Description
This is a PPT about bond pricing and mathematics.
Bond Mathematics
Types of Yields Coupon Yield Nominal yield is the annual interest rate percentage payable specified in the indenture and printed on the face of the bond certificate. Nominal is fixed and not related to market value. Current Yield It is calculated by dividing annual interest income by the current market price. This yield takes into account the relationship between the interest received and the actual investment made. • Current yield is greater than the coupon rate if the bond is selling at a discount. • Current yield is less than the coupon rate if the bond is selling at a premium. • Current is equal to the coupon rate if the bond is available at a par value.
Yield to Maturity It is an average rate of return involving collective consideration of a bond
(a) Interest rate (b) Current price (c) Number of years remaining until maturity.
If the bond is available at a discount then, Coupon + Prorated Discount Yield to Maturity = (Face Value + Purchase Price)/2 If a bond is available at premium then, Coupon - Prorated Premium Yield to Maturity = (Face Value + Purchase Price)/2
Use of Time Value of Money Concept in Computation of YTM and Bond Pricing Bond Pricing The value of a bond - or any asset, real or financial - is equal to the present value of the cash flows expected from it. Hence determining the value of a bond requires : • An estimate of expected cash flows.
• An estimate of the required return.
• The coupon interest rate is fixed for the term of the bond. • The coupon payments are made every year and the next coupon payment is receivable exactly a year from now. • The bond will be redeemed at par on maturity.
C M ??? ? t n (1 ? r ) t ?1 (1 ? r )
Where ? = value (in rupees) n = number of years C = annual coupon payment (in rupees) r = periodic required return M = maturity value t = time period when the payment is received
?
C/2 M ??? ? t (1 ? r / 2) (1 ? r / 2) 2 n t ?1
= C/2 (PVIFAr/2,2n) + M(PVIFr/2,2n)
2?
Where ? = value of the bond C/2 = semi-annual interest payment r/2 = discount rate applicable to a half-year period M = maturity value 2n = maturity period expressed in terms of halfyearly period
Price Yield Relationship
A basic property of a bond is that its price varies inversely with yield. The reason is simple. As the required yield increases, the present value of the cash flow decreases; hence the price decreases. When the required yield decreases, the present value of the cash flow increases; hence the price increases.
Interest rate risk is measured by the percentage change in the value of a bond in response to a given interest rate change of the maturity period of the bond and its coupon interest rate. Current price = Present value of Present value of of bond interest payments + principal repayment
C M ?? ? t (1 ? r ) (1 ? r ) n
An examination of this formula reveals that :
Longer maturity period
Greater sensitivity of price to changes in interest rates
Lesser sensitivity of price to changes in interest rates
Larger coupon (interest) payment
Sensitivity to bond prices to changes in market rates :
1. There is an inverse relationship between bond prices and yields.
2. An increase in yield causes a proportionately smaller price change than a decrease in yield of the same magnitude. 3. Prices of long-term bonds are more sensitive to interest rate changes than prices of short-term bonds. 4. As maturity increases, interest rate risk increases but at a decreasing rate.
5. Prices of low-coupon bonds are more sensitive to interest rate changes than prices of high-coupon bonds. 6. Bond prices are more sensitive to yield changes when the bond is initially selling at a lower yield.
The duration of a bond is the weighted average maturity of its cash flow stream, where the weights are proportional to the present value of cash flows. Formally, it is defined as : Duration = [PV(C1)x1 + PV (C2)x2 + … PV (Cn)xn]/V0 where PV(C t) = present value of the cash flow receivable at the end of year t (t = 1,2, …, n) V0 = current value of the bond of the bond issue is used as the discount rate
Duration is a key concept in bond analysis for the following reasons : • It measures the interest rate sensitivity of a bond • It is a useful tool for immunising against interest rate risk. Duration and Volatility The proportional change in the price of a bond in response to the change in its yield is as follows :
?P ? (1 ? y ) ? ?D ? ( ) P 1? y
where ?P / P = proportional price change D = duration of the bond y = yield
The following rules relate to duration :
1. The duration of a zero coupon bond is the same as its maturity.
2. For a given maturity, a bond’s duration is higher when its coupon rate is lower. 3. For a given coupon rate, a bond’s duration generally increases with maturity.
4. Other things being equal, the duration of a coupon bond varies inversely with its yield to maturity.
5. The duration of a level perpetuity is :
(1 + yield)/yield
For example, at a 9 per cent yield, the duration of a perpetuity that pays Rs. 100 per year forever will be equal to : (1.09/.09) = 12.11
Duration and Immunisation If the interest rates goes up it has two consequences for a bondholder : (i) the capital value of the bond falls, and (ii) the return on reinvestment of interest income improves. Interest rate change has two effects in opposite directions. If the investor chooses a bond whose duration is equal to his investment horizon. For example, if an investor’s investment horizon is 5 years he should choose a bond that has a duration of 5 years if he wants to insulate himself against interest rate risk. If he does so, whenever there is a change in interest rate, losses (or gains) in capital value will be exactly offset by gains (or losses) on reinvestments.
Limitations to Using Modified Duration
There are limitations in using Modified Duration in predicting the price/yield relationship. It is only valid for small changes in yield; for parallel shifts in the yield curve; and for small time horizons. The price/yield relationship estimated from the Modified Duration of a bond is linear however the actual price/yield relationship is a curve. There is therefore an error when using Modified Duration to estimate price movements. The following graph shows that the larger the change in yield, the greater the degree of error in the price change calculated using modified duration. This error is primarily accounted for by convexity.
Error in estimating price change for a change in yield, using modified duration
Actual Relationship Price Po Price/yield relationship from modified duration
Yield
Convexity
Modified Duration indicates how the price of a bond varies for small changes in yield. However, for large changes in yield, two bonds that have the same yield and the same Modified Duration can behave quite differently. This is due to the `error’ in using modified duration. This error is explained by Used in convexity convexity, which is the second derivative of a bond’s price with respect to yield. conjunction with Modified Duration,
provides a more accurate approximation of the percentage price change, resulting from a specified change, in a bond’s yield, than using modified
doc_596277839.ppt
This is a PPT about bond pricing and mathematics.
Bond Mathematics
Types of Yields Coupon Yield Nominal yield is the annual interest rate percentage payable specified in the indenture and printed on the face of the bond certificate. Nominal is fixed and not related to market value. Current Yield It is calculated by dividing annual interest income by the current market price. This yield takes into account the relationship between the interest received and the actual investment made. • Current yield is greater than the coupon rate if the bond is selling at a discount. • Current yield is less than the coupon rate if the bond is selling at a premium. • Current is equal to the coupon rate if the bond is available at a par value.
Yield to Maturity It is an average rate of return involving collective consideration of a bond
(a) Interest rate (b) Current price (c) Number of years remaining until maturity.
If the bond is available at a discount then, Coupon + Prorated Discount Yield to Maturity = (Face Value + Purchase Price)/2 If a bond is available at premium then, Coupon - Prorated Premium Yield to Maturity = (Face Value + Purchase Price)/2
Use of Time Value of Money Concept in Computation of YTM and Bond Pricing Bond Pricing The value of a bond - or any asset, real or financial - is equal to the present value of the cash flows expected from it. Hence determining the value of a bond requires : • An estimate of expected cash flows.
• An estimate of the required return.
• The coupon interest rate is fixed for the term of the bond. • The coupon payments are made every year and the next coupon payment is receivable exactly a year from now. • The bond will be redeemed at par on maturity.
C M ??? ? t n (1 ? r ) t ?1 (1 ? r )
Where ? = value (in rupees) n = number of years C = annual coupon payment (in rupees) r = periodic required return M = maturity value t = time period when the payment is received
?
C/2 M ??? ? t (1 ? r / 2) (1 ? r / 2) 2 n t ?1
= C/2 (PVIFAr/2,2n) + M(PVIFr/2,2n)
2?
Where ? = value of the bond C/2 = semi-annual interest payment r/2 = discount rate applicable to a half-year period M = maturity value 2n = maturity period expressed in terms of halfyearly period
Price Yield Relationship
A basic property of a bond is that its price varies inversely with yield. The reason is simple. As the required yield increases, the present value of the cash flow decreases; hence the price decreases. When the required yield decreases, the present value of the cash flow increases; hence the price increases.
Interest rate risk is measured by the percentage change in the value of a bond in response to a given interest rate change of the maturity period of the bond and its coupon interest rate. Current price = Present value of Present value of of bond interest payments + principal repayment
C M ?? ? t (1 ? r ) (1 ? r ) n
An examination of this formula reveals that :
Longer maturity period
Greater sensitivity of price to changes in interest rates
Lesser sensitivity of price to changes in interest rates
Larger coupon (interest) payment
Sensitivity to bond prices to changes in market rates :
1. There is an inverse relationship between bond prices and yields.
2. An increase in yield causes a proportionately smaller price change than a decrease in yield of the same magnitude. 3. Prices of long-term bonds are more sensitive to interest rate changes than prices of short-term bonds. 4. As maturity increases, interest rate risk increases but at a decreasing rate.
5. Prices of low-coupon bonds are more sensitive to interest rate changes than prices of high-coupon bonds. 6. Bond prices are more sensitive to yield changes when the bond is initially selling at a lower yield.
The duration of a bond is the weighted average maturity of its cash flow stream, where the weights are proportional to the present value of cash flows. Formally, it is defined as : Duration = [PV(C1)x1 + PV (C2)x2 + … PV (Cn)xn]/V0 where PV(C t) = present value of the cash flow receivable at the end of year t (t = 1,2, …, n) V0 = current value of the bond of the bond issue is used as the discount rate
Duration is a key concept in bond analysis for the following reasons : • It measures the interest rate sensitivity of a bond • It is a useful tool for immunising against interest rate risk. Duration and Volatility The proportional change in the price of a bond in response to the change in its yield is as follows :
?P ? (1 ? y ) ? ?D ? ( ) P 1? y
where ?P / P = proportional price change D = duration of the bond y = yield
The following rules relate to duration :
1. The duration of a zero coupon bond is the same as its maturity.
2. For a given maturity, a bond’s duration is higher when its coupon rate is lower. 3. For a given coupon rate, a bond’s duration generally increases with maturity.
4. Other things being equal, the duration of a coupon bond varies inversely with its yield to maturity.
5. The duration of a level perpetuity is :
(1 + yield)/yield
For example, at a 9 per cent yield, the duration of a perpetuity that pays Rs. 100 per year forever will be equal to : (1.09/.09) = 12.11
Duration and Immunisation If the interest rates goes up it has two consequences for a bondholder : (i) the capital value of the bond falls, and (ii) the return on reinvestment of interest income improves. Interest rate change has two effects in opposite directions. If the investor chooses a bond whose duration is equal to his investment horizon. For example, if an investor’s investment horizon is 5 years he should choose a bond that has a duration of 5 years if he wants to insulate himself against interest rate risk. If he does so, whenever there is a change in interest rate, losses (or gains) in capital value will be exactly offset by gains (or losses) on reinvestments.
Limitations to Using Modified Duration
There are limitations in using Modified Duration in predicting the price/yield relationship. It is only valid for small changes in yield; for parallel shifts in the yield curve; and for small time horizons. The price/yield relationship estimated from the Modified Duration of a bond is linear however the actual price/yield relationship is a curve. There is therefore an error when using Modified Duration to estimate price movements. The following graph shows that the larger the change in yield, the greater the degree of error in the price change calculated using modified duration. This error is primarily accounted for by convexity.
Error in estimating price change for a change in yield, using modified duration
Actual Relationship Price Po Price/yield relationship from modified duration
Yield
Convexity
Modified Duration indicates how the price of a bond varies for small changes in yield. However, for large changes in yield, two bonds that have the same yield and the same Modified Duration can behave quite differently. This is due to the `error’ in using modified duration. This error is explained by Used in convexity convexity, which is the second derivative of a bond’s price with respect to yield. conjunction with Modified Duration,
provides a more accurate approximation of the percentage price change, resulting from a specified change, in a bond’s yield, than using modified
doc_596277839.ppt