This week we learn the basics of how to calculate the price of bonds and stocks. Since bonds and stocks both have cash flows going into the future, we can apply our present value formula for multiple cash flows. The present value computed will be our expected price or value of the bonds or stocks.
The company or government that issues the bond is the borrower. They sell the bonds to raise the borrowed money which will have to be paid back when the bond maturity or term is over (end of loan). The bondholder is the investor who purchased the bonds – they are the lender. Bonds are typically interest only loans, meaning the issuer makes periodic interest payments (coupon payments) and at the end of the loan they pay not only the final interest payment but also the amount borrowed in one big payment. The amount borrowed has several names: face value, par, principal or maturity value. The interest rate to calculate the interest only payment (coupon payment) is called the coupon rate. Annual coupon payment = coupon rate x face value Semi-annual coupon payment = coupon rate x ½ x face value
Bond Price = PV Bond Yield is the APR of the discount rate r. If bond has annual coupon payments then Bond yield = r, but if the bond has semiannual coupon payments then bond yield = 2 x r. Current yield is the total coupon payments made in one year divided by the current price. It shows on current return and can change over time as bond prices change. Notice in the PV formula that the par value for the bond is repaid until the very end. This is different than most home mortgages, where you pay back a small p[art of the principal every month.
9 PV=-1081.95 The sign is negative on the calculator’s pv calculation, because it balances cash flows – what you pay out initially is balanced by what comes back to you over time. So your answer for the price of this bond would be $1,081.95. The following slide shows the +/- nature of these bond cash flows.
This is how the cash flows look for the bond in the previous slide.
13 The semiannual compounding causes us to change several inputs to the calculator: N, I/Y, PMT N = number of periods = since the bond had a 3 year maturity , but now includes semiannual compounding, the number of periods = 3 years x 2 periods/year = 6 periods The discount rate must be converted to the periodic rate, since discount rates (Yield to maturities) are quoted in APR terms, we can just divide the annual rate by 2 to get the semiannual rate = periodic rate = annual rate / 2 periods/year = 2.25/2 = 1.075. The annual coupon = coupon rate x par Semiannual coupon = coupon rate x par / 2 = .05 x 1000/2 = 25 FV, which is par, will stay the same PV = price = 1,082.37
16 The more frequent compounding of the semi-annual coupons also slightly increased the value of the bond.
This is an fundamental rule on bonds you need to remember: As bond discount rates rise, bond prices fall. The bond discount rate can also be referred to as the yield to maturity and is affected by other interest rates in the debt markets.
Notice how interest rates do change over time. Most of us think about bonds as being the safe place to invest and they are generally safer than stocks, but there is risk. The risk is that if interest rates change – the price of the bond that you own will change.
Zero coupon bonds are what they are called – bonds that pay no coupon. You buy the bond for say 321.97, in this case, and in ten years you receive face value of $1,000. Just use the present value formula on the future face value payment to calculate the price of this bond: PV = 321.97 You could also calculate the price using the time value of money calculator keys: N=10 I/YR = 12 PMT = 0 FV = 1000 Compute PV You should get the same answer.
21 Here we are given a bond price of 1081.95 for the 5% annual coupon bond with 3 years to maturity and $1000 face value. Note: Some problems may not specify the face value – you can assume that face value equals $1000 unless otherwise stated. To solve this problem: N = 3 PV = -1081.95 -notice the negative sign on the pv input, forget this and you will get an error, remember cash flows in calculator must balance PMT = .05 x 1000 = 50 FV = 1000 Compute I/Y = 2.15% this is the yield to maturity
This is a very nice way of communicating interest rates (yields – not coupons) and how they vary with bond maturity.
If you buy a bond you receive your coupons and part of your return, but also any price appreciation in the bond. This formula is used to calculate your total return on owning a bond. An interesting point is that as long as interest rates and your bond’s yield to maturity do not change, your return on a bond will equal its yield to maturity. Remember the yield to maturity is our discount rate and the discount rate is basically our required or expected return on an investment.
By law many insurance companies are not allowed to invest is junk bonds due to their high risk.
Here is a description of the ratings categories. The ratings themselves are developed from regression models that use historical default data to predict when a bond will go into default. The higher the probability, the lower the credit rating. Recent history with mortgage backed securities has shown us that these ratings are not perfect. Also, the fact the ratings agencies are paid by the companies that they give ratings on – leaves many to think that there is too much conflict of interest.
Here are the average yields on bonds within each of the credit rating categories. The difference between yields is the default spread. Notice how the default spread varies over time. In bad economic times, the spread is high because investors worry more that risky companies will go into default.
Chapter 6-bonds (1)
Valuing Bonds Chapter Six
Bond Terminology Bond • A security that obligates the issuer to make fixed payments to the bondholder until the maturity of the bond (fixed income security) Face Value (par, principal or maturity value) • Payment made at the maturity of the bond. In the US it is usually $1,000 Coupon • The interest payments made to the bondholder. In the USA coupons are paid twice a year Coupon Rate • Total annual interest payment as percent of bond face value. Only used to calculate coupon. Not used as discount rate, but upon issuance the coupon rate is usually close or equal to the discount or required rate of return.
Bond Terminology and Pricing Bond Price • Equals the present value of all future coupon payments and the future par value payment. Bond Yield • The annualized (APR) discount rate used to calculate the bond’s present value. Also known as the yield to maturity or required rate of return. Current Yield • Annual coupon payments divided by bond price. coupon coupon ( coupon + par )PV = + + .... + (1 + r ) 1 (1 + r ) 2 (1 + r ) t
Bond Pricing Example What is the price of a 5.0 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 2.15%. 50 50 1,050PV = + + (1.0215) (1.0215) (1.0215) 3 1 2PV = $1,081.95 Calculator: N=3, I/Y=2.15, PMT=.05x1,000=50, FV=1,000, PV=?
Bond Pricing Example What is the price of the bond if the required rate of return is 2.15% and the coupons are paid semi- annually? 25 25 25 1,025PV = 1 + 2 + ... + 5 + 6 (1.01075) (1.01075) (1.01075) (1.01075)PV = $1,082.37 Calculator: N=3 years x 2 payments/year=6, I/Y=2.15/2 payments/year = 1.075, PMT=.05x1,000/2 payments/year=25, FV=1,000, PV=?
Semi-Annual Coupons How did the calculation change, given semi-annual coupons versus annual coupon payments? Time Periods Discount RatePaying coupons twice a year, Since the time periods are instead of once doubles the now half years, thetotal number of cash flows to discount rate is also be discounted in the PV changed from the annual formula. rate to the half year rate.
Bond Pricing Example What is the price of the annual bond if the required rate of return is 5.0 %? 50 50 1,050 PV = 1 + 2 + 3 (1.050) (1.050) (1.050) PV = $1,000
Bond Pricing Example What is the price of the annual bond if the required rate of return is 8 %? 50 50 1,050 PV = + + (1.08) (1.08) (1.08) 3 1 2 PV = $922.69
Interest Rate Risk The value of an existing bond falls as interest (discount) rates rise 1,200 1,100 1,000 900 B p d n o $ e c r ) ( i 800 700 0 2 4 6 8 10 12 14 16 Interest rate (%)
Interest Rate Risk The value of an existing bond falls as interest (discount) rates rise. The intuition is that, when buying a bond, we agree on a fixed interest payment. Afterwards, when current interest rates go up, our bond pays less interest than newer bonds, making our bond worth less.
Pricing of Zero-Coupon Bonds Price a 10 year zero-coupon bond at a 12% discount rate. PV = CFt / (1 + r)t = 1000 / (1 + .12)10 Price = 1000 / (1+0.12)10 = $321.97
Bond Yield or Yield to Maturity (YTM) When a normal coupon bond is initially issued its coupon rate should be close to or equal to its yield to maturity (discount rate) But market conditions like changing interest rates can cause the bond prices and yields to change over time while the coupon rate stays constant If given the bond price, you can calculate its YTM by solving for r below. coupon coupon ( coupon + par )PV = + + .... + (1 + r ) 1 (1 + r ) 2 (1 + r ) t
Bond Yields Example What is the YTM of a 5.0 % annual coupon bond, with a $1,000 face value, which matures in 3 years? The market price of the bond is $1,081.95. 50 50 1,050 PV = + + (1 + r ) (1 + r ) 1 2 (1 + r ) 3 PV = $1,081.95 YTM = 2.15%
The Yield CurveTerm Structure of Interest Rates - A listing of bond maturity dates and the interest rates that correspond with each date.Yield Curve - Graph of the term structure.
The Yield Curve Treasury zero coupon bonds (strips) are bonds that make a single payment. The yields on Treasury strips in February 2008 show that investors received a higher yield on longer term bonds. 6 5 4 Yield % 3 2 1 0 11 13 15 17 19 21 23 25 27 29 1 3 5 7 9 Maturity (years)
Bond Rates of ReturnRate of Return – actual earnings of aninvestor during a certain period of time perdollar invested in a bond. (coupons + change in price)Rate of return = investment (coupons + Price 1 – Price 0 )Rate of return = Price 0
Bonds and Credit RisksCredit risk is the risk of default by the issuer, which is theinability to pay coupons or face value at maturity. USGovernment debt has no credit risk by convention.Investors require higher yields on riskier bonds.The default premium is the difference between the yields ongovernment bonds and corporate bonds with the samematurity, but more default risk.
Default Risk and Ratings• Rating companies – Moody’s Investor Service – Standard & Poor’s – Duff and Phelps• Rating Categories – Investment grade = BBB, Baa and above – Speculative grade or Junk bonds = BB and below
Bond Ratings StandardMoody s & Poors SafetyAaa AAA The strongest rating; ability to repay interest and principal is very strong.Aa AA Very strong likelihood that interest and principal will be repaidA A Strong ability to repay, but some vulnerability to changes in circumstancesBaa BBB Adequate capacity to repay; more vulnerability to changes in economic circumstancesBa BB Considerable uncertainty about ability to repay.B B Likelihood of interest and principal payments over sustained periods is questionable.Caa CCC Bonds in the Caa/CCC and Ca/CC classes may already beCa CC in default or in danger of imminent defaultC C C-rated bonds offer little prospect for interest or principal on the debt ever to be repaid.