1. Copyright: M. S. Humayun1 Financial Management Lecture No. 15 Bond Valuation & Yield Numerical Examples Batch 4-3

2. Copyright: M. S. Humayun2 Bond Valuation Formula Recap Old PV Formula (Note: NPV = -Io + PV): n PV= CF t / (1+r D ) t =CF1/(1+r D )+CF2/(1+rD) 2 +…+CFn/(1+r D ) n +PAR/(1+r D ) n t =1 Apply NPV Formula to Bond Valuation: –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. Different from the Par (or Face) Value which is printed on the Bond paper. –r D = 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 ! –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.

3. Copyright: M. S. Humayun3 Bonds - Savings Certificate Example Multiple Compounding 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. After 1 year, you will be repaid your Principal Investment (or Par Value) of Rs 100,000. –Assume your Required Return (r D ) is 10% pa. –What is the Present Value of this Investment to you?

4. Copyright: M. S. Humayun4 Bonds - Savings Certificate Example Old Approach: 1 Annual Coupon 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: NPV = -Io + CF1 / (1+ i) + CFI1 / (1+ i) = -100,000 + 12,000/(1+0.10) + 100,000/(1+0.10) = -100,000 + 10,909 + 90,909 = 1,818 (NOTE: PV = NPV + Io = 10,909 + 90,909 = 101,818) But this is NOT the correct exact answer to our present example because it ignores monthly compounding.

5. Copyright: M. S. Humayun5 Bonds - Savings Certificate Example 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. 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. 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. 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

6. Copyright: M. S. Humayun6 Bond Cash Flow Diagram Savings Certificate Example 0126 12 Time (Months) 0126 12 Time (Months) Coupon Annuity : Par Receipt : (at Maturity) (Monthly) Combined Cash Flow Diagram 012 Rs 1,000 Rs 100,000 2 Cash Flow Arrows at SAME point in time can be added.

7. Copyright: M. S. Humayun7 Bonds - Savings Certificate Example Multiple Compounding Calculate the PV of Coupons from the FV Formula for Annuities (with multiple compounding within 1 year): –FV = CCF (1 + r D /m ) nxm - 1 r D /m –Use Monthly Basis for this example. n = 1 year m = 12 months –CCF = Constant Cash Flow = Rs 1,000 = Monthly Coupon –r D = Annual Nominal Required Rate of Return for investment in Bond (Debt) = 10% pa. –Periodic Monthly Required Rate of Return is r D /m = 10/12 = 0.833 % = 0.00833 p.m. m = 12 months 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. You can compute their PV’s separately and then add them up later

8. Copyright: M. S. Humayun8 Bonds - Savings Certificate Example 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 + r D /m ) nxm = 12,566 / (1.00833) 12 = +Rs 11,374 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) 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. NOTE: Our answer is slightly higher than what we got when we used Annual compounding (Rs 101,818).

9. Copyright: M. S. Humayun9 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. YTM similar to IRR from Capital Budgeting when we set NPV=0 and solved for “i”! Set the PV Equation for Bond Valuation equal to the Present Market Price and solve for “r D ”. Use Trial and Error or Iteration. The value of “r D ” that gives PV = Market Price is the YTM for that Bond. PV = Bond Market Price = CF t / (1+r D ) t = CF1/(1+r D )+CF2/(1+r D ) 2 +…+CFn/(1+r D ) n +PAR/(1+r D ) n All variables are known (ie. CF, PAR, and n) EXCEPT r D. Set PV equal to the Actual Present Market Price of Bond and solve for r D YTM = r D

10. Copyright: M. S. Humayun10 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? 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 (r D ) for Investors rises. But, Coupon Rate fixed by Bond Issuer at time of issue. The Expected (or Promised) Rate of Return for Investors is the Yield to Maturity (or YTM).

11. Copyright: M. S. Humayun11 Bond YTM - TFC Example 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 r D = Minimum Return Required by the Investors investing in the Bond Market = YTM. This is unknown in the equation. PV = 900 = 150 / (1+ r D ) + 150 / (1+r D ) 2 + 1,000 / (1+r D ) 2 900 = 150 / (1+ r D ) + 1,150 / (1+r D ) 2. Use Trial & Error r D > 15%: Try r D = 20%: PV = 924 (close) Try r D = 21%: PV = 909 (closer) YTM = 21.7%: (Gives PV=Rs 900)

12. Copyright: M. S. Humayun12 Other Bond Yields YTM: Expected (or Promised) Rate of Return for Investors in the Market provided you hold Bond to Maturity. BUT, if the Bond is Called (by the Issuer) before the Maturity or if you sell the Bond before Maturity, then YTM Equation will change: n PV= CF t / (1+r D ) t =CF1/(1+r D )+CF2/(1+rD) 2 +…+CFn/(1+r D ) n +CALL/(1+r D ) n t=1 Where CALL = PAR Value + 1 Year’s Worth of Coupon Receipts YTM =Total or Overall Yield = Interest Yield + Capital Gains Yield TFC Example Total Yield = YTM = +21.7% Interest Yield or Current Yield = Coupon / Market Price TFC Example Interest Yield = Rs 150 / Rs 900 = +16.7% pa Capital Gains Yield = YTM - Interest Yield TFC Example Capital Gains Yield = 21.7% - 16.7% = +5 %