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Summary of Last Lecture

Summary of Last Lecture. Bonds Valuation and Theory. BONDS VALUATION AND YIELD ON BONDS. Learning Objectives:. After going through this lecture, you would be able to have an understanding of the following topics Bond Valuation Yield of Bonds. Bonds basics Revisited:.

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Summary of Last Lecture

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  1. Summary of Last Lecture • Bonds Valuation and Theory

  2. BONDS VALUATION AND YIELD ON BONDS

  3. Learning Objectives: • After going through this lecture, you would be able to have an understanding of the following topics • Bond Valuation • Yield of Bonds

  4. Bonds basics Revisited: • In previous lecture, we studied that bonds are long term debt instruments. Like Stocks, bonds are also direct claim securities which means that the value of these bonds is determined by the future cash flows that bond holders will receive.

  5. Bonds basics Revisited: • These cash flows are of two basic types • 1. Cash inflow: in the form of coupon receipt with regular interval over the life of the bond • 2. Other cash flow is the par value of the bond which you will receive at the maturity date of the bond.

  6. Present Value formula for the bond: • PV= Σ CFt / (1+rD)t =CF1/(1+rD)+CFn/(1+rD)2 +..+CFn/ (1+rD) n +PAR/ (1+rD) n • NPV = Intrinsic Value of Bond or Fair Price (in rupees) paid to invest in the bond. It is the Expected or Theoretical Value and needs to be compared to the Market Price. It is different from the Par (or Face) Value which is printed on the Bond paper.

  7. Present Value formula for the bond: • There are basically 2 kinds of Cash Flows: • (1) Annuity from Fixed Regular Coupon Receipts (CF= Coupon Rate x Par Value) and • (2) Single Cash Flow from Par Value (or Initial Investment) Returned to the Investor on maturity.

  8. Present Value formula for the bond: • rD = Bondholder’s (or Investor’s) Required Rate of Return for investing in Bond (Debt). DIFFERENT from the Coupon Rate and the Market / Macroeconomic Interest Rate!

  9. Present Value formula for the bond: • In this equation rD: It represents the required rate of return. It is the return which is required by the investor based on his opportunity cost.

  10. Present Value formula for the bond: • In case of Pakistan, the investor required a higher return on bond then the rate of markup offered by the PLS account in bank. It is different from the Coupon Rate and the Market /Macroeconomic Interest Rate.

  11. Present Value formula for the bond: • Coupon or CF: • It is a fixed rate and it is equal to (CF= Coupon Rate x Par Value) Par value of the bond is fixed but the market price varies with the change in the supply and demand, perception of investor for that bond.

  12. Example: • Defense Savings Certificates: Suppose that you invest in a Defense Savings Certificate whose Par Value is Rs 100,000. The Bond Issuer is the Government of Pakistan. The Certificate has small detachable coupons. You (as the Bondholder or Investor) can present one Coupon at the end of every month and receive Rs 1,000 cash.

  13. Example: • After 1 year, you will be repaid your Principal Investment (or Par Value) of Rs 100,000. Assume your Required Return (rD) is 10% pa. What is the Present Value of this Investment to you?

  14. Example: • In the previous lectures, we have solved simple version of similar example we solved a problem similar to this where we had to calculate the NPV of the Defense Savings Certificate with 1 Annual Coupon payment after 1 year. We arrived at the following approximate answer:

  15. Example:

  16. Example: • But this is not the correct exact answer to our present example because it ignores monthly compounding.

  17. Accurate Solution - Monthly Compounding: • The Accurate solution to the Savings Certificate Example with Monthly Coupons requires us to use a monthly cash flow diagram and do monthly discounting.

  18. Accurate Solution - Monthly Compounding: • There is an Annuity Stream of 12 Coupons (Cash Inflows) of Rs 1,000 each at the end of every month. There is a final Cash Inflow worth the Par Value of Rs 100,000 at the end of the 12th month.

  19. Accurate Solution - Monthly Compounding: • The Cash Flow Diagram for Bonds is a Combination of 2 Flows: (1) an Annuity Stream (of Coupon Receipts) every month for 12 months and (2) One Par Receipt at the end of the 12th month.

  20. Accurate Solution - Monthly Compounding: • You can draw their individual Cash Flow Diagrams and then add them up later. You can compute their PV’s separately and then add them up later.

  21. Accurate Solution - Monthly Compounding: • Cash flows from coupons represents by the upward pointing arrows which represents cash inflows. • In combined diagram, at the end of the year there are two upward pointing arrows. One for coupon rate and the other is for the payment of par value of bond.

  22. Accurate Solution - Monthly Compounding:

  23. Accurate Solution - Monthly Compounding: • Calculate the PV of Coupons from the FV Formula for Annuities (with multiple compounding within 1 year): • FV = CCF (1 + rD/m )nxm - 1/rD/m • Use Monthly Basis for this example. n = 1 year m = 12 months CCF = Constant Cash Flow = Rs 1,000 = Monthly Coupon

  24. Accurate Solution - Monthly Compounding: • Coupon Annuity Cash Flow Receipts • FV = 1,000 x [(1.00833)12- 1] /0.00833 = +Rs 12,566 (at the end of 1 year) • PV (Coupons Annuity) = FV / (1 + rD/m) nxm • = 12,566 / (1.00833)12 = +Rs 11,374

  25. Accurate Solution - Monthly Compounding: • Final Par Value Cash Flow Receipt • FV = 100,000 (at the end of 1 year) • PV (Par) = 100,000 / (1.00833) ^ 12 = +Rs 90,522 • PV = PV (Coupons Annuity) + PV (Par) = 11,374 + 90,522 • = + Rs 101,896 (Final Answer)

  26. Accurate Solution - Monthly Compounding: • So this Certificate is worth Rs 101,896 to you today. It is worth more than the Market Price (Rs 100,000). So it is a good investment.

  27. Accurate Solution - Monthly Compounding: • NOTE: Our answer is slightly higher than what we got when we used Annual compounding (Rs 101,818).when we consider multiple compounding the present value of the bond increases.

  28. Accurate Solution - Monthly Compounding: • Its NPV is greater than zero so on the basis of our capital budgeting techniques you should invest in that project.

  29. Accurate Solution - Monthly Compounding: • Now, we consider over all rate of return on a bond. We have studied expected price of the bond. These two are complimentary. When bonds trader talk about he overall return on a particular bond they referred to yield to maturity.

  30. Bond Yield to Maturity (YTM): • We can calculate the Value of our Investment in Bonds. But how can we compute its Rate of • Return? Both are important whether you are talking about Investment in Real Assets or Securities. • The most common way to compare the Overall Rate of Return of different Bonds is to compare their YTM’s.

  31. Bond Yield to Maturity (YTM): • In capital budgeting, you can calculate IRR using the NPV equation. Similarly, you can calculate it by setting the PV Equation for Bond Valuation equal to the Present Market Price and solve for “rD”. • Use Trial and Error or Iteration. The value of “rD” that gives PV = Market Price is the YTM for that Bond.

  32. Bond Yield to Maturity (YTM): • PV = Bond Market Price = CFt / (1+rD)t • CF1/ (1+rD) +CF2/(1+rD)2+…+CFn/(1+rD)n+ PAR/(1+rD)n • All variables are known (ie. CF, PAR, and n) EXCEPT rD. Set PV equal to the Actual Present Market Price of Bond and solve for rD • YTM = rD

  33. Bond YTM – Example: • Term Finance Certificate (TFC): The TFC (a kind of Bond) of Company ABC is traded in the Karachi Stock Exchange for Rs 900. The Par Value of the TFC is Rs 1,000. The Coupon Rate is fixed at 15% pa. Coupons are paid annually. The TFC will Mature after exactly 2 Years (it is a 5 Year Bond issued 3 Years ago). What is the Overall Expected Rate of Return (ie. YTM) offered by this TFC?

  34. Bond YTM – Example: • Market Price (Rs 900) is LESS than its Par Value (Rs 1,000). This Bond is selling at a Discount. Why? • Possibly Interest Rate Risk. Market Interest Rate rises above TFC’s Fixed Coupon Rate so Market Price of the TFC falls below Par. Note: when Market Interest Rates rise, Required Rate of Return (rD) for Investors rises. But, Coupon Rate fixed by Bond Issuer at time of issue.

  35. Bond YTM – Example: • The Expected (or Promised) Rate of Return for Investors is the Yield to Maturity (or YTM). • Compute the Overall Return (or YTM) for the TFC using the Old IRR-like Approach: • PV = Market Price = Rs 900 • Par Value =Rs 1,000. Receive this after 2 Years (remaining life) • Annual Coupons =Coupon Rate x Par =15%x1, 000 = Rs 150 • rD = Minimum Return Required by the Investors investing in The Bond Market = YTM.

  36. Bond YTM – Example: • rD = Minimum Return Required by the Investors investing in The Bond Market = YTM. • This is unknown in the equation. • PV = 900 = 150 / (1+ rD) + 150 / (1+rD)2 + 1,000 / (1+rD)2 • 900 = 150 / (1+ rD) + 1,150 / (1+rD)2 . Use Trial & Error • rD > 15%: Try rD = 20%: PV = 924 (close) • Try rD = 21%: PV = 909 (closer) • YTM = 21.7%: (Gives PV=Rs 900)

  37. Bond YTM – Example: • YTM: YTM is the expected rate of return for which the bond holder holds the bond until maturity but if the bond holder before maturity is called by the issuer or if the holder of the bond decides to sell the bond before maturity then your answer will change .all the calculation will remain the same only par value is replaced as n • PV=Σ CFt / (1+rD)t =CF1/(1+rD)+CF2/(1+rD)2+…+CFn/(1+rD)n+CALL/(1+rD)n • t=1 • Where CALL = PAR Value + 1 Year’s Worth of Coupon Receipts

  38. Bond YTM – Example: • YTM =Total or Overall Yield = Interest Yield + Capital Gains Yield • TFC Example Total Yield = YTM = +21.7%

  39. Bond YTM – Example: • Interest Yield or Current Yield = Coupon / Market Price • TFC Example Interest Yield = Rs 150 / Rs 900 = +16.7% pa

  40. Bond YTM – Example: • Capital Gains Yield = YTM - Interest Yield • TFC Example Capital Gains Yield = 21.7% - 16.7% = +5 %

  41. Bond YTM – Example: • n = Maturity or Life of Bond (in years) • FV=CCF[(1+rD/m)^ n*m-1]/rD/m • N=1 year ,m= no. of intervals in a year =12 • CCF=constant cash flow =1000=monthly coupon .we can plug the values in this formula to know what • the future value of annuity is going to be?

  42. Bond YTM – Example: • Take a look at the coupon annuity : • FV=1000[(1-0.00833)12-1]/0.0083=+12566 at the end of one year what is the present value of this coupon annuity. • PV=FV/(1+rD/m )n*m • =12566(1.00833)12 • =+11374

  43. Bond YTM – Example: • This is the present value of cash flow from coupon. Now we need to calculate the present value of face value at maturity suppose face value =100,000 then • PV(PAR)=100,000/(1.00833) ^ 12 • =+90,522 • Now, we combine the present value of coupon interest and present value of par both • i.e.=11374+90522= Rs.101896.

  44. Bond YTM – Example: • When we compare the answer with annual cash flows where coupon was not compounded monthly . It is greater because monthly compounding increase future cash flows as well as the present value . • 2nd thing is that this NPV is grater than the initial investment which is Rs.100,000, so we should undertake this project because the NPV is greater now.

  45. Bond YTM – Example: • The next area is the rate of return so, the important thing in this regard is yield to maturity, this is abbreviated as YTM. It is easy to understand because we have discussed IRR in capital budgeting. Where we set NPV=0 and calculated for r. Here market price is the YTM of the bond and then solving for the variable rD=required rate of return .

  46. Bond YTM – Example: • So, let’s try to understand YTM using a very simple example ,the example that we will pick out is that of term finance certificate or TFC which is by the stock exchanges of Pakistan for Rs.900. Let’s assume that its par value is Rs.1000 • Fixed or coupon interest rate is 15 p.a. and it is paid annually ,total life of the TFC is 5 years, 3 years have already passed and it will mature 2 years from now what will be over all expected rate of return .

  47. Bond YTM – Example: • So, let’s see the equation if we compute the over all yield here we can equate • PV=market value is the YTM for the bond the PV=900 which is market price PAR=1000. • Annual coupon rate =coupon rate *par =15/100*1000=150 • rD=minimum return required by the investor in the bond market =YTM it is unknown ?

  48. Bond YTM – Example: • PV= 900=150/(1+rD )+150/ (1+rD) ^ 2+1000/(1+rD) ^ 2 • We also know that the value of rD should be more than 15% you will try different values for example if you try 20% you will come up with PV=24 (close),try rD=21.7% PV=900 so, • YTM =21.7% =900

  49. Bond YTM – Example: • Therefore 21.7% is the yield to maturity for this TFC because rD=YTM .YTM is the expected rate of return for which the bond holder holds the bond until maturity but if the bond holder before maturity is called by the issuer or if the holder of the bond decides to sell the bond before maturity then your answer will change. All the calculation will remain the same only par value is replaced as call value.

  50. Bond YTM – Example: • So, Call=par value +I, year coupon receipts • Another thing to keep in mind is that YTM has two components first is YTM=interest yield on bond +capital gain yield on bond from his example • YTM= 21.7% so,let’s calculate the interest yield • INTEREST YIELD =annual copoun interest /market price • =150/900 =16.7% so, • CAPITAL YIELD =YTM –INTEREST YIELD • =21.7%-16.7%=5%

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