4.1.2.1 Zero Bond Rates and Interpolation

The simplest way to obtain an appropriate interest rate for discounting a given future cash flow would be to find a comparable zero coupon instrument with a redemption date that falls on the same date as the cash flow. In general it is unlikely that such an instrument would be available for an arbitrary date. Fortunately this can be easily overcome. If no zero coupon instruments with a fitting date exists we can use interpolation to arrive at an approximate rate from one zero coupon instrument that matures before the cash flow we want to price and one that matures after it. Formula 4-8 provides a method for Linear Interpolation that you can use to calculate an approximate rate for a given cash flow date that lies between the maturity dates of two zero coupon instruments such as either Zero Coupon Bonds or money market instruments that do not pay coupons (such as commercial paper).

In Example 4.1 you can see how Formula 4-8 can be used with some specific zero coupon bonds to obtain an interpolated interest rate for a point in time that lies between the maturity dates of the two chosen reference bonds.

4.1.2.2 Bootstrapping and Interpolation

Although in some cases you may have two or more suitable Zero Coupon Bonds for which you may be able to obtain accurate market prices, in general there are not enough zero coupon instruments to be able to construct complete yield curves from the shortest maturities all the way to long-term maturities above 10 years and reaching 15, 20, or even 30 years.

One way to get around this problem is to recover zero coupon rates from the prices of coupon bearing bonds using a procedure called bootstrapping.

To illustrate what bootstrapping is, let’s consider the case where you have obtained a collection of coupon bonds each paying a six monthly coupon. The shortest maturity in this collection is a bond with only the last coupon and the redemption still outstanding, and thus six months or less from maturity. The next maturity in the collection should be a bond with two coupons still outstanding and thus a year or less from redemption. The remaining bonds in the collection in increasing order of time to maturity each should have one more additional coupon period remaining the one preceding it, starting with one that has three coupons outstanding.

Bootstrapping works by solving first for the interest rate implied by the price for the bond with only one coupon remaining. The interest rate obtained in this way can then be substituted back into the calculation for the implied interest rate for the last coupon and redemption of the bond with two coupons outstanding, which now becomes solvable.

After solving for the implied one year zero coupon interest rate we can then substitute this together with the six month rate back into the calculation for the implied interest rate for the last coupon and redemption of the bond with three coupons outstanding. This will let us then solve for the zero coupon interest rate applicable to a term of 1.5 years.

If we now continue in the same way we will be able to obtain a zero coupon rate for each maturity in the set. Formula 4-9 provides a method for performing the bootstrap calculation in one step using matrix notation. Denote a discount factor d(t) as a function of a term t specific interest rates r(t) as

Formula 4-9 then shows that a vector of discount factors d multiplied by the matrix of coupon and redemption cash flows F results in the vector p of observed bond prices. This can be rearranged to solve for discount factors in d, which can be obtained by multiplying the transpose of Matrix F with the vector p. This method is implemented in the code for Lab Exercise 4.2 on page 89.

Nota Bene. The above method assumes that the series of cash flow dates coincide between the instruments used. This is a general shortcoming of the Bootstrapping Method. For the method to work, real cash flows have to be properly adjusted to fit the generic cash flow schedule used in the method.

Once we have obtained a set of zero coupon rates we can again interpolate rates between the maturities of the set of zero coupon instruments we used as input. We could again use linear interpolation. Alternatively we can use cubic interpolation, which will result in a smoother overall curve. Formula 4-10 provides a method for cubic interpolation that we can use for calculating intermediate points either in order to obtain rates for use in discounted cash flow valuation or to draw the full yield curve.

Again, it is helpful to let the above logic sink in using the Lab Exercise below of the companion website.

4.1.2.3 Parametric Yield Curve Estimation

Although the bootstrap method is easy to implement, it has one important drawback: Since it fits the resulting yield curve exactly to the data we have used as input it does not deal well with the errors that are unavoidable in the prices used. Bootstrapping will thus tend to be significantly influenced by any errors in individual bond prices. To alleviate this problem we can use a parametric yield curve estimation approach that fits a suitable parametric function to the observed data using a Nonlinear Least Squares estimation algorithm.

One of the earliest such approaches is the method set out by Nelson & Siegel (1987). Formula 4-11 is implemented in Lab Exercise 4.3, Parametric Yield Curve Estimation, which uses a Nonlinear Least Squares estimation algorithm to fit this formula to a given data set of fixed income instruments.

Once the parameters α₁, α₂, α₃, β have been estimated we can then use Formula 4-12 to calculate the interest rate at any point t in time as long as t > 0 and smaller than or equal to the value of t for the time of the cash flow with the longest time to maturity in the set of bonds that we used to estimate the parameters of α₁, α₂, α₃, β.

Lab Exercise 4.3 implements both the estimation of the parameters α₁, α₂, α₃, β: for Formula 4-11 and the actual calculation of point on the estimated yield curve using estimated values for parameters α₁, α₂, α₃, β as inputs to Formula 4-12 for a range of equally spaced values for t.

4.1.2.4 Sources of Ready-made Yield Curve Data

After taking a look at the procedures for estimating yield curves and the resulting interest rates we need for valuation it should be clear that this is no trivial task, even in the very simplest of circumstances and taking the simplest possible method. It therefore often pays to obtain yield curve data ready from an external source such as either a commercial data vendor or a specialised calculation service.

4.2.1.1 Zero Bonds and Equivalent Securities and Loans

The simplest kinds of debt instruments are zero coupon bonds and equivalent securities and loans. A zero coupon bond is issued at a deep discount to its par or redemption value and then redeemed in one single payment that constitutes both a return of capital and a component that represents the required interest income for the entire life of the bond for the money originally received by the issuer from the investor when the instrument was issued or the loan taken out.

The cash flows involved can be illustrated in a Cash Flow Timeline diagram. Figure 4.5a shows such a diagram for a zero coupon bond or equivalent security or loan.

Figure 4.5a: Cash flow timeline for a zero coupon bond or equivalent security or loan.

Figure 4.5b shows the data model building blocks involved when capturing data in a Zero Coupon Bond.

Figure 4.5b: Defining a zero coupon bond.

To value a Zero Coupon Bond or equivalent security or loan you can use the basic formula for calculating the present value of future cash flows by applying an appropriate interest rate for discrete compounding periods such as an annual interest or an interest rate for compounding at six month intervals.

We reproduce the formula in Formula 4-1, introduced earlier.

Formula 4-1 also works for shorter or longer compounding periods. For instance if interest is supposed to be compounded every six months then the rate r_{six month} should be used instead of r_annual. Likewise instead of T_{in Years} you need to use T_{in Half-Years} and for T_{in Years} the equivalent six months figures can be calculated as follows: T_{in Half-Years} = 2 * T_{in Years} and r_{six month} = (1/2) * r_annual = 0.5 * r_annual.

Formulas 4-17 and 4-18 show how r can be adapted to compounding periods other than a year in the general case. Use Formula 4-16 to translate annual rate into one for another period and Formula 4-18 to translate the time to maturity from a value measured in years into one measured in other types of periods like months or quarters. After converting both the rate of interest and the number of periods to maturity you can then again use Formula 4-1, but this time substitute r_period for r_annual and T_{in Periods} for T_{in Years}.

Often you may have the interest rate as a rate for continuous compounding rather than for a given compounding period like a year. In this case you will need to use a slightly different formula. We have already introduced this in Formula 4-4, which shows how to calculate the price of a zero bond or equivalent if you have the interest rate used for continuous compounding. For convenience we have repeated Formula 4-4 here again.

4.2.1.2 Fixed Coupon Bonds and Equivalent Securities and Loans

A large number of bonds and some equivalent instruments as coupon paying certificates of deposit have coupon or interest payments at regular intervals throughout the life of the instrument in addition to a redemption payment returning the capital at the end of the life of the instrument.

Figure 4.6a shows the cash flows involved on a cash flow timeline diagram. Looking at this diagram you may come to the conclusion that in fact the instrument is very much like a series of zero bonds—one for each coupon and the redemption—all packaged into a single unit. This in fact is a very useful view since it lets us reuse the pricing formulas we used for zero bonds. The price of the coupon bond in this case is simply the sum of the prices of the individual coupon and the redemption cash flow all added together.

Figure 4.6a: Cash flow timeline for a coupon bond or equivalent security or loan.

Figure 4.6b and Figure 4.6c show the data model building blocks involved in capturing data on a Fixed Coupon Bond or an equivalent instrument.

Figure 4.6b: Coupon Bond composition.

Figure 4.6c: Coupon stream details.

If we need a quick shorthand method for calculating the price of a coupon paying bond we can also use Formula 4-5, which we introduced at the beginning of the chapter.

However, as this formula does not allow us to use different interest rates for the cash flows that occur at different points it can only approximate the price. It can be very useful if we need a fast and easy way to calculate an approximate price. For higher accuracy, however, we need to determine the details of each cash flow of the bond, and treating each like a separate Zero Coupon Bond, by calculating a price for each using the appropriate interest rate from a suitable yield curve.

4.2.1.3 Bonds with Floating Coupons and Other Features

Another common form of a bond are Floating Rate Notes. They pay floating or resettable interest rates. In that case instead of paying a given coupon of say X% over the life of the instrument it pays a coupon that is based on some market interest rate like the LIBOR. At periodic intervals the rate is reset using a formula such as LIBOR + Y% and the following coupon or coupons until the next rate reset are then based on this rate as fixed at the last reset date. In the next chapter we will show how the formulas used for pricing Zero Coupon Bonds can again be applied to Floating Rate Notes. In Figure 4.7 you can see an illustration of how a floating rate without a call feature can be represented using the building block method. Figure 4.8 shows a building block representation of a simple zero bond in comparison.

Figure 4.7: Defining a callable Fixed Coupon Bond using building blocks.

Figure 4.8: Building block view of a Zero Coupon Bond.

Floating Rate Notes also sometimes have a cap or a floor that defines the maximum or minimum to which the rate can be reset. Here, as we will show later our zero bond formulas are no longer sufficient on their own, and we need to make use of option pricing tools to price the added feature. Again we will look at how to price these features in the following chapter.

Both fixed and floating coupon instruments also often offer the issuer the ability to call; that is to redeem the instrument early; and some bonds allow the investor to put the bonds back to the issuer; that is, they can ask the issuer to redeem their bonds early. Redemption put features for investors and redemption call features for issuers represent options that again could be calculated separately from the value of the rest of the bond and just added to the total price for the instrument.

Other features that can be added to a bond can usually be dealt with in a similar way.

4.2.2.1 Valuation Models for Dividend Paying Equities

All equities ultimately derive their value from some form of equitable share of the cash flows generated by one or more underlying assets. Usually, this is a business or in the case of fund units a portfolio of financial assets or real assets like real estate properties. Equities are claims on the residual cash flows from income and capital after all other obligations such as debt interest and repayment and taxation have been met. Equities can thus again be valued with tools derived from those we used for valuing debt.

Figure 4.9: Building block view of a callable Fixed Coupon Bond.

In a very simple case we might find an equity that has a long track record, paying a steadily growing divided paid for by a business in a very stable industry and very stable economic environment with profits growing at a steady rate that could be maintained indefinitely. The steadily growing dividend cash flows of this scenario are illustrated in Figure 4.10. In this case we can use the simple dividend discount model for equity valuation shown in Formula 4-19.

Figure 4.10: Growing dividends or free cash flows used for valuing equity.

Many times you will find that even for a very simple rough estimate the assumptions of the simple dividend discount equity valuation model are unrealistic.

A very common case in valuation are firms that grow at very fast rate for some years and then are expected to settle down at a rate of growth that is sustainable indefinitely. In this case you need to value the equity by valuing the expected divided for each of the first n years using the simple debt valuation model from Formula 4-1, then calculate a value for the dividends from the year after year n until forever using simple dividend discount equity valuation from Formula 4-19, but using the correct values for year n. Then add all these valuations to a single total value for the equity. This valuation model is shown in Formula 4-20. Figure 4.11 illustrates the cash flow pattern of the two stage growth model.

Figure 4.11: Cash flow growth in the two stage growth model.

If the dividend growth is the same for the first n year you can take a slight short-cut using the model shown in Formula 4-21.

The model in Formula 4-20 is the most common in practical use since more accurate valuations can often be obtained from detailed forecasts that can be made for a certain number of years, whereas the value of cash flows from years beyond the forecast period is estimated using the simple steady state growth equity valuation model.

4.2.2.2 Valuation Models for Equities Based on Free Cash Flows

Up to now we have taken the basis for equity valuation to be the dividend stream they pay. In practice, however, you will find many equities do not pay a regular dividend. This is often the case with younger companies experiencing strong growth. In this case we can still value the equity but need to use a replacement for the dividend cash flows. This replacement is a value called Free Cash Flow to Equity, which is the net cash flow for the company in each period after paying all other claims like operating and other cash-based expenses, interest, and principal repayments on debt due as well as taxes.

We will look at how to calculate and value free cash flows in more detail in the following chapter.

4.2.3.1 Valuing Forwards and Futures of Financial Assets without Income

Prices for futures and forwards are not necessarily exactly equal because of the difference in actual cash flow patterns between the two. In practice, however, this difference is usually ignored and a single formula is used for both. The basis for valuing both forwards and futures is the fact that they are essentially debt instruments denominated in terms of a certain quantity of an asset. In Figure 4.7 you can see an illustration of how a floating rate not with a call feature can be represented using the building block method. Figure 4.8 shows a building block representation of a simple zero bond in comparison.

To illustrate this consider the following alternatives for both the buyer and the seller of the forward or future:

The buyer of a forward could borrow money and buy the asset in the spot market. If the seller did not want to offer a forward, he or she could just sell the asset in the spot market and invest the proceeds in the money market, earning the given market interest rate for deposits or short-term instruments. Figure 4.13 illustrates the building block structure of a simple forward contract without any income and cost features.

Figure 4.13: Building block view of a simple asset-based forward or futures contract.

Formula 4-22, which is the model for valuing futures and forwards on assets not paying any income and not having any storage cost, thus is based on the value of the asset in the spot market and the interest that could be earned on the equivalent cash amount in the money market at the risk-free rate.

4.2.3.2 Valuing Forwards and Futures of Financial Assets with Income or Storage Costs

Some assets (e.g., certain equities) actually pay a known or expected income over the period of the forward or future. Other assets such as oil or other commodities will require storage at a known cost. In both cases we will need to extend our valuation model to account for the cost. If the asset receives some income between the start of the forward and the time delivery takes place then we need to subtract the appropriately discounted income from the plain forward price since the income reduces the funding costs for the provider of the forward. If the asset requires the holder to pay expenses for storage, the appropriately discounted expenses need to be added to the plain forward price since the expenses add to the funding costs for the provider of the forward. Formula 4-23 shows the extended forward valuation model that includes both an adjustment for income received and expenses paid. Figure 4.14 illustrates the building block structure of a forward contract with income and costs.

Figure 4.14: Building block view of a forward contract with income or storage costs.