direction interest rate move but it can be –ve if the bond has embedded options.
Thus, approximate bond price change (using duration and convexity), i.e.
Approx. price change = -1 x Duration x dy + Convexity x dy2
Modified Convexity vs. Effective Convexity
With modified convexity the cash flows do not change due to a change in interest rates.
Effective Convexity, on the other hand, assumes that cash flow does change due to a change in interest rates.
When bonds have options, it is best to use effective convexity just like you should use effective duration. For option-free bonds, either convexity measure will be a positive value, whereas when it comes to bonds with options, the effective convexity could be negative even if the modified convexity is positive.
Price value of a basis point (PVBP)
A measure of bond price volatility that shows the extent to which the price of a bond will change when the required yield changes by one basis point. That is, the difference between the initial price and the price if yield is changed by 1bp, a variation of dollar duration.
PVBP = Duration x Price / 10,000
PVBP = Duration x Price / 10,000
Approximate % change in bond price
Approximate % change in bond price = (-) (duration)(dy)
Duration tends to:
· underestimate the increase in price that occurs with a drecrease in yield; overestimate the decrease in price that comes with an increse in yield.
· The difference is due to the curvature of the actual price path.
· The larger the change in yield, the larger the error.
· For small changes, estimated and actual price changes are equal or very close.
Duration
Duration is a measure of the slope of the price-yield function, steeper at low interest rate and flatter at high interest rate for non-callable bonds. It represents the percentage change in price for a 100 basis point change in yield.
Effective Duration
Duration is the approximate percentage change in price for a 100 basis point change in rates.
Effective Duration = (V_- V+) / (2 x Vo x dy in decimal).
Note:
· go down or up by same no. of basis points
Modified Duration
Modified duration is the approximate percentage change in a bond’s price for a 100 basis points change in yield, assuming that the bond’s expected cash flow does not change when the yield changes. This works for option-free bonds such as Treasuries but not with option-embedded bonds because the cash flows may change due to a call or prepayment.
Macaulay duration
Macaulay’s duration is the weighted average number of years remaining to receive the present value of a bond.
It gives the analysis a short cut to measure modified duration. But because modified duration is flawed by not incorporating the change in cash flows due to an embedded option, so are Macaulay durations.
Modified duration = Macaulay’s Duration/ (1 + yield/k)
Macaulay duration = Modified duration x (1 + BEY/2).
Notes:
· The duration of zero coupon bond = its maturity;
· Duration of a floater coupon bond = the time to the next reset date
When is Effective Duration a Better Measure?
When a bond has an embedded option, the cash flows can change when interest rates change because of prepayments and the exercise of calls and puts. Effective duration takes into consideration the changes in cash flows and values that can occur from these embedded options.
Duration of a portfolio
Duration of a portfolio equals the weighted average of the durations of the bonds in the portfolio.
Effective Duration
Duration is the approximate percentage change in price for a 100 basis point change in rates.
Effective Duration = (V_- V+) / (2 x Vo x dy in decimal).
Note:
· go down or up by same no. of basis points
Modified Duration
Modified duration is the approximate percentage change in a bond’s price for a 100 basis points change in yield, assuming that the bond’s expected cash flow does not change when the yield changes. This works for option-free bonds such as Treasuries but not with option-embedded bonds because the cash flows may change due to a call or prepayment.
Macaulay duration
Macaulay’s duration is the weighted average number of years remaining to receive the present value of a bond.
It gives the analysis a short cut to measure modified duration. But because modified duration is flawed by not incorporating the change in cash flows due to an embedded option, so are Macaulay durations.
Modified duration = Macaulay’s Duration/ (1 + yield/k)
Macaulay duration = Modified duration x (1 + BEY/2).
Notes:
· The duration of zero coupon bond = its maturity;
· Duration of a floater coupon bond = the time to the next reset date
When is Effective Duration a Better Measure?
When a bond has an embedded option, the cash flows can change when interest rates change because of prepayments and the exercise of calls and puts. Effective duration takes into consideration the changes in cash flows and values that can occur from these embedded options.
Duration of a portfolio
Duration of a portfolio equals the weighted average of the durations of the bonds in the portfolio.
Duration/convexity approach
To provide an approximation of the actual interest rate sensitivity of a bond or bond portfolio, simplier than full valuation approach.
Full Valuation Approach
To measure the interest rate risk by re-valuing the bond or portfolio for a given interest-rate change scenario, referred to as a scenario analysis. Work well for periodic reports but impractical for managing risk of large portfolio.
Steps:
1. Start with current market yield and price
2. Estimate chagnes in yields
3. Revalue bonds
4. Compare new value to current value
It take the changes in cash flows into account.
Steps:
1. Start with current market yield and price
2. Estimate chagnes in yields
3. Revalue bonds
4. Compare new value to current value
It take the changes in cash flows into account.
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