Bonds in Finance – Valuation Concepts & Methods
Bonds are a way of raising funds whenever a corporation or government needs to make a long-term investment. Such ongoing capital raising endeavors have given rise to the bond market, the value of which far exceeds the capitalization of companies listed on the New York Stock Exchange or NASDAQ. The U.S. government is a major participant in the bond market, issuing a variety of bills, notes and bonds, which, because they pay interest, are important investment instruments for individuals and institutions alike. In what follows, we will examine some of the essential concepts that bond investors must consider before purchasing bonds and the way that bonds are valued.
The Opportunity Cost of Money
Current methods of bond valuation are based on the concept of opportunity cost, which is essentially the cost or price of choice. In a world of finite resources, choosing one thing always means forgoing another. If we have spent our last dollar on a hamburger, we won’t be able to buy pie, as well. Therefore, the pie is the opportunity cost or “price” we pay for the hamburger. In economics, the opportunity cost of a resource is defined as the value of the next (or second) best use of a resource, assuming that the best use is the one chosen.
The extension of this concept gives us the time value of money, since money invested one way has lost an opportunity, i.e. incurred an opportunity cost, to be invested in some other way. At the very least, money may be deposited in a bank account where it will earn interest at rate r, say. After one year, the amount in the account will increase in the way given by the compound interest formula: A = P(1 + r)^{t}. (A is the final amount; P is the amount deposited; r is the annual rate of interest and t is the number of years)
This formula can be rearranged to compute how much has to be deposited to achieve a certain sum, after a certain time, at a set rate:
~ P = A/(1 + r)^{t} ~
From this perspective, the principal amount, P, is the present value of a future amount.
Similarly, when an investor purchases a particular bond, an opportunity cost arises, which is measured by the next best way the funds could have been used. Generally, this opportunity cost is measured by the value that may have been derived if the investor had purchased a bond of similar risk. As a result, the value or price of any bond depends on the value offered by other bonds. Since the income or return from a bond is determined by its coupon rate, the coupon rate will be compared to that offered by similar investments. As a result, the value or price of a bond varies with changes in interest rates. How it does, is illustrated below.
Interest Rates and Bond Prices
The value of a bond is the present value of its cash flows: the periodic interest payments and the repayment of principal. To find this value, these cash flows are discounted by the current interest rate, to adjust for the time value of money. For example, consider a 3-year bond paying an annual coupon of 5% when current interest rates are 4.5%. The value of the bond would be calculated as follows:
Present Value (PV) = CF^{1}/(1 + r)^{1} + CF^{2}/(1+ r)^{2} + CF^{3}/(1+ r)^{3}. CF^{1 }refers to cash flow – the coupon payment – in Year 1; CF^{2 }to Year 2 cash flow and so on. These cash flows are discounted at the current interest rate, r, compounded annually, (1 + r)^{1}, (1+ r)^{2} etc, as in the table below.
Year 1 | 50 | 47.85 |
Year 2 | 50 | 45.79 |
Year 3 | 1,050.00 | 920.11 |
PV | 1,013.74 |
The present value of this bond is $1,013.74. The price would be expressed as 101.374, that is 101.374% of its face value of $1,000.
In the U.S., most bond coupon payments are semi-annual. This means that for the bond above, the coupon rate would be 2.25% for the 6-month period. The bond with a semi-annual coupon will have a slightly higher value.
1 | 25 | 24.45 |
2 | 25 | 23.91 |
3 | 25 | 23.39 |
4 | 25 | 22.87 |
5 | 25 | 22.37 |
6 | 1,025.00 | 896.9 |
PV | 1,013.89 |
Instead of three, there are now six payments. In Year 1, payments 1 and 2 pay the coupon after six months and a year. Payments 3 and 4 are the coupon payments for Year 2. Payment 5 is the coupon for Year 3. Payment 6 is the final coupon plus the repayment of principal. All cash flows are discounted at the rate of 2¼%.
How Bond Prices Vary with Interest Rates
Bond prices are affected by interest rates. For example, if economic conditions change and bonds similar to our 3-year bond start paying 6%, its value would fall to $972.91.
1 | 25 | 24.27 |
2 | 25 | 23.56 |
3 | 25 | 22.88 |
4 | 25 | 22.21 |
5 | 25 | 21.57 |
6 | 1,025.00 | 858.42 |
PV | 972.91 |
The inverse relationship between interest rates and bonds applies generally. Bond prices fall when interest rates rise and rise when interest rates fall.
Current Yield of a Bond
The current yield of a bond depends on the price paid for it. For example, if we sell our 3-year bond with an annual 5% coupon at its last value of $972.91, the purchaser would earn a current yield of 50/972.91 = 5.139%. The current yield is simply the annual coupon as a proportion of the price. In addition, if he holds the bond until maturity, he is sure to have a capital gain, since he will receive $1,000 on the date of maturity. The capital gain is $1,000 – $972.91 = $27.09. This capital gain improves the overall yield but only if the bond is held to maturity. The current yield including capital gains and losses is known as the yield-to-maturity. Generally, since interest rates fluctuate and bond prices vary, a bond sold before maturity will, most likely, have either capital gains and losses for the purchaser and seller.
Yield-to-Maturity
The Yield-to-Maturity (YTM) = Current Yield +/- Capital Gains/Losses
For a bond selling at face value, the current yield equals the yield-to-maturity, for there are no capital gains or losses: the face value of the bond is $1,000 and the bondholder will receive $1,000 at maturity. In addition, the rate received equals the coupon rate; the $50 annual coupon is exactly 5% of $1,000. Naturally, a bondholder can always sell before the date of maturity, at which time he will receive a price based on the current market value of the bond, as we did when the 3-year bond was sold.
However, any bond not selling at face value will have a capital gain or loss, if held to maturity. Consequently, an investor who buys a bond at a premium must absorb a capital loss over the life of the bond, so the return on these bonds is always less than the bond’s current yield. If he buys a bond at a discount, he will enjoy a capital gain over the life of the bond, providing a return that is greater than its current yield.
The yield-to-maturity is the discount rate – the current rate level of bonds of similar risk – at which the price of the bond equals the present value of its cash flows. For a bond selling at face value, the YTM will match the coupon rate. Consider a 3-year bond paying an annual coupon of 7%, when current rates are 7%. The present value of this bond equals its face value at 7%.
PV ($) at 7% = 70/(1.07) + 70/(1.07)^{2} + 1070/(1.07)^{3} = 65.42 + 61.14 + 873.44 = $1,000.
However, if the bond was purchased for $1,100, its YTM would fall substantially to 3.44% because of the capital loss of $100.
While it’s relatively easy to calculate present value when the discount rate is known, there is no simple general procedure for doing the reverse, i.e., computing the discount rate for a given present value. It is usually a matter of trial and error. The difficulty arises because of the compounding factors that result in mathematical exponents, i.e. powers of the interest rate term, (1.07)^{1}, (1.07)^{2}, (1.07)^{3}, etc. However, these difficulties do not arise for one-period bonds, and the YTM is easily calculated.
For example, for a 1-year bond with a coupon rate of 7% that was purchased for 101, the YTM would be 6.36%. At 101, a bond with a face value of $1,000 cost $1,100. Its yield-to-maturity is YTM = 70/1100 = 0.0636 = 6.36%
Example of Bond Financing for a Project
Sources of Funds | $ |
Par Amount | 60,000,000 |
Net Amount of Premium | 850,000 |
Total | 60,850,000 |
Uses | $ |
Project Fund Deposits | |
Project Fund | 50,000,000 |
Other Fund Deposits | |
Capitalized Interest Fund | 5,500,000 |
Debt Service Reserve Fund | 5,000,000 |
Delivery Date Expenses | |
Costs of Issuance | 150,000 |
Underwriter’s Discount | 200,000 |
Total | 60,850,000 |
As the table above indicates, although the project cost is $50 million, bonds with a face value of $60 million are issued at 101.42, which results in total proceeds of $60, 851,000.
The extra cash covers a variety of expenses associated with bond issuance. The net amount of premium is the amount above par value received, since the bonds were sold at 101.42. The Capitalized Interest Fund is part of the proceeds set aside to ensure that funds are available to pay interest on the bonds before the project begins to generate cash flows. A similar fund that usually covers principal, as well as interest, called a Debt Service Reserve Fund is typical. A Debt Service Reserve Fund will usually ensure that enough funds are on hand to cover at least one-year’s worth of debt service payments. Both the Capitalized Interest Fund and the Debt Service Reserve Fund are usually invested in short-term securities, the income from which will increase the funds.
Bond issuance costs money. Expenses include fees to a rating agency, to legal professionals, to trustees, and to the underwriters.
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