# Investment Analysis. Bond Value Bond Yields Bond Pricing Relationships Duration Convexity Bond Options. Bonds part 2. Financial analysis

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1 Bond Value Bond Yields Bond Pricing Relationships Duration Convexity Bond Options Investment Analysis Bonds part 2 Financial analysis

2 - 2 Duration Duration = the average maturity of the bond s promised cash flows A measure of the price risk of a bond Direct measure of the sensitivity of a bond s price to a change in its yield Macaulay duration The weighted average of the lengths of time until each payment is received, with the weights proportional to the present value of the payment Other measures of duration

3 - 3 Macaulay duration formula T t t t T t t t k V k V t V t bond s cash flows 0 B k V t T t t t = B 0 Coupon payments and par value

4 - 4 Duration as a risk measure Better volatility measure than maturity Connects maturity with a cash flow model Simple risk measure Interest s immunization tool Measure of expected bond s price changes Depends on: Coupon interest rate Bond s cash flow model Maturity Market interest rate

5 - 5 Measure of price changes The price of a coupon bond Its differential d B 0 taking We get T t I ( k t k t b ) t B ( 0 k T It ( k b t ) T And deviding by price d B B 0 0 k dk b d ) t k ( k b ) T

6 - 6 Modified duration * k First derivation of price with respect to yield divided by price Price sensitivity measure to a change in interest rates Bond is more sensitive to a change in interest rates when its cash flows are farther from the present When duration is longer, a bond is more sensitive to a change in interest rates * B0 k B 0

7 - 7 Duration in the light of immunization Immunization an asset/liability management approach that structures investment in bonds to match (offset) liabilities weighted average duration When the durations of the asset and the liabilities of a portfolio are the same, the portfolio is inherently protected against interest rate changes Flat yield curve assumption Possible modification according to the yield curve

8 - 8 Duration for more than one coupon payment Annual coupon payment More than one coupon payment per year number of payment periods B k V T B k V B k V b T b b B k V n n T B k V n B k V n Tn b n Tn n b n n b n n

9 - 9 A yield curve problem Macaulay duration is based on the assumption of flat yield curve Fisher-Weil duration Assumes a parallel changes in interest rates Cox-Ingersoll-Ross duration B k k k V T B k k V B k V b b b T b b b

10 - 0 Convexity Risk measure for investing in bonds The relationship between yield changes and bond price changes is convex Duration approximates a change in a bond s price just for small incremental changes It is necessary to modify duration as a bond s sensitivity measure The curvature of the price-yield relationship of a bond The rate of change of the slope of the price-yield curve, expressed as a fraction of the bond price

11 Price - Relationship between Duration and Convexity Duration Bond s price Duration tangent to the bond s price function Bond s price function is convex (it opens upward) Convexity measures a curvature of price function yield

12 - 2 Calculation of Convexity 2nd derivative of price with respect to interest rate 2nd derivative (the rate of change of the slope) of the price-yield curve divided by bond price: Convexity /B d B dy / 2 T 2 B0 k t CF t k t t 2 t Measuring bond s price changes B0 * k k B 2 0 2

13 - 3 Bond Options Redeemable bonds Callable bonds Redeemable bonds at the option of the holder Put bonds Attached warrants Detachable warrant Convertible bonds An option to exchange each bond for a specified number of shares of common stock

14 - 4 Call Provision Issuer can buy-back if rates decline. That helps the issuer but hurts the investor Company would Call if k d is below the coupon rate and bond sells at a premium. Use open market purchase if k d is above coupon rate and bond sells at a discount Borrowers are willing to pay more, and lenders require more, on callable bonds (higher interest rate) Most bonds have a deferred call and a declining call premium

15 - 5 The effect of the call Real bond s value A bond s value without the option Convex declining curve Value of the call Call price Option for issuer benefit N B 0 Minimal value of callable bond min (bond s value + premium; call price) C k b

16 - 6 The value of a callable bond Traditional analysis Monitor yield to maturity At the same time monitor yield to call B 0 C The value of the call A call option to the issuer Allows the issuer to repurchase its debt at the lower price

17 - 7 The value for holder Reverse from the value of the issuer Written uncovered option Obligation to sell at the redemption price (call price) exercise price C B 0

18 - 8 The value of callable bond Value of redemption Value for the issuer Allows repurchasing of debt at the lower price Allows lowering the cost of debt The value of callable bond Premium above lower bound Real value the call option value C N B 0 k b

19 - 9 Redemption at the option of the Option to redeem holder The holder of the puttable bond has the right to demand early repayment of the principal Put option of the owner put bond It is exercised when interest rates rise Protects bond s price fall Redemption price Price of a puttable bond is always higher than the price of a straight bond Sweetener of the issue

20 - 20 The effect of redemption Real bond s value B 0 Redemption value Premium paid for a put bond The option for the holder Redeemable bond s minimum value Real value premium Redemption price N C k b

21 - 2 The value of redeemable bond Redemption value Put option of the owner Allows its holder to sell the bond at higher price Will be exercised when the bond s price is lower than redemption price C B 0

22 - 22 The value of redeemable bond B 0 The value of redeemable bond Premium above the lower N bound of value C Price of a straight bond + price of put option k b

23 Callable bond reedemable at the - 23 option of the holder B 0 Straight value Influence of the call N Influence of the reedem k b

24 - 24 Warrants Type of security, usually issued together with bond, that entitles the holder to buy a proportionate amount of common stock at a specified price Essentially call options issued by a firm Warrant can be detached from bond after bonds are sold Freely transferable securities Are traded on the major exchanges

25 - 25 Warrant valuation Value of a straight bond Bond s value after the warrant is detached Lower value than the equal bond without a warrant Premium paid for the warrant Lower coupon rate Option value Value of detachable warrant Value of call option

26 - 26 Distinction to call option Call option Right to buy underlying asset (stock) at exercise price within a certain period or specific date Option writer person or financial institution (investors) Short-term instrument Warrant Right to buy stock at exercise price for a period of years Written by a firm to its own stocks They are added to the bond issue and last for a number of years creates the potential for an increase in outstanding shares of stock if exercise occurs

27 - 27 Warrant s elements Exercise price Could be stepwise exercise price Raising the price after some period of time Exercise ratio Bond s par value divided by exercise price Does not have to stand if we have exercise price and vice verse Maturity Usually the same as bond s

28 - 28 Warant s value model premium Maksimum option value C max P 0 E minimum option value C max( P E; 0 min 0 )

29 - 29 Warant with exercise price 40 Stock price Exercise price Exercise ratio Minimum warant s price Warant s market price Premium

30 - 30 The effect of stepwise exercise price Payoffs and profits Stock price Exercise price Exercise price date of change time

31 - 3 The option to convert An option to exchange each bond for a specified number of shares of stock of the firm at the fixed price conversion price, or for the fixed number of shares conversion ratio The conversion ratio gives the number of shares for which each bond may be exchanged. Distinction to warrant It can not be detached from the bond Complex valuation model Deletes the difference between stocks and bonds

32 - 32 Option to convert valuation Value components Real bond s value Stock s value Premium It is possible to choose bigger value between a bond s value and a stock s value The option to acquire stocks through the value held in bonds Capital dilution when option is exercised

33 - 33 Convertible bond s elements Conversion price Conversion ratio Bond s par value divided by exercise price Does not have to stand if we have exercise price and vice verse Maturity Usually the same as bond Call provision Sometimes as an extra clause

34 - 34 Value model Real bond s value Increases with a stock s price rise Slow increase B C Stock s value Proportional to a bond s price Minimum value of convertible bond Premium N P t

35 - 35 Convertible bond Constant growth of stock s price Option to convert Conversion value Option to call Stock price Conversion value Conversion ratio Conversion value Redemption price V S P t c c R c c N C V R N c c c T P 0 P 0 P t g g Par value Conversion price Stock s price ts Annual growth rate Annual call premium R c

36 maturity - 36 Value model Bond s par value value Real bond s value Redemption price Stock price Convertible bond s value S k V k max (P t ; B t ) N Lower bound of convertible bond s value P t B t Firm s value

37 - 37 Example year Par value Real bond s value Redemption price Stock price Conversion value , , , , , , , , , , ,6.753

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