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# Bond and share valuation

The presentation highlights some shortcut formulas that can speed up PV computations if a project have a particular set of cash flow patterns and the opportunity cost of capital is constant

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### Bond and share valuation

1. 1. BOND AND SHARE VALUATION Richard Wamalwa Wanzala HFIN 4104: Corporate Finance Theory
2. 2. OUTLINE  INTRODUCTION  BOND VALUATION  Bonds  Return from Bonds  Coupon rate  Current yield  Spot interest rate  Yield to maturity or redemption yield:  Shortcut Method for Arriving at YTM  Yield to call  Bond Duration  Macaulay's Duration (D) HFIN 4104: Corporate Finance Theory
3. 3. OUTLINE – CONT’D  EQUITY MARKET  Equity Instruments  SHARE VALUATION  Share Valuation Models  Discount Rate  Dividend Discount Models  One year holding period model  Multiple Year Holding Period Model  Constant Growth Model (CGM)  Constant Growth Model (CGM)  Zero Growth Models (ZGM)  H-Model HFIN 4104: Corporate Finance Theory
4. 4. INTRODUCTION  Debt markets are used by both firms and governments to raise funds for long-term purposes, though most investment by firms is financed by retained profits. Bonds are long- term borrowing instruments for the issuer. Major issuers of bonds are governments (Treasury bonds in US, gilts in the UK, Bunds in Germany) and firms, which issue corporate bonds. HFIN 4104: Corporate Finance Theory
5. 5. BOND VALUATION HFIN 4104: Corporate Finance Theory
6. 6. BONDS  Bonds/ debentures are fixed income securities containing acknowledgement of indebtedness by the issuing company and are similar in nature except for the issuing company.  The bond/ debentures will contain a promise to pay interest for a specified period and then to repay the principal at the end of specified period, on a given date of maturity. HFIN 4104: Corporate Finance Theory
7. 7. TYPES OF BONDS • Callable and putable bonds • Convertible bonds • Eurobonds • Floating rate notes (FRNs) • Foreign bonds • Index-linked bonds. • Junk bonds • Covered bonds HFIN 4104: Corporate Finance Theory
8. 8. RETURN FROM BONDS Bonds returns are expressed in the following five forms:  Coupon rate  Current yield  Spot interest rate  Yield to maturity or redemption yield:  Yield to call HFIN 4104: Corporate Finance Theory
9. 9. COUPON RATE  It is the rate of interest fixed for a bond and printed on the bond certificate. Interest payable is calculated at the coupon rate on the face value of the bond. If the face value of the bond is, say Shs. 1000 and if the coupon rate is fixed at, say 9% p.a. the annual interest payable by the company to the bond holder is Shs. 90 per bond. HFIN 4104: Corporate Finance Theory
10. 10. CURRENT YIELD • It is the rate of return available from a bond on its current price in the secondary market. • It fluctuates depending on the bond price. • It only measures the annual rate of return accruing to an investor who purchase the bond from the secondary market. Illustration • A bond has a face value of KShs. 1,000 and a coupon rate of 9 per annum. The bond is currently selling in the secondary market at a price of KShs. 800. Calculate the current yield. HFIN 4104: Corporate Finance Theory
11. 11. CURRENT YIELD – CONT’D: SOLUTION  Annual interest payable (calculated at the rate on the face value)  Annual interest payable (calculated at the rate on the face value) HFIN 4104: Corporate Finance Theory *100 Pr Annual Interest Payable on the Bond Current Yield Current ice  9 *100 100 .90 90 *100 800 11.25% KShs Current Yield    
12. 12. SPOT INTEREST RATE (SIR)  It is the annual rate of return on a bond that offers only one cash inflow to the investor. A zero coupon bond offers only one cash inflow to the investor on its maturity and hence the rate of return offered by a zero coupon bond calculated on an annualized basis is as “SIR”.  Mathematically, SIR of a zero coupon bond is the discount rate that equates the present value of the single cash inflow available to the investor on maturity of the bond to the price of the bond. HFIN 4104: Corporate Finance Theory
13. 13. SIR – CONT’D: ILLUSTRATION • A zero coupon bond of face value KShs. 1,000 is issued at discounted price KShs. 600. The bond has a maturity period of 5 years. Find the Spot Interest Rate of the bond. Solution: • Let be the spot interest rate • The face value of KShs. 1,000 will be received after a period of 5 years • Amount receivable after five years hence = KShs. 1,000 HFIN 4104: Corporate Finance Theory % . .i pa
14. 14. SIR – CONT’D  Thus if the present value is at the end of 5 years a sum KShs. 1,000 would be available. HFIN 4104: Corporate Finance Theory 1 4 . 1,000* 1 1 1 3 . 1,000* 1 1 1 1 2 . 1,000* 1 1 Amount receivable after years hence KShs i Amount receivable after years hence KShs i i Amount receivable after years hence KShs i i                              5 1 1 1 1 1 1 . 1,000* 1 1 1 1 1 1 1 1 1 Pr . 1,000* 1 1 1 1 1 1 . 1,000* 1 i Amount receivable after one year hence KShs i i i i esent Value KShs i i i i i KShs i                                                        5 1 1,000 , 1 i       
15. 15. SIR – CONT’D  Thus: HFIN 4104: Corporate Finance Theory 5 5 5 1 600 1, 000 * 1 1 600 1 1, 000 1 0.6 1 10.76% Present value Discounted price of bond i i i                     
16. 16. YIELD TO MATURITY  It is the internal rate of return earned from a bond, holding the bond till its maturity.  It is calculated by equating bond the cost of bond to the present value of cash inflows from the bond held till maturity. Hence, the YTM is the discount rate makes the cost of bond equal to the present value of cash inflows from the bond held till maturity. HFIN 4104: Corporate Finance Theory
17. 17. YTM – CONT’D Illustration  Bond of face value KShs. 1,000 with a coupon rate of 8% p.a. and present value of KShs. 700 has a maturity period of 5 years. Calculate the (YTM) on the bond. HFIN 4104: Corporate Finance Theory    1 mi . 1 n 1 t n n t Annual Interest rece Cost of b ivable Ter al Value of a B ond annual interest receivable plus terminal value of bo ond Cost of Bond where i yield to maturity t time peri nd od n no of year i i s to maturity          
18. 18. YTM – CONT’D: SOLUTION  Annual interest rate at the coupon rate of 8% = KShs. 80 (i.e. on the face value of KShs. 1,000) HFIN 4104: Corporate Finance Theory    1 1 1 min t n t n Annual Interest receivable Ter al Value of a Bond Cost of Bond i i                                    1 2 3 4 5 5 1 2 3 4 5 5 Pr ., .700 80 80 80 80 80 1,000 700 1 1 1 1 1 1 1 1 1 1 1 1,000 80 1 1 1 1 1 1 Cost of Bond esent Value viz KShs i i i i i i i i i i i i                                                   
19. 19. YTM – CONT’D: SOLUTION  The value of the YTM can be arrived at any trial and error method assuming a certain yield to maturity are reworking with a different value until the value of the RHS matches with the value on the LHS of the equation. (Try YTM= 9.5%; 10%; 17%; and 18%). HFIN 4104: Corporate Finance Theory               1 2 3 4 5 5 1 1 1 1 1 0.18 1 0.18 1 0.18 1 0.18 1,000 700 80 1 1 0.18 1 0.18 1,000 80 0.8474 0.7182 0.6086 0.5158 0.4371 2.2878 250.17 437.10 687.27                                      
20. 20. YTM – CONT’D: SOLUTION  The value of RHS is now less than the value of LHS. The YTM lies in between 17% and 18% which can be arrived at by interpolating between 17% and 18%. HFIN 4104: Corporate Finance Theory       18 17 17% 712.4 700 712.4 687.27 1 17% 12.04 247.70 17% 0.05% 17.05% YTM YTM                    
21. 21. SHORTCUT METHOD FOR ARRIVING AT YTM HFIN 4104: Corporate Finance Theory     / / 2 I FV C n YTM FV C where I annual interest FV face value of bond C current price of bond n bond period        
22. 22. SHORTCUT … CONT’D Illustration  Bond of face value KShs. 1,000 with a coupon rate of 8% p.a. and present value of KShs. 700 has a maturity period of 5 years. Calculate the Yield To Maturity (YTM) on the bond using shortcut method. Solution HFIN 4104: Corporate Finance Theory         / / 2 80 1, 000 700 / 5 1, 000 200 / 2 80 60 0.1647 850 16.47% I FV C n YTM FV C Appprimate value of YTM            
23. 23. YIELD TO CALL  YTC is the discount rate that makes the present values of cash inflows to call (i.e. the annual interests till the call date and the specified call price of the bond at which it is redeemed) equal to the cost of the bond.  YTC can be calculated on a similar lines as a YTM as calculated HFIN 4104: Corporate Finance Theory
24. 24. BOND DURATION • It is defined as the time period at which the price risk and the investment risk of the bond are equal in magnitude but opposite in direction (Nagarajan and Jayabal, 2011) . Illustration • A 5 years bond with a face value of KShs. 100 has a coupon rate of 10%. What is the terminal value that will be available to the investor in this bond, if the market interest rate is 12%? HFIN 4104: Corporate Finance Theory
25. 25. BOND DURATION – CONT’D At the end of year 1 At the end of year 2 At the end of year 3 At the end of year 4 At the end of year 5 cashflow to the bond owner 10.00 10.00 10.00 10.00 10.00 100.00 (redeemed value of the bond) 110.00 HFIN 4104: Corporate Finance Theory         4 3 2 1 10 1 0.12 10 1 0.12 10 1 0.12 min 10 1 0.12 110 .163.53 Ter al Value KShs                
26. 26. MACAULAY'S DURATION (D)  It is “the weighted average of time periods to maturity” and is given by the following formula: HFIN 4104: Corporate Finance Theory         1 1 . : int 1 / or " " in 1 t n t t t t n t t t t C D C where t time period of each cashflow C erest and principal payment that occurs in period t i market erest rate n maturity per i i iod of the bond           
27. 27. MACAULAY'S … CONT’D Illustration  A new bond is issued by a company with a maturity period of 5 years and a coupon rate of 12%. The face value of the bond is KShs. 100 and the bond is redeemable at par after its maturity period of 5 years. The market interest rate is 12%. Prove that the duration of the bond is less than its period of maturity HFIN 4104: Corporate Finance Theory
28. 28. MACAULAY'S … CONT’D t 1 12.00 10.714 10.714 2 12.00 9.566 19.132 3 12.00 8.541 25.623 4 12.00 7.626 30.504 5 112.00 63.552 317.769 HFIN 4104: Corporate Finance Theory    1 t tC itC    1 . t tt C i 99.999 403.733
29. 29. MACAULAY'S … CONT’D  Thus: HFIN 4104: Corporate Finance Theory 403.733 99.999 4.03733 ( . . 4.04 ) Duration i e years  
30. 30. EQUITY MARKET HFIN 4104: Corporate Finance Theory
31. 31. EQUITY MARKET • Equity markets are markets which organize trading nationally and internationally in such instruments, as common equity, preferred shares, as well as derivatives on equity instruments. • The purpose of equity is the following: a. A new issue of equity shares is an important source of external corporate financing; b. Equity shares perform a financing role from internally generated funds (retained earnings); c. Equity shares perform an institutional role as a means of ownership. HFIN 4104: Corporate Finance Theory
32. 32. Equity Instruments a. Common or ordinary share (stock) is an equity share that does not have a fixed dividend yield. It represent partial ownership of the company and provide their holders claims to future streams of income, paid out of company profits and commonly referred to as dividends. b. Preferred share is an equity security, which carries a predetermined constant dividend payment. It is a financial instrument, which represents an equity interest in a firm and which usually does not allow for voting rights of its owners. HFIN 4104: Corporate Finance Theory
33. 33. EQUITY INSTRUMENTS – CONT’D The decision to issue equity against debt is based on several factors: • Tax incentives. In many countries interest payments are tax deductible, however dividends are taxed. Thus the tax shield of debt forms incentives to finance company by debt. • Cost of distress. Increase of company leverage, increases the risk of financial insolvency and may cause distress as well as lead to bankruptcy. Thus companies tend to minimize their credit risk and increase the portion of equity in the capital structure. • Agency conflicts. When a company is financed by debt, an inherent conflict arises between debt holders and equity holders. Shareholders have incentives to undertake a riskier operating and investment decisions, hoping for higher profits in case of optimistic outcomes. Their incentives are mainly based by limited liability of their investments. In case of worst outcome debt holders may suffer more, in spite of their priority claims towards company assets. • Signaling effect. The companies, which issue equity to finance operations, provide signals to the market, that current share selling price is high and company is overvalued. HFIN 4104: Corporate Finance Theory
34. 34. EQUITY INSTRUMENTS – CONT’D Types of Preferred shares • Cumulative preference shares • Non-cumulative preferred shares • A redeemable preferred share • Convertible preferred shares • Participating preferred shares • Stepped preferred shares • Specific adjustable rate preferred shares • Auction rate preferred shares (ARPS) or Single point adjustable rate shares (ARPS) • Preferred equity redemption cumulative stocks (PERCS) HFIN 4104: Corporate Finance Theory
35. 35. SHARE VALUATION HFIN 4104: Corporate Finance Theory
36. 36. SHARE VALUATION MODELS  Share valuation means arriving at the intrinsic value of shares. A share possess intrinsic value because it offers returns to the shareholder in the form of dividends and capital appreciation. HFIN 4104: Corporate Finance Theory
37. 37. DISCOUNT RATE  The discount rate is the rate of return required by the investor on his investment in the share. This rate of return is arrived at by taking into account the risk involved in the investment. The discount rate consist of two components: risk free interest rate and risk premium for the share concerned HFIN 4104: Corporate Finance Theory
38. 38. DIVIDEND DISCOUNT MODELS • Based on the holding period and the dividend expected to be received, share valuation models can be grouped into two: a. Holding period model • One year holding period model • Multiple year holding period model b. Dividend growth models • Constant growth model • Multiple growth model HFIN 4104: Corporate Finance Theory
39. 39. ONE YEAR HOLDING PERIOD MODEL Assumptions  The investor purchases the share now  The investor intends to hold the share for a period of one year  The investor intends to dispose off the share at the end of one year The present value (or intrinsic value) of the share is given by the following relationship: HFIN 4104: Corporate Finance Theory     1 1 1 1 0 1 1 , 1 1 o where S present value of the share D dividend expected to be received at the end of the first year S expected selling price of the share at the end of the fir D S S k k k st year rate of retu          ‘ ’rn required by the investor also called capitalization rate
40. 40. ONE YEAR… CONT’D: ILLUSTRATION 4.2  An investor desires to purchase the equity share of a company from the secondary market. The investor prefers to hold share for one year and dispose off the share after on year. The investor expects to get a dividend of KShs.5.00 per share next year and he is hopeful of selling the share in the secondary market at a price of KShs.70 after one year. He expects a return of 20% on his investment, considering the level of risk associated with it. Calculate the present value of the share to the investor. HFIN 4104: Corporate Finance Theory
41. 41. ONE YEAR… CONT’D: SOLUTION • Hence, present value of the share for the investor is given the following relationship Therefore, intrinsic value of the share for the investor KShs. 62.50 HFIN 4104: Corporate Finance Theory       1 1 . 5.00 . 70.00 02 % Dividend expected after one year D KShs Expected sales realization after one year S KShs Return required by the investor k            1 1 1 1 1 1 1 1 5.00 70 1 0.20 1 0.20 4.17 58.93 .62.50 o D S S k k KShs           
42. 42. MULTIPLE YEAR HOLDING PERIOD MODEL Assumptions  The investor purchases the share now  The investor intends to hold the share for a certain number of years  The investor will dispose off the share at the end of the holding period HFIN 4104: Corporate Finance Theory
43. 43. MULTIPLE YEAR…CONT’D  Accordingly, the present value of the share as per multiple years holding period model is given by the following relationship HFIN 4104: Corporate Finance Theory         31 2 1 2 3 ... 1 1 1 1 n n o n D D SD D S k k k k           0 1 2 3, , ,. : .., n n where S present value of the share D D D D annual dividend that will be received in the respective years S expected sales price of the share at the end of the holding period k rate of retu       rn required for the investor n holding period in years
44. 44. MULTIPLE YEAR…CONT’D: ILLUSTRATION  An investor desires to purchase the equity share of a company from the secondary market. The investor prefers to hold the share for a period of four years and dispose off the share after four years. He expected to get a dividend of KShs. 6.00, KShs. 6.50, KShs. 7.50 and KShs. 9.00 per share in the next four years respectively. He is hopeful of selling the share in the secondary market at a price of KShs. 120 after the end of four years. He expects a return of 22% on his investment considering the level of risk associated with it. Calculate the present value of the share to the investor. HFIN 4104: Corporate Finance Theory
45. 45. MULTIPLE YEAR…CONT’D  Solution HFIN 4104: Corporate Finance Theory                 31 2 1 2 3 1 2 3 4 ... 1 1 1 1 6.00 6.50 7.50 9.00 120 1 0.22 1 0.22 1 0.22 1 0.22 4.92 4.38 4.13 58.23 Pr .71.66 n n o n D D SD D S k k k k esent value of the share KShs                        
46. 46. CONSTANT GROWTH MODEL (CGM)  Assumptions  The investor buys and holds the share forever  The dividends from the share grow at a constant rate.  The discount rate (used for arriving at the present value of the share) is greater than the dividend growth rate. HFIN 4104: Corporate Finance Theory
47. 47. CGM – CONT’D  the present value of the share is the sum of present value of all future dividends.  When “n” approaches infinity the above formula can be simplified as:  Since the present value can also be written as HFIN 4104: Corporate Finance Theory                 1 2 3 1 2 3 3 1 2 3 1 1 1 1 ... 1 1 1 1 n o n D g D g D g D g S k k k k                  0 1 – o D g S g k    – oS g D k   1 0 ,1D D g 
48. 48. CGM – CONT’D LIMITATIONS OF GORDON DIVIDEND MODEL  The use of this model is restricted to firms that have been growing and are expected to grow at a stable growth rate forever and are expected to offer dividends at the stable growth rate forever.  If the dividend growth rate in more than the required rate of return, the model cannot be used since the value of the stock will become negative  If the dividend growth rate is equal to the required rate of return, the model cannot be used since the value of the stock will become infinity. HFIN 4104: Corporate Finance Theory
49. 49. CGM – CONT’D: ILLUSTRATION • M/s. Good tread company ltd, has declared a dividend of KShs. 4.50 per equity share for the current year. As a policy, the company has been enhancing its dividend by 12% every year. The company is expected to follow its dividend policy in the future also. An investor is interested to invest in the equity shares of the company as a long term investment. The investor expects a return of 18% on his investment considering the level of risk associated with the share of the company. Estimate the intrinsic value of the share for the investor. HFIN 4104: Corporate Finance Theory
50. 50. CGM – CONT’D: SOLUTION  The intrinsic value of the share is given by: HFIN 4104: Corporate Finance Theory       1’ . 4.50 18% . . 12% . . Current year s dividend D KShs per share Return required for the investor k p a Growth rate of dividend g p a            0 4.50 1 0.1 1 – 0.18 012 . 84 2 o D g k g KShs S      
51. 51. CONSTANT GROWTH MODEL (CGM) Two stage growth model Assumptions  The time period (in which the investor will be getting streams of dividends) is divisible into two different growth stages.  During the initial growth stage (stage -1) the growth rate of dividend is variable from year to year.  During the latter growth stage (stage II) the dividend growth rate will remain constant from year to year. This stage will have indefinite time duration. HFIN 4104: Corporate Finance Theory
52. 52. MGM – CONT’D  The intrinsic value of the share is the sum of the present values of the dividend flows from the two stages; Stage 1  Stage II  Intrinsic value HFIN 4104: Corporate Finance Theory           1 1 1 1 2 3 . . ... 1 1 1 1 N NI DD D D P V k k k k                 1 . 1– . N N II P g k V k D g       . . . .I II PV PV        1 . . 1 1 1 – NtN o t Nt i e D gD k S k kg                        
53. 53. MGM – CONT’D: ILLUSTRATION • A Company has paid a dividend of KShs. 150 per share during the current year. The company is expected to pay a dividend of KShs. 2.00 per share during the next year. The company is expected to pay a dividend of Analysts forecasts a dividend of KShs. 3.00 and KShs. 3.50 per share during the subsequent two years. After three years, the company is expected to pay dividends that are expected to grow at the rate of 10% every year. The investor expects a return of 20% on his investment. Calculate the intrinsic value of the share. HFIN 4104: Corporate Finance Theory
54. 54. MGM – CONT’D: SOLUTION  I. Present value of stage growth model  II. Present value of dividends from the fourth year to infinity HFIN 4104: Corporate Finance Theory       1 2 3 1 0.20 1 0.2 2.00 3.00 3. 0 1 0.20 50 . 5.78KShs              – 0.20 – 0 3. .1 50 1 0 0 .10 . 38.50 k D KShs g    
55. 55. MGM – CONT’D: SOLUTION  Present value of KShs. 38.50 receivable after three years is arrived at as below: HFIN 4104: Corporate Finance Theory     3 3 38.50 38.50 . 22.80 1 1 0.20 PV 5.78 22.28 . 28.06 o k of dividends from Stage I PV Intrinsic value of the share S of dividends from Stage II KShs KShs               
56. 56. ZERO GROWTH MODELS (ZGM)  The dividend per share will remain constant every year, forever.  The above assumption means that the dividend stream is a long-term annuity  Symbolically it means:  The intrinsic value of the share as per zero growth models is given by: HFIN 4104: Corporate Finance Theory o D Intrinsic value S k        1 2 3 4...D D D D D D    
57. 57. ZGM – CONT’D: ILLUSTRATION  A company pays a dividend of KShs. 20 per equity share. The dividend is expected to remain the same. An investor requires a return of 16% for investment in equity shares of the company. Calculate the value of equity share.  Solution HFIN 4104: Corporate Finance Theory 20 0.16 . 125 o D Intrinsic value of the equity share S k KShs   
58. 58. H-MODEL (TWO STAGE DIVIDEND MODEL) Assumptions of H-Model  Current above-normal growth rate is  The dividend growth rate falls down gradually over a period of years  After a period of years, the dividend growth rate falls down to . HFIN 4104: Corporate Finance Theory         0 01 f c f o f f The intrinsic value of the share under the above conditions is given by the following relationship : D g D g g S k – g k – g .H    cg 2H 2H fg
59. 59. H-MODEL – CONT’D: ILLUSTRATION  The equity share of a company offers dividend of KShs. 4.00 at present. The present dividend growth rate is 40%. Analysts predict that the dividend growth rate will decline linearly over a period of 12 years after which it will stabilize at 15%. An investor requires a return of 18% for his investment in the equity share of the company. What is the intrinsic value of the equity share? HFIN 4104: Corporate Finance Theory
60. 60. H-MODEL – CONT’D: SOLUTION  The dividend pattern resembles HFIN 4104: Corporate Finance Theory        4.00 1 – – 1 0.15 4.00 6 0.40 0.15 0.18 – 0.15 0.18 – 0. 153.33 200 . 353 1 .33 5 o c fo o f f D g gD g Intrinsic value of the equity share S k g H KSh g s k             H model
61. 61. THANK YOU!!! HFIN 4104: Corporate Finance Theory