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EBS – Bond Pricing, YTM, Forward Interest Rates Bond pricing and yield to maturity Pricing the bond The bond price is obtained by discounting the bond cash flows by the set of spot rates covering the timing of the cash flows. If you have a 9% coupon, 5 year bond, the cash flows will be 9, 9, 9, 9, 109. The 9 is the interest that the bondholder gets each year (this is 9% of the par or face value, which will be 100). The 109 at the end is the repayment of the original 100 plus the last coupon payment of 9. If the set of spot rates for the next 5 years is 6%, 6.25%, 6.5%, 6.95% and 7.5% what is the bond price? The bond price is the discounted value of the bond’s cash flows, so the calculation would be: Bond price = 9 + 9 + 9 + 9 + 109 (1.06) (1.0625)2 (1.065)3 (1.0695)4 (1.075)5 Bond price = 106.72 The bond price is the present value of the bond’s cash flows, discounted by the set of spot rates. Terminology Face value: also known as ‘Par value’ or ‘Nominal value’. This is usually the issue price of the bond. In the UK this will be £100, in the US and Europe it will be $1000 or €1000. Coupon: this is the income return that the bondholder receives. The coupon is usually expressed as a percentage, for example, 6%. This is the coupon rate. The income that a 6% bond would deliver in the UK would be £6 and in the US and Europe, $60 or €60. The bondholder will receive this every year through until the maturity of the bond. Maturity: this is when the bond is due for repayment, for example, the 6% bond above might have a seven year maturity. What that means is that the company has to pay back the sum borrowed originally (ie the face value of the bond for all the bondholders of that bond issue). The redemption value of the bond is the same as the face value of the bond (£100, $1000, or €1000). Market price of bond: the bond will be sold at face value (£100, $1000, or €1000), but once it is on the market, the bond price may move away from that face value. Why would it do that? The two major factors that affect the bond price are; the term structure of interest rates and the creditworthiness of the borrower. The term 1
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EBS – Bond Pricing, YTM, Forward Interest Rates structure of interest rates is the set of interest rates that hold from overnight borrowing through to 30 year borrowing. The cash flows of the bond will be discounted by the appropriate rates for each time period from the term structure. So this means that the bond price is affected by the set of interest rates that dictate the direction of the economy. If the economy is in recession, interest rates will be low and as a result, bond prices will be high (because they are being discounted by the low interest rates). The second factor that affects the bond price is the creditworthiness of the borrower. If this deteriorates, then the interest rates for the borrower will go up. So the individual term structure for the borrower will see a widening spread over the risk free rate of interest (this effect can be seen with Greece, Ireland and Portugal in the Eurozone – their borrowing rates are over 10% higher than Germany’s for the key ten year borrowing rate). Types of bond: Fixed rate bond, this is the most popular, it has a fixed coupon and the borrower pays that rate through to maturity. Floating rate bond, this bond has a coupon that is reset every six months according to the level of LIBOR (London Inter-Bank Offer Rate). What this means is that the return the bondholder receives is variable. If interest rates go up, the bondholder will receive more interest income. Zero coupon bond, this is a bond that pays no income whatsoever (0%). You might say, why would anyone buy that? Zero coupon bonds are issued at a discount, for example a three year bond might be sold to investors for £86.15. The investor will receive no income, but they will get £100 back at the end of three years. So you can work out the annual return received from the bond; 100/86.15 to the third root. This is 1.160766(1/3) = 1.05095, so the annualized return on the zero coupon bond would be 5.095%. This bondholder does not receive an annual return, rather a capital gain after three years – which is taxed at a different rate from the income tax on a normal bond – a reason for some people to buy these bonds. A bond represents the debt of a company. Bank borrowings are also company debt. Smaller companies borrow from a bank, larger companies are able to sell bonds. Larger companies can have many bonds outstanding, for example, Vodafone has over 30 bonds in issue at the current time (Oct 2013). You can see this by looking at their debt investor page, http://www.vodafone.com/content/index/investors/debt_investors.html Vodafone have between 35 and 40 bonds in issue at any time, you can see the maturities range from very soon (sometimes they don’t update the page, so old issues hang about for a while) through to Feb 2032 (19 years from just now). The face value of Vodafone debt in Feb 2012 was roughly $33.5bn or £20.6bn. The market value of Vodafone equity at that time was £89.5bn (share price 174.95p), so gearing by market value was 18.7% (debt/debt+equity). Gearing by book value will be higher as the market value of Vodafone’s equity is higher than the book value of equity. 2
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EBS – Bond Pricing, YTM, Forward Interest Rates Calculating the YTM Once you have the bond price you can work out the yield to maturity. The YTM is simply the internal rate of return (IRR) of the bond. It is the same calculation as IRR. The YTM is a very important figure for a company. It represents the replacement cost of debt for the business. If, for example, you took the Vodafone example and they wanted to undertake a five year project partly funded with debt. How do you find out what the debt rate is? Well you would look for 5 year Vodafone bonds, find out the price of these bonds and work out what the YTM is on the bonds. If Vodafone want to sell any new 5 year bonds, the YTM that has just been calculated on the existing 5 year bonds will be the rate that will be offered on the new 5 year bonds. This YTM figure could very easily be different from the coupon rate on the existing 5 year bonds. This is because the price of the bond has changed over time, there may be a different economic environment to when that bond was issued and Vodafone’s credit quality may have improved or deteriorated. YTM You regard the price of the bond as a cash outflow, the price you pay for the bond today. You would lay out the cash flows as follows: -106.72 = 9 + 9 + 9 + 9 + 109 (1+YTM) (1+YTM)2 (1+YTM)3 (1+YTM)4 (1+YTM)5 You have to guess what the YTM rate will be. This is the rate that will discount the bond’s cash flows so that they exactly equal 106.72. If you buy the bond for 106.72 and hold it for 5 years you will lose £6.72, but you will earn £9 income per year. The YTM is a return that is a blend of the income you get and the capital gain or loss you make on the bond. With this bond you know that you will lose slightly more than £1 per year on the capital (a loss of £6.72 coming from 100 -106.72, divide the loss by 5 giving 1.344). You know the YTM, the average return on the bond will be less than 9%, because you will be making a capital loss on the bond. If you subtract the 1 from the bond coupon that will give you a starting guess of 8%. So try discounting at 8%. year 1 2 3 4 5 Bond price -106.72 9 9 9 9 109 Discount at 8% discount factor 0.926 0.857 0.794 0.735 0.681 table A1.1 look up 8% for figures discounted cash flow 8.3 7.7 7.1 6.6 74.2 sum of all cash flows = 104.0 bond price at 8% 3
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EBS – Bond Pricing, YTM, Forward Interest Rates The bond price works out at 104. This is lower than 106.72. This means the interest rate is too high, so try a lower interest rate. What we are trying to do is to get a value that is too high and a value that is too low. If we do that we know that the YTM will be somewhere inbetween the two interest rates and we can interpolate to get the correct YTM. So try 7% to get a higher value (remember that if you lower the interest rate you are increasing the present value of the cash flows. Discounting at 7%. year 1 2 3 4 5 Bond price -106.72 9 9 9 9 109 Discount at 7% discount factor 0.935 0.873 0.816 0.763 0.713 table A1.1 look up 7% for figures discounted cash flow 8.4 7.9 7.3 6.9 77.7 sum of all cash flows = 108.2 bond price at 7% The bond price works out at 108.2, this is too high. But we know that 106.72 is inbetween 104 and 108.2. So we know that the YTM is somewhere between 7% and 8%. We interpolate to get an accurate answer. YTM = r1 + ((NPV1/(NPV1 + NPV2)) * (r2 – r1)) Where r1 = first interest rate guess (here 8%) r2 = second interest rate guess (here 7%) NPV1 = actual bond price – first guess value (here 106.72 – 104 = 2.72) NPV2 = actual bond price – second guess value (here 106.72 – 108.2 = 1.48) *** you ignore the sign when calculating these values – all you want is the distance between the points. You have all the numbers for the calculation YTM = 8 + ((2.72/(2.72 + 1.48) * (7 – 8)) YTM = 8 + ((0.6476) * (-1)) YTM = 8 + (-0.6476) YTM = 7.3524% And that’s all there is to it. Try one for yourself. There is a bond priced at £95.32, with four years to maturity and a coupon of 4%. What is the YTM? (Answer at the end of the file/document). 4
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EBS – Bond Pricing, YTM, Forward Interest Rates Forward interest rates Terminology Spot rates: exist from today until some point in the future Forward rates: exist from a point in the future to a further point in the future Forward interest rates are used by companies to lock in an interest rate today that will take effect at some point in the future. For example, a company knows that it will have a borrowing requirement in six months time. It does not need the cash just now, but it is worried that interest rates might rise between now and six months time when they need to borrow. So, what it can do is to take out a forward interest rate contract, which locks in a future borrowing rate today. The company can fix the interest rate it borrows at in six months time today. This is done using the term structure of interest rates that exists in the market today. From that the bankers will be able to calculate what the forward rate will be that the company pays. This means the company has got certainty over its borrowing rate, they do not need to worry whether the rate will be higher or lower in six months. Nomenclature In labeling these rates we use s for spot rates, and f for forward rates. Preceeding the s or f will be a number that represents the start of the interest rate. There is a number that follows the s or f, this represents the end of the interest rate. For instance 2f3 is a forward rate that starts at the end of year 2 and runs to the end of year 3. 1f4 is a three year forward rate that starts at the end of the first year and runs to the end of year 4. In calculating these forward rates, the relevant rates to take are contained in the label for the forward rate. For example 2f3 To calculate 2f3, we must calculate (1 + 2f3) (1 + 2f3) = (1 + 0s3)3 (1 + 0s2)2 The forward rate will be calculated from the set of spot rates that are available in the market place. You can see from the calculation that the 0s3 part comes from the 3 that is after the f in 2f3, and the 2 comes from the 2 that is before f in the 2f3. Similarly for 1f4 (1 + 1f4)3 = (1 + 0s4)4 (1 + 0s1) In this case your answer will be a three year return, and you will need to take the answer to the third root to annualize the rate (all the interest rates are annualized). 5
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EBS – Bond Pricing, YTM, Forward Interest Rates If you are given forward rates and asked to calculate the relevant spot rates it is just a case of rearranging the formula; Eg, You are given 0s1, 1f2, 2f3. From this you can work out 0s2 and 0s3. (1+1f2) = (1+0s2)2 (1+0s1) Now if you have 1f2 and 0s1, but not 0s2 you have to rearrange and solve for 0s2; (1+0s2)2 = (1+0s1)*(1+1f2) Remember the answer you get will be a two year return so you need to take the answer to the second root to express as an annualized rate. For 0s3, think of (1+2f3) = (1+0s3)3 (1+0s2)2 Here we are missing 0s3. We have just worked out 0s2 so we can use it to calculate 0s3. The formula is (1+0s3)3 = (1+0s2)2*(1+2f3) Remember the answer you get will be a three year return so you need to take the answer to the third root to express as an annualized rate. This process could be continued as long as you have the relevant forward rates, then you could calculate the missing spot rates. This kind of methodology would be used to solve Case study 1.7 6
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EBS – Bond Pricing, YTM, Forward Interest Rates Example: If you have 1, 2, 3, and 4 year spot rates of 4%, 4.5%, 5%, and 5.5% respectively, what is the one year forward rate starting in one year’s time? What is the two year forward rate starting in two years time? What is the one year forward rate starting in three years time? First you need to label the forward rates, they are; 1f2, 2f4, and 3f4. Once you have the correct labels for the forward rates it is easy to set up the calculation. (1 + 1f2) = (1 + 0s2)2 (1 + 0s1) = 1.0452 1.04 = 1.05002 1f2 = 5.002% (1 + 2f4)2 = (1 + 0s4)4 (1 + 0s2)2 = 1.0554 1.0452 = 1.134429 This is a two year return, you need to annualize, so take the second root: = 1.06510 2f4 = 6.51% For 3f4 the answer is 7.014%. 7
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EBS – Bond Pricing, YTM, Forward Interest Rates If you have one year spot rate of 6% and the one year forward starting at the end of year one of 6.5% and the one year forward starting at the end of year two of 7%, what are the two and three year spot rates? For the two year spot rate: (1+1f2) = (1+0s2)2 (1+0s1) Here you have 1f2 and 0s1, but not 0s2 you have to rearrange and solve for 0s2; (1+0s2)2 = (1+0s1)*(1+1f2) (1+0s2)2 = (1.06)*(1.065) (1+0s2)2 = 1.1289 This is a two year return so we square root to annualize; (1+0s2)2 = 1.062497 Subtract 1 to get your spot rate = 6.25% For the three year spot rate: (1+2f3) = (1+0s3)3 (1+0s2)2 Here we are missing 0s3. We have just worked out 0s2 so we can use it to calculate 0s3. The formula is (1+0s3)3 = (1+0s2)2*(1+2f3) (1+0s3)3 = 1.1289 * 1.07 (1+0s3)3 = 1.207923 This is a three year return so we take the third root to annualize; (1+0s3)3 = 1.06499 Subtract 1 to get your spot rate = 6.5% Practice these types of question at the Self Assessment Area in the EBS Finance Course website , Cases 1.6 and 1.7. Practice Questions: 8
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EBS – Bond Pricing, YTM, Forward Interest Rates 1. There are four zero coupon bonds that are priced at £95.45, £90.62, £85.74 and £81.16, maturing in one, two three and four years time. Calculate the spot rates that will hold for years one, two, three and four. From these spot rates calculate the one-year forward rates that would start in years one, two and three. Calculate the two-year forward rate that starts in year two. There is a coupon paying bond that has four years to redemption. The coupon is £6.50, what is the price of the bond? 9
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EBS – Bond Pricing, YTM, Forward Interest Rates Solution The spot rates are found by dividing the price of the zero coupon into the par value (100) and taking to the root of the number of years. ZC bond par =100 year price spot yield full return 1 95.45 0.04767 1.04767 2 90.62 0.05048 1.10351 3 85.74 0.05262 1.16632 4 81.16 0.05357 1.23213 {Full marks for correct answers. Give credit if simple error has resulted in wrong answer. Less marks if they don’t know how to obtain spot rates.] The forward price is gained, for example using the one year forward starting in year one by using the following formula: (1 + 1f2) = (1 + 0s2)2 (1 + 0s1) in this case = 1.10351 1.04767 = 1.05330 subtract 1.0 to obtain the forward rate = 5.33% The rest of the forward rates can be obtained in a similar manner. The full set of forward rates is set out below: forward rates 1f2 1.05330 5.330% 2f3 1.05692 5.692% 3f4 1.05643 5.643% 2f4 1.11656 5.67% The 2f4 rate will produce a two year return, this must be square rooted to give an annualised forward rate. The bond price is obtained by discounting the bond cash flows by the set of spot rates. coupon: 6.5 term: 4 year: 1 2 3 4 value: 6.5 6.5 6.5 106.5 10
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EBS – Bond Pricing, YTM, Forward Interest Rates disc factor 1.0477 1.1035 1.1663 1.2321 disc c/f = 6.20 5.89 5.57 86.44 £104.10 Bond Price Yield to Maturity (YTM) calculation from earlier (p.4), YTM = 5.33%. If you have a 4 year bond priced at £95.32 and a coupon of 4%, you know the YTM will be greater than 4%, because the bond will be redeemed at £100 (so you make a gain of £4.68). If you roughly work out the income and capital gain on an annual basis, that will give you a starting point for working out the YTM. Rough YTM = Annual (income + capital gain) = 4 (income) + 1.17 (capital gain 4.68/4) = 5.17. Take this and use percentage either side, eg 4% and 6%. Work out bond price using 4% discount rate = £100 (this is the original coupon rate, so bond will = par value at this discount rate). Difference between this and current bond price = 4.68. Now work out bond price using 6% discount rate = £93.07. Difference = 2.25 Now interpolate using the YTM formula from the formula sheet; YTM = r1 + [(NPV1/(NPV1 + NPV2))* (r2-r1)] YTM = 4 + [(4.68/(4.68 + 2.25))* (6 - 4)] YTM = 4 + (0.675 * 2) YTM = 4 + 1.35 YTM = 5.35% The answer of 5.33% was calculated using excel, so will be more accurate, but using 4% & 6% is really pretty close. 11
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