What is the Bangladesh standard on Audit(BSA-300) ? Describes in details

Bond Valuation Definition, Formula with practical Examples

Bond valuation is the process of determining the fair value or intrinsic worth of a bond. It is essential for investors and financial analysts to assess whether a bond is a good investment, and to compare different bonds with varying characteristics. Bonds are debt securities issued by corporations, governments, or other entities to raise capital. When you buy a bond, you are effectively lending money to the issuer, and in return, you receive periodic interest payments (coupon payments) and the principal amount (face value) back at maturity.

There are several factors that come into play when valuing a bond:

1. Coupon rate:

   This is the annual interest rate specified on the bond, which is used to calculate the periodic coupon payments.

2. Maturity date: 

   The date on which the bond will reach its full face value and be redeemed by the issuer.

3. Face value (par value): 

   The nominal or original value of the bond, which will be returned to the bondholder at maturity.

4. Market interest rates: 

   The prevailing interest rates in the market at the time of valuation. Bonds with fixed coupon rates will be more or less attractive based on how their coupon rate compares to the current market rates.

      There are two primary methods of bond valuation are:

1. Present Value (PV) Method: 

   This method discounts all the future cash flows (coupon payments and face value) of the bond back to the present at a discount rate that represents the bond's required rate of return. The present value of these cash flows represents the fair value of the bond.

2. Yield to Maturity (YTM)(r): 

   The YTM is the total return anticipated on a bond if it is held until it matures. It takes into account the current market price of the bond, its coupon rate, and the time remaining until maturity. The YTM is the discount rate that equates the present value of the bond's cash flows to its current market price.

If the bond's current market price is higher than its fair value (determined by the PV method), the bond is said to be trading at a premium. Conversely, if the market price is lower than the fair value, the bond is trading at a discount.

Keep in mind that bond valuation can be more complex for bonds with special features such as variable coupon rates, embedded options, or convertible features. Additionally, credit risk associated with the issuer's ability to pay its obligations should also be taken into account while valuing a bond.

There are  three way to issue Bond such as-  1. Bond issue at par  2. Bond issue at Premium  3. Bond issue at discount There are some matter are consideration for Bond valuation 1. Maturity Value 2. Required rate of return 3. Amount of Interest 4. Period of repayment (Years) 5. Bond issue at discount /  at premium 6. Flotation cost consideration  (if any) 7.NSV = Net sale value calculation  [when flotation cost exist] 8. Flotation cost charged only Face value [whether or not, Bond issue at discount or at a premium] 9.Coupon rate / discount rate {where consider interest implies against coupon rate/Discount rate denote(r)} 10.Multiple compound interest =Annually, Half-yearly, quarterly, monthly, Bi- monthly 11.Bi monthly = once in every two month [when 1 year , m = 6] Abbreviation YTC =Yield to call CY  = Current Yield MV / RV =Maturity value Int=Amount of interest r =Yield to maturity /required rate of return/ discount rate N=Numbers of year Bi- monthly = once in every two months semimonthly/Twice a month = occurring twice a month M = Yearly interest factor( wheres, M implies,divided to the rate of interest & multiply the number of years) YTM (r)=Yield to Maturity/ required rate of return/ discount rate  PVIFA = Present value interest factor Annuity or, {1-1/(1+r)n/r}  PVIF = Present value interest factor respectively or, {1/(1+r)n } RV / MV= Redeemable value N=Number of years CP=call price SV=sales value FV= Face value Perpetual bond = where no mention any year & maturity time ,but continuing  forever.....  FC = Flotation cost Some Important Formulas of Bond Valuation 1. Bond Valuation (Bv)=Int{1-1/(1+r)n/r}+MV/(1+r)n (Normal way) 2. Bond Valuation (Bv)=MV/(1+r)n (When determined O coupon bond) 3. Bond Valuation (Bv)=Total interest/Discount rate(when no mention time period) 4. Bond Valuation (Bv) = (Int*PVIFA) + (RV*PVIF)(when Calculate BV by using (PVIFA & PVIF ) 5.PVIFA ={1-1/(1+r)n /r} 6.PVIF ={1/(1+r)n } 7.Bond value(Bv) =(Int*PVIFA)+(RV*PVIF) (When PVIFA & PVIF  exist in the  question) 8. Bond value (Bv)=Int/r 9. Current Yield =(Int/sales value)*100 10. YTM = Int+RV-SV/N/ RV+SV/2*100   (When flotation cost  not exist) 11. YTM=Int+RV-NSV/N/ RV-NSV/N*100 (When flotation cost  exist) 12. YTC=Int+CP-SV/N/ CP+SV/N*100     (When call price & call years is given)  13. Capital Gain / Loss =YTM- Current Yield 14. FV=SV (1+r) n [ when coupon rate not exist in the question)  15.Perpetual bond(Bv) = Int /r [where no mention any year & maturity time ,but continuing  forever.....]  16.Sale Value(SV) = (Interest / Current Yield) [ whether no mention sales value] Note- YTM value half year, then convert full year Let’s start with Practical Example-01. Uttara Motors bond have 10 years remaining to maturity.  Interest is paid annually .The Bond have a Tk.1000 par value, and the coupon interest rate 8%.  The Bond has a yield to Maturity of 9%. Requirement: 1. What is the current market price of these Bond? Here, FV=Tk.1000, MV=Tk.1000 , Interest =1000*8% = TK.80, r= 9% or 0.09, N=10 years We know that, Bond Valuation (Bv)=Int{1-1/(1+r)n/r}+MV/(1+r)n                                     =80 {1-1/(1+0.09)10/0.09}+1000/(1.09)10                                     = 80 {1-0.4224/0.09}+1000/2.3674                                     = 80* {0.5776/0.09}+422.40                                      = (80*6.4177)+422.40                                      = 513.42+422.40                                      = Tk.935.82(Ans) Practical Example-02 Here, FV=Tk.1000, MV=Tk.1000 , Interest =1000*10% = TK.100, r= 15% or 0.15, N=12 years Bond Face value= Tk.1000,  Coupon rate-10% Discount rate-15% Maturity period -12 Years Requirement: 1. Determined the value of these Bond? We know that, Bond Valuation (Bv)=Int{1-1/(1+r)n/r}+MV/(1+r)n                                     =100{1-1/(1+0.15)12/0.15}+1000/(1.15)12                                     = (100*5.4206)+186.91                                     = 542.06+186.91                                      =Tk.728.97(Ans) Practical Example-3 Here, FV=Tk.2500, MV=Tk.2500 , SV=Tk.1500 , m=2, Interest =(2500*8%)/02  = TK.100 ,  r = (12.50%/2)=6.25 or 0.0625 , N=(10*2) = 20 years A Tk.2500 per value bond bearing 8% coupon rate having exactly 10 years remaining to maturity, currently sells at Tk.1500.  Interest is paid semi annually. If the nominal required rate of return is 12.50%, find the intrinsic value of the bond. We know that, Bond Valuation (Bv)=Int{1-1/(1+r)n/r}+MV/(1+r)n                                     =100 {1-1/(1+0.0625)20/0.0625}+2500 /(1.0625)20                                      = (100*11.2408)+743.63                                      = 1124.08+743.62                                      = Tk.1867.71(Ans) Practical Example-4 Following are the information relation to a bond, face value Tk.1,00,000 Coupon rate =14%,  Discount rate =16% maturity period =8 years Requirement: 1. What will be the price of the bond if interest in paid bi- monthly? Here,  FV=Tk.1,00,000 , MV=Tk.1,00,000, m= 6 , Interest =1,00,000*14%/6 = TK.2,330  r= 16% or 0.16/6  or 0.0267 , N=8*6 =48 times We know that,  Bond Valuation (Bv)=Int{1-1/(1+r)n/r}+MV/(1+r)n                                     =2330 {1-1/(1+0.0267)48 /0.0267}+1,00,000/(1.0267)48                                     =2330 {1-0.2823 /0.0267}+1,00,000/3.5423                                    = 62,630.40+28,230.25                                      = Tk.90, 860.65 (Ans) Practical Example-5 Calculate the value of the bond from the following information using PVIFA and PVIF respectively,  Interest of the Bond=Tk.250 Maturity life-12 years, Redeemable value Tk.1500, Expected rate of return =15%.  At first we have to  calculate PVIFA & PVIF  to solve this problem  Here,  FV=Tk.1,500 , MV=Tk.1,500, m= 6 , Interest = TK.250, r= 15 % or 0.15   , N= 12 years PVIFA ={1-1/(1+r)n /r}             = {1-1/(1+.15)12 /.15}             = 5.4206 PVIFA =  {1/(1+r)n }             = {1/(1+.15)12}             =0.1869   Bond Valuation (Bv)=(Int*PVIFA)+(RV*PVIF)                                       =(250*5.4206)+(1500*0.1869)                                                                          = 1355.15+280.35                                        = Tk.1,635.50 (Ans)              Practical Example-6 XYZ LTD a zero coupon Bond with a face value of Tk.1000   having a 10 years maturity,  Requirement: Calculate the value of the bond if the required rate of return is 12%. Here,MV=Tk.1000 R=12% N=10 years  Bond value(Bv)=MV/(1+r)n)                             =1000/(1+.12)10                             =1000/3.1058                            =Tk.322(Ans) Practical Example-7 A bond face value of Tk.2,000 and a coupon rate of 12%,   the bond are perpetual bond, that is they do not have any defiant maturity.  Calculate the value of perpetual bond if required rate of return is 10%. Bond value(Bv)=Int/r                             =240/0.10                            =Tk.2400(Ans) Practical Example-8 A bond face value of Tk.1000 and a coupon rate of 12%, is currently selling for Tk.800  what is the current Yield of the bond. Current Yield =(Int/sales value)*100                          =(120/800)*100                           =Tk.15%(ans) Practical Example-9 XYZ corporation bond have 12 years remaining to maturity interest is paid annually. The bond have a Tk.1000 per value and the coupon interest rate is 10%.  The bond sell at a price of Tk.850 .  Requirement: 01.what is their Approximate yield to maturity(YTM)? Here, FV = 1000, RV = 1000, SV = 850, Int-1000*10%,Tk.100, N = 12 years ,YTM =? We know that YTM=Int+RV-SV/N/ RV+SV/N*100 (When flotation cost  not exist)          =100+1000-850/10/ 1000+850/2*100 (When flotation cost  not exist)          =100+15/925*100           =(115/925)*100           =12.43%(Ans) Practical Example-10 Corporate practice bd Ltd. bonds will mature in 10 years.  The bond has face value of Tk.1000 and has an 10% annual coupon.  The bond has currently yield of 12%.    Requirement: What is the bond Yield to maturity (YTM?) Ans:- Here, FV = 1000, RV = 1000, Int-1000*10%,Tk.100, N = 10 years ,Current yield =12% or 0.12 , SV = (Interest / Current yield) = (100/0.12)=833.33 YTM = {Int+RV-SV/N/ RV+SV/2}*100          ={100+1000-833.33/10/ 1000+833.33/2}*100          ={(100+16.67/916.67)}*100           =(116.67/916.67)*100           =12.73%(Ans) Practical Example-11. Wimax LTD has a 14% debenture with face value of Tk.100 that matures at par in 15 years.  The debenture is callable in five years (05) at Tk.114. It currently sells for Tk.105.  Requirement: Calculate the bond yields to maturity and Yield to call.  Ans:- Here, Fv=100 ,RV=100 ,Int=100*14%=Tk.14, SV=105 ,N=15,CP=114, N(call price) = 5 We know that , YTM=Int+RV-SV/N/ RV+SV/2*100          ={14+100-105/15/ 100+105/2}*100          ={(100-0.33/102.50)}*100           =13.34 %(Ans) We know that , YTC= {Int+CP-SV/N/ CP+SV/2}*100 (When call price & call years is given)           =14+{(114-105/5)} / {(114+105 /2)}*100          = (100+1.80/109.50)*100          = 14.43% Practical Example-12. An investor recently purchase a bond with Tk.21,000 face value . 12.50% coupon rate and 8.50 years remaining to maturity.  The investor paid Tk.21,500 for the bond.  If the bond can be called two years from now at a price of Tk.22,160.    Requirement:  01. What is its YTC?   Here, FV=Tk. 21,000, SV=Tk.21,500,Int=12.50%- (21000*0.125)=Tk.2625 ,N=8.50 CP=Tk.22,160,     We know that,  YTC= Int+({CP-SV/N)} / ({CP+SV/2)} *100           = 2,625+({22,160-21,500 /8.50 )} /  ({22,160 + 21,500 /2 )}*100           = 2625+77.64 /21,830)*100           =12.38 % (Ans) Practical Example-13. Each of the bonds shown in the following table pays interest annually Bond Face value (TK) Sales Price(TK) Maturity period A 1000 350 10 B 2000 800 10 C 3000 2000 5 D 5000 3000 5               Requirement: 01.Calculate Yield to maturity for each bond Here,  (i) We notice that, there is no mention coupon rate      (ii) FV = Tk.1,000, SV =Tk350, N= 10 years ,YTM (r) =?      For Bond –A                         FV     =   SV (1+r)n      [when no mention coupon rate]                         Or,  1000 = 350(1+r)10                         Or,  350(1+r)10 = 1000                         Or, (1+r)10  = 1000/350                         Or, (1+r)10  = 2.8571                         Or, (1+r)=10√ 2.8571                         Or, (1+r)=1.110689                         Or, r =1.1106-1                         Or, r =0.1106                          = 11.06%(ans) Practical Example-14 ABC Company’s band which is currently sells for Tk.1,080 has a 10% coupon interest rate . and Tk.1000 per value pays interest annually and has 10 years to maturity. Requirement: 01.Calculate the approximate yield to maturity (YTM). also 02. Calculate the YTM for a 10 Years Zero coupon bonds sold at TK.500   Ans:  Req- (01) Here, FV = 1000, RV = 1000, Interest =Tk.1,000*10%, = Tk.100, N = 10 years ,SV =Tk.1,080 ,YTM =? We know that,   YTM= Int+ ( RV-SV/N) / (RV+SV/2) *100   [when already mention coupon rate]             =100+{(1000-1080/10)}/ ({1000+1080/2)} *100           =100+(-8) /1040 *100            =( 92/1040)*100              =8.85 %(Ans) Req- (02) Here, FV = 1000, RV = 1000, N = 10 years ,SV =Tk.500 ,YTM (r) =? We know that,              FV=SV (1+r) n     [when no mention coupon rate]               Or, 1000=500(1+r) 10               Or, 500(1+r) 10=1000               Or, (1+r) 10=1000/500               Or, (1+r) 10=2                Or, (1+r) =10√2               Or (1+r)=1.07177                Or, r =1.07418-1                Or,  r = 0.07177               Or, r = 7.18%(Ans) Practical Example-15 A Tk.1,000 bond is currently selling for Tk. 900. The coupon rate is 14%  and the appropriate discount rate is 15%. Requirement :calculate: 1.The value of the bond should it be bought 2. What is the current Yield? 3. What would be the Yield to maturity (YTM), if the maturity period is 5 years? We know that,  Hare, FV= Tk.1000,SV=Tk.900,Int=1000*14%=Tk.140,r=15% or, 0.15,YTM= ? 1.Bond value                           = Int/r  [When no mention N = Time period  ]                                                   = 140/0.15                                                   =Tk.933 2.Current yield = Int/sales price*100 [When no mention N = Time period  ]                              =140/900*100                              =15.56% 3. we know that,    FV= Tk.1000, RV= Tk.1000, SV=Tk.900,Int=1000*14%=Tk.140,r = 15% or, 0.15, N= 5 years YTM= ?                      YTM  =  Int+RV-SV/N/ RV+SV/2*100                                  = 140 +1000-900 /05/ 1000+900/2*100                                  = 140+20/950*100                                  = 160/950*100                                  = 16.84% Practical Example-16 A 9% 12 years bond of Tk.5,000 bond has been issued @ 5% discount and redeemed @ 3% premium.  If flotation cost is 2% of the face value. Requirement: 01.Calculates its approximate YTM. Here,        FV = Tk.5000,  Int = 5000*9% =Tk.450,         SV = {5000-(5000*5%) = 5000-250 =Tk.4750,       RV= {5000+(5000*3%) = 5000+150 =Tk.5,150       FC = 5000*2% =Tk.100,       NSV = SV-FC= 4750-100 = Tk. 4650       N =12,   YTM = ?  We know that,                       YTM = Int+RV-NSV/N/ RV+NSV/2*100 [When exist flotation cost]                               = { 450 +(5150-4650 /12) /  (5150+4650/2)}*100                               = { (450+41.67) / (4900)} *100                               = (491.67/4900)*100                               = 10.03% (Ans) Practical Example-17 The ABC  company bond have 4 years remaining to maturity.  The bond has a face value of Tk.1,000 at 10% coupon rate.   Requirement:-  01.What is the yield to maturity at a current market price of Tk.1100? 02. would you pay Tk.1,200 for one of these bonds,  if the appropriate rate of interest was 12%. Give explanation of your answer. We know that,  Req-01.    YTM = Int+{(RV-SV/N) / (RV+SV/2)}*100                           = 100+{(1000-1100/4 ) / (1000+1100/2)}*100                           = 100+(-)25/1050 *100                           = (75/1050)*100                           = 7.14 %(Ans)   Req-02. Here, FV=Tk.1000,NV=Tk.1000,Int =1000*12% =Tk.120,N=4,r=12%, BV =? We know that, Bond Valuation (Bv) = Int{1-1/(1+r)n/r}+MV/(1+r)n                                    = 120 {1-1/(1+0.12)4 /0.12}+1,000/(1.12)4                                    = 120 {1-0.635518 /0.12}+1,000/1.5735                                     = 364.48+635.52                                     = Tk.1000 (Ans) Explanation: No, we would not pay Tk.1200 for the buy of bond,  because its current worth Tk. 1,000 is less than the purchase price Practical Example-18 From the following information calculate the value of bond if interest is paid 1.Yearly 2.Bi-Monthly Face value of the bond =Tk.2000 Coupon rate =13% Discount rate =15%  Maturity period =7 years 1. If interest paid yearly Bond Valuation (Bv) = Int{1-1/(1+r)n/r}+MV/(1+r)n (Normal way)                                    = 260{1-1/(1+0.15)7/0.15}+2000/(1+0.15)7                                          = 260{1-0.635518/0.12 +2000/1.5735                                         = 1081.74+751.88                                         = 1833.58 (Ans)                  Practical Example-19-(CMA Exam-January-2023): The corporate practice bd Ltd. considering buying a machine that would be cost Tk.5,40,000.  The machine will be depreciated over five years by the straight line depreciation method and  will be worthless at the end of five years. The company can lease the machine with year end payments of Tk. 1,45,000. The company can issue a bond at 9% interest rate. If the corporate tax rate is 35%.   Requirement:(Try yourself)    (i) Should the company buy or lease the Machine?     Practical Example-20-(CMA Exam-January-2022):   The Hacking software has 6.2% coupon bonds on the market with 9 years to maturity.The bond makes semiannual payments and currently sell for 104% of par.   Requirement:(Try yourself)   (i) What is the current yield of the bond? (ii) What is the YTM?   (iii)  What is the effective annual yield?   Practical Example-21-(CMA Exam-January-2022):   The wimax com Ltd. is considering issuing a new 10 years bond in the domestic market. The interest rate on the bond is 20% .Interest will be paid semi-annually . The directors are considering the appropriate price at which the new bonds should be sold.The market required return is 25%.   Requirement:(Try yourself)   (i) Compute the price investors would be willing to pay for each Tk.100 face value bond. (ii) Explain how changes in average interest rate affect the value of bonds.  Practical Example-22: Let’s find the value of a corporate bond with an annual interest rate of 5%, making semi-annual interest payments for 2 years, After which the bond matures and the principal must be repaid.  Face value of Bond is $ 1,000. Assume a YTM of 3%: Ans: F = $1,000 for corporate bond Coupon rate annual = 5%, so, Coupon rate semi-annual = 5% / 2 = 2.5% C = 2.5% x $1000 = $25 per period n = 2 years’ x 2 times = 4 periods for semi-annual coupon payments r = YTM of 3% / 2 for semi-annual compounding = 1.5% Present value of semi-annual payments = 25 / (1.015)1 + 25 / (1.015)2 + 25 / (1.015)3 + 25 / (1.015)4 = 96.36 Present value of face value = 1000 / (1.015)4 = 942.18  Practical Example-23: Complex Systems has an outstanding issue of $1,000-parvalue bonds with a 12% coupon interest rate.  The issue pays interest annually and has 16 years remaining to its maturity date a. If bonds of similar risk are currently earning a 10% rate of return, how much should the Complex Systems bond sell for today? b. Describe the two possible reasons why the rate on similar-risk bonds is below the coupon interest rate on the Complex Systems bond. c. If the required return were at 12% instead of 10%, what would the current value of Complex Systems’ bond be? Contrast this finding with your findings in part a and discuss. Ans: a. Bo = I x (PVIFAkd%,n) + M x (PVIFkd%,n) Bo = 120 x (PVIFA10%,16) + M x (PVIF10%,16) Bo = $120 x (7.824) + $1,000 x (.218) Bo = $938.88 + $218 Bo = $1,156.88 Calculator solution: $1,156.47 b. Since Complex Systems' bonds were issued,  there may have been a shift in the supply-demand relationship for money or a change in the risk of the firm. c. Bo = I x (PVIFAkd%,n) + M x (PVIFkd%,n) Bo = 120 x (PVIFA12%,16) + M x (PVIF12%,16) Bo = $120 x (6.974) + $1,000 x (.163) Bo = $836.88 + $163 Bo = $999.88 Calculator solution: $1,000 When the required return is equal to the coupon rate, the bond value is equal to the par value. In contrast to a. above,  if the required return is less than the coupon rate, the bond will sell at a premium (its value will be greater than par)