Bonds and Their Valuation

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What is Bond: a bond is an IOU issued by a borrower (US government, state and local governments or corporations). The price (value) of a bond on the date of issuance is usually equal to the face value (par value or maturity value, denoted by M in this chapter) of the bond. When bond investor buys a bond, she/he expects to receive interest payment known as coupon payment each year until the date of maturity of the bond. On the date of maturity she/he is entitled to receive the face value of bond. For example, suppose you buy a bond with $1,000 face value, a 10% coupon rate (denoted by c in this chapter) and a 15-year maturity (denoted by n in this chapter). In this case, you receive coupon payment (interest payment) of $100 each year for 15 years (assuming bond calls for annual payments). On the date of maturity you would receive $1,000 and the last coupon payment. The type of bond that we explained above is called” fixed coupon payment” bond with annual 2

payments (there are bonds that call for semiannual coupon payments). Bonds are generally classified as "fixed-income" securities.

 Bond’s Characteristics 1. Par value (M) 2. Coupon interest rate(c) 3. Coupon pmt = I = M x c 4. Required Rate of Return (rd) 5. Maturity 6. Years to maturity (n) 7. Issue date 8. Default risk 9. Special provisions: call provision ( call premium = rd(c) - rd (nc) >0) 10. Call price 11. Call Protection 3

 Types of Bonds(in general) 1. Treasury bonds and notes. 2. Corporate bonds 3. Municipal bonds 4. Eurobonds 5. Foreign Bonds 

Bond Valuation Model(Fixed Coupon Rate Bond)

Value of a bond is present value of its expected cash flows (Is and M):  1 1   n ( 1  r ) d VI   rd  

  M +  (1  rd ) n  

Example 1: The Morrissey Company's bonds mature in 7 years, have a par value of $1,000, and make an annual coupon payment of $70.

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The market interest rate for the bonds is 8.5%. What is the bond's price?

 1 1   n ( 1  r ) d VB  I   rd   1  1   (1  0.085) 7 V  $70  0.085  

  M +  (1  rd ) n  

  1000  $923.22 + 7 ( 1  0 . 085 )  

What is $70?  $70 is coupon payment (= interest income to the bond investor) and is calculated as follows: 1,000 x 0.07= $70 Example 2 Suppose we have a bond with the following characteristics M =$1,000, c = 10%, rd=8% and n = 3 years. The fundamental value of this bond is calculated as:

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 1 1  (1  rd ) n  VB  I  rd  

  + M  (1  rd ) n  

 1 1  3 ( 1  0 . 08 ) V  $100   0.08  

   + 1000  257.71  793.83  1051.54  (1  0.08) 3  

Example 3: Jack Weber calls his broker to inquire about purchasing a bond of Just-in-Time Technologies. His broker quotes a price of $1,180. Jack is concerned that the bond might be overpriced based on the facts involved. The $1,000 face value bond pays 14% interest (coupon rate) annually, and it has 25 years to maturity. The present interest rate on similar bonds is 12 percent. What is the fundamental value this bond? Is this bond overpriced? Please explain. M = $1,000; I = $140(=> coupon rate of 14%); n= 25 years; rd = 12% 1   1   (1  0.12) 25  1000  VB  140  1098.04  58.82  $1,156.86 25 0.12   (1  0.12)   Current market price =$1,180.00

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 Yield to Maturity (YTM) YTM is the discount rate that forces sum of present values of the future expected cash flows of a bond to be price of the bond, if bond is bought today and is held until the maturity. YTM can be viewed as internal rate of return of investment on this bond. It is the promised yield(ex-ante) if bond is purchased to-day and held to maturity. Or Yield to Maturity (YTM) = ex-ante yield; is the interest rate (discount rate/ex-ante yield) that equates the PV of all cash inflows from a bond to the price of bond, if it is held until the maturity.

Example 1: Ezzell Enterprises’ non-callable bonds currently sell for $1,165. They have a 15-year maturity, an annual coupon of $95, and a par value of $1,000. What is their yield to maturity?  1 1  (1  YTM )15 1,165  95   YTM  

N PV PMT FV I/YR

   + 1,000  (1  YTM )15  

15 $1,165 $95 $1,000 7.62%=YTM

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Example 2: Suppose a bond is now selling for $877.07(VB= $877.07) with coupon rate of 8% (c = 8%), maturity value of $1,000(M =$1,000), and years to maturity of 10 years (n = 10 i.e. years that is bond has 10 years to maturity), what is the YTM of this bond? Solution: 1   1   (1  YTM )10  1000 877.07  80  10 YTM   (1  YTM )  

,

Using financial calculator (HP, 10B), we key the following: PV= -877.07; FV=1000;PMT = 80; N=10; we solve for I/YR which is YTM = 10% Using financial calculator => YTM = 10%

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 Change in Bond’ value over time. Value of Bond(VB)

----------------------------------------------------------------------------------------------------------rd(c=10%) today(n=3) n=2 n=1 n=0(Just before maturity) 8%

$1,051.54

$1,035.76

$1,018.52

$1,000.00

10%

1,000.00

1,000.00

1,000.00

1,000.00

12%

951.96

966.21

982.14

1,000.00

How bond values in the above table are calculated? Here are 2 examples: a) Consider the first bond in the above table ($1,051.54). This bond has the following characteristics: M =$1,000, c = 10%, rd=8% and n = 3 years and its value is calculated as:  1 1  (1  rd ) n VB  I   rd  

  M +  (1  rd ) n  

 1 1  (1  0.08) 3 VB  $100   0.08  

  1000 +  257.71  793.83  1051.54  (1  0.08) 3  

b) Consider the last number in the 2nd column of the above table:  1 1  (1  rd ) n  VB  I  rd  

  M +  (1  rd ) n  

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 1 1  (1  0.12) 3  VB  $100  0.12  

   + 1000  240.18  711.78  951.96  (1  0.12)3  

Value (VB) rd=8%c=10% Years to Maturity

 Discount bonds: rd > c => V < M  Premium bonds: rd < c => V > M  Par bonds: rd = c => V = M  Total Rate of Return on Bond Held from t to t+1 : Total yield = Current Yield (CY) + Capital Gains Yield (CGY)

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RET  Total rate of return  Total yield  note :

I VB ( t )

VB ( t )



VB (t 1)  VB ( t ) VB ( t )

 iC  Current yield

VB ( t 1)  VB ( t ) VB ( t )

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 g  Capital gains yield

If stays the same(constant): t0 t1 M=$1,000 M=$1,000 I =$150 I =$150 rd = 15% rd = 15% n=15 n=14 VB0 =$1,000 VB1=$1,000

t2 M=$1,000 I =$150 rd = 15% n=13 VB2 =$1,000

RET  Total rate of return  Total yield  

150 1,000  1,000   15.00%  (0.0%)  15% 1,000 1,000

If interest rate falls: t0 t1 F=$1,000 F=$1,000 I =$150 I =$150 rd = 15% rd = 10% n=15 n=14 VB0 =$1,000 VB1 =$1,368.33

t2 F=$1,000 I =$150 rd = 10% n=13 VB2 =$1,355.16

RET  Total rate of return  Total yield  

150 1,355.16  1,368.33   10.96%  (0.96%)  10% 1,368.33 1,368.33

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If interest rate rises: t0 t1 F=$1,000 F=$1,000 I =$150 I =$150 rd = 15% rd = 20% n=15 n=14 VB0 =$1,000 VB1 =$769.48

t2 F=$1,000 I =$150 rd = 20% n=13 VB2 =$773.37

RET  Total rate of return  150 773.37  769.48    19.49%  (0.51%)  20% 769.48 769.48  Reading Quotations: I.

Treasury Bonds and Notes

II.

Corporate Bonds

 Yield to Call(YTC)  Yield to Call (YTC):Callability is a provision: the bondholder would not have the option of holding callable bond if bond called by issuer.  Rationale: XYZ corporation issues a set of annual bonds with M=$1,000;coupon rate =12%; n=15 year ;N=5 years. After 5 years interest on similar bonds falls to 6%; corporation then issues now bonds with the rate of 6% and saves interest cost: $120 – $60 = $60 per bond per year 12

 Calculation of YTC: 1  1  N   1  YTC  VB  I  YTC  

  CP  3  (1  YTC ) 

Where N is number of years until bond becomes callable; and CP is call price Example 1: Consider the following annual callable bond: Original Maturity = 10 years, issued 1 years ago, coupon rate = 8%, M = $1000, Price 110.961% of the face value and call price = $1,080 with call protection period of 5 year. To calculate YTC, we have:  1 1  1  YTC 4 $1109.61  $80  YTC  

    $1080  (1  YTC ) 4  

I use my financial calculator and I key the followings: PV= -1,109.61; FV=1080;PMT = 80; N=4; I solve for I/YR then YTC = 6.61% Example 2 : Consider the following callable bond:

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Original Maturity = 20 years, issued 2 years ago, c = 10.5%, M = $1000, Price 115.174% of the face value (i.e.V = $1,151.74 calculated as 1.15174 x 1000=$1,151.74), call price = $1,100, with call protection period of 5 years. To calculate YTC, we solve the following equation:

 1 1   3  1  YTC   $1,151.74  $105  YTC  

    $1100  (1  YTC ) 3  

Using financial calculator (HP, 10B), we key the following: PV= -1,151.74; FV=1100;PMT = 105; N=3; we solve for I/YR which is YTC = 7.73% Example 3: Sadik Inc.'s bonds currently sell for $1,280 and have a par value of $1,000. They pay a $135 annual coupon and have a 15-year maturity, but they can be called in 5 years at $1,050. What is their yield to call (YTC)?

1  1   1  YTC 5 V  135 YTC   

  1050  (1  YTC ) 5   

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N PV PMT FV I/YR = YTC

5 $1,280 $135 $1,050 7.45%

 Note that, generally, if a bond sells at a premium (i.e. its coupon rate is higher than current YTM of similar bonds) a call is likely. It follows you should expect to earn: - YTC on premium bonds - YTM on par and discount bonds  Value of Bond with Semi-annual Coupon Payments: 1  1   r (1  d ) 2 n  I 2 VB   rd 2  2 

   M   (1  rd ) 2 n  2 

Example 1 Assume that you are considering the purchase of a 15-year bond with an annual coupon rate of 9.5%. The bond has face value of $1,000 and makes semiannual interest payments. If you require an 11.0% nominal yield to maturity on this investment, what is the maximum price you should be willing to pay for the bond?

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Par value $1,000 Coupon rate 9.5% Periods/year 2 Yrs to maturity 15 Annual rate 11.0% =================== N = periods Periodic rate PMT/period FV PV

30 5.50% $47.50 $1,000 $891.00

Example 2: M= $1000, c=10%, n=20 years, rd =YTM=12% 1   1   (1.06) 40  1000  V  $50  $849.52 40 . 06 ( 1 . 06 )      YTM(ex-ante yield) of semi-annual bond is

solution the following equation: 1  2n 1  YTM    1   I  2   VB   YTM 2  2   

        

M YTM   1   2  

2n

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Consider a 3-year, 5%, $1000 per value semi-annual bond at a 97.291% price, what is YTM (Ex-ante yield) of this bond?

1   1  6   YTM     1   50  1000 2    972.91   6  YTM 2  YTM     1   2 2      

Using MY financial calculator (HP10-B):

PV= -972.91; FV=1000; PMT = 25; N=3x2; we solve for I/YR which is YTM/2 = 3.00% => YTM=6% How about YTC in the case of semi-annual payment? easy: 1  2N 1  YTC    1   I 2   V  YTC 2  2  

   CP   2N  YTC    1   2    

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Example 1: Hood Corporation recently issued 20-year bonds. The bonds have a coupon rate of 8 percent and pay interest semiannually. Also, the bonds are callable in 6 years at a call price equal to 115 percent of par value. The par value of the bonds is $1,000. If the yield to maturity is 7 percent, what is the yield to call?

Solution: First, calculate the price of the bond as follows: 1   1  220   0.07     1   80  1000 2    Pb    $1,106.78 220  0.07 2  0 . 07    1   2 2      

Now, we can calculate the YTC as follows, recognizing that the bond can be called in 6 years at a call price of 115% x1,000 = 1,150: 1  1  26  YTC    1   80   2  $1,106.78  YTC 2   2  

    1.15  1000 26  YTC    1   2    

Using my calculator: N = 6 x 2 = 12, PV = -1,106.78, PMT = 40, FV = 1,150, and solve for I/YR = ? = 3.8758% x 2 = 7.75%.

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Example2: Keenan Industries has a bond outstanding with 15 years to maturity, an 8.75% coupon paid semiannually, and a $1,000 par value. The bond has a 6.50% nominal yield to maturity, but it can be called in 6 years at a price of $1,050. What is the bond’s nominal yield to call? First, use the given data to find the bond's current price. Then use that price to find the YTC. Coupon rate 8.75% YTM 6.50% Maturity 15 Par value $1,000 Periods/year 2 Determine the bond's price PMT/period $43.75 N 30 I/YR 3.25% FV $1,000.00 PV = Price $1,213.55

Yrs to call Call price

6 $1,050.00

Determine the bond's YTC N 12 PV $1,213.55 PMT $43.75 FV $1,050.00 I/YR 2.64% Nom. YTC 5.27%

Zero-coupon Bonds (Zeros): Zero coupon bonds are type of bonds that do not offer any coupon payments. This means that investor (bondholder) receives only the face value (maturity value) of the zero-bond on the date of maturity. The value of a zero is calculated using the following formula:

M Vzero  (1  rd ) n Example: suppose you are looking at a zero with the face value of $5,000 and 10 years to maturity. 19

What is the value of this zero if your required rate of return ( rd) is 8%?

Vzero 

$5,000  $2,315.97 10 (1  0.08)

Perpetual Bonds: Perpetual bonds are type of bonds that never mature. They call for coupon payments (interest payments) forever. The value of a perpetual bond is calculated using the following formula: V perp 

I rd

Example: suppose you are looking at a perpetual bond that calls for $50 interest payment per year. What is the maximum price that you are willing to pay for this bond if your required rate of return is 7%?

V perp 

$50  $714.29 0.07

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 Interest Rate Risk of Bond The effect of Δs in i on the realized rate of return of a bond causing its return to deviate from it YTM is referred as “interest rate risk”. 1.

Price risk: the effect of Δs in interest rate on price of a bond is referred as “price risk”. That is the risk of interest rate rising causing market price of bond falling if bond is sold prior to it maturity thus resulting in a lower realized rate of return (ex-post rate of return).

2.

Reinvestment risk: the effect of Δs in interest rate on reinvestment income of a bond is referred as “reinvestment risk”. That is the risk of interest rate falling and having to reinvest the coupon payments at a lower rate than YTM resulting in lower realized rate of return (ex-post rate of return).

 Short Term Bonds Versus Long Term Bonds Short Term Bonds: Reinvestment risk Long Term Bonds: Price Risk = Reading and understanding T-notes and Tbonds quotations from WSJ

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