Discount rate relationship

Exhibit 3.1 depicts this inverse relationship between an option-free bond s price and its discount rate (i.e., required yield). There are two things to infer from the price/discount rate relationship depicted in the exhibit. First, the relationship is downward sloping. This is simply the inverse relationship between present values and discount rates at work. Second, the relationship is represented as a curve rather than a straight line. In fact, the shape of the curve in Exhibit 3.1 is referred to as convex. By convex, it simply means the curve is bowed in relative to the origin. This second observation raises two questions about the convex or curved shape of the price/discount rate relationship. First, why is it curved Second, what is the import of the curvature ... 

Price/Discount Rate Relationship for an Option-Free Bond... 

As for the importance of the curvature to bond investors, let s consider what happens to bond prices in both falling and rising interest rate environments. First, what happens to bond prices as interest rates fall The answer is obvious bond prices rise. How about the rate at which they rise If the price/discount rate relationship was linear, as interest rates fell, bond prices would rise at a constant rate. However, the relationship is not linear, it is curved and curved inward. Accordingly, when interest rates fall, bond prices increase at an increasing rate. Now, let s consider what happens when interest rates rise. Of course, bond prices fall. How about the rate at which bond prices fall Once again, if the price/discount rate relationship were linear, as interest rates rose, bond prices would fall at a constant rate. Since it curved inward, when interest rates rise, bond prices decrease at a decreasing rate. In Chapter 4, we will explore more fully the implications of the curvature or convexity of the price/discount rate relationship. 

Therefore, in Equations (6.12) and (6.13) we use both nominal cash flows and discount rates. Conversely, in Equation (6.15) the real yield is the appropriate discount rate for discounting real cash flows. This relationship is tme only in the perfect indexation scenario without indexation lag. 

We will value our 4-year, 6% coupon bond under three different scenarios. These scenarios are defined by the relationship between the discount rate or required yield and the coupon rate. In the first scenario, we will consider the case when the annual discount rate and the coupon rate are equal. For the second scenario, we will value the bond when the discount rate is greater than the coupon rate. The last scenario assumes the discount rate is less than the coupon rate. 

The preceding three scenarios illustrate an important general property of present value. The higher (lower) the discount rate, the lower (higher) the present value. Since the value of a security is the present value of the expected future cash flows, this property carries over to the value of a security the higher (lower) the discount rate, the lower (higher) a security s value. We can summarize the relationship between the coupon rate, the required market yield, and the bond s price relative to its par value as follows ... 

The answer to the first question is mathematical and lies in the denominator of the bond pricing formula. Since we are raising one plus the discount rate to powers greater than one, it should not be surprising that the relationship between the level of the price and the level of the discount rate is not linear. 

It is clear from the bond price formula that a bonds yield and its price are closely related. Specifically, the price moves in the opposite direction from the yield. This is because a bonds price is the net present value of its cash flows if the discount rate—that is, the yield required by investors— increases, the present values of the cash flows decrease. In the same way, if the required yield decreases, the price of the bond rises. The relationship between a bond s price and any required yield level is illustrated by the graph in FIGURE 1.5, which plots the yield against the corresponding price to form a convex curve. 

The relationship between discount factors and the spot rates for the same periods can be used to calculate forward rates. Say the spot rate for period 1 is known. The corresponding discount rate can be derived using (7.9), which reduces to (7.11). 

The relationships among the various annual costs given by Eqs. (9-1) through (9-9) are illustrated diagrammaticaUy in Fig. 9-1. The top half of the diagram shows the tools of the accountant the bottom half, those of the engineer. The net annual cash flow Acp, which excludes any provision for balance-sheet depreciation Abd, is used in two of the more modern methods of profitability assessment the net-present-value (NPV) method and the discounted-cash-flow-rate-of-return (DCFRR) method. In both methods, depreciation is inherently taken care of by calculations which include capital recoveiy. 

Comparisons on the basis of interest can be summarized as (1) the net present value (NPV) and (2) the discounted-cash-flow rate of return (DCFRR), which from Eqs. (9-53) and (9-54) is given formally as the fractional interest rate i which satisfies the relationship... 

FIG. 9-13 Relationship between payback period and discounted-cash-flow rate of return. 

If there is no inflation, then the middle hne pertains. Because there is no inflation, the nominal (DCFRR) is equal to or identical with the real discounted-cash-flow rate of return, as can be seen from the relationship expressed in Eq. (9-113). 

It is also possible to combine (MSF) considerations with evaluation of the true discounted-cash-flow rate of return (DCFRR) by using Eq. (9-62). The relationship of Eq. (9-59) is independent of inflation if all money values are based on those prevailing in the startup year. For this case, Fig. 9-34 shows the true (DCFRR) reached in a given time, expressed as the number of elapsed payback periods for various values of the payback period. 

The discounted cash flow rate of return (DCFRR) or internal rate of return (IRR) is the percentage interest i at which the NPV becomes equal to zero. Fig. 5.2-4 illustrates the relationship between NPV and DCFRR (IRR), the intersection of the NPV curve with the axis of percentage interest i corresponds to the DCFRR (IRR) for NPV = 0. The investment is considered advantageous if DCFRR > 15 - 20 % (depending on the risk). 

Time value of money has been integrated into investment-evaluation systems by means of compound-interest relationships. Dollars, at different times, are given different degrees of importance by means of compounding or discounting at some preselected compound-interest rate. For any assumed interest value of money, a known amount at any one time can be converted to an equivalent but different amount at a different time. As time passes, money can be invested to increase at the interest rate. If the time when money is needed for investment is in the future, the present value of that investment can be calculated by discounting from the time of investment back to the present at the assumed interest rate. 

FIO. 9-1 Relationship between annual costs, annual profits, and cash flows for a project. Ago = annual depreciation allowance Acf = annual net cash flow after tax Aa = annual cash income Age = annual general expense Agp = annual gross profit An = annual tax Ame = annual manufacturing cost A ci = annual net cash income Afjfjp = annual net profit after taxes Afjp = annual net profit As = annual sales Arc = annual total cost (DCFRR) = discounted-cash-flow rate of return (NPV) = net present value. 

In an economic evaluation of a project, it is often necessary to evaluate the present value of funds that will be received at some definite time in the future. The present value (PV) of a future amount can be considered as the present principal at a given rate and compounded to give the actual amount received at a future date. The relationship between the indicated future amount and the present value is determined by a discount factor. Discounting evaluates each year s flow on an equal basis. It does this by means of the discount, or present value factor, and the reciprocal of the compound interest factor (1 -(- i)" with... 

Using this relationship, we build the yield curve by obtaining the relevant reference rate for regular intervals on the term structure, and then obtaining the discount factors for each point along the term structure. This set of discount factors is then used to extract the yield curve. 

As discussed in chapter 1, yield to maturity is the interest rate that relates a bonds price to its future returns. More precisely, using the notation defined above, it is the rate that discounts the bond s cash flow stream C to its price P(t, T). This relationship is expressed formally in equation (3.7). 

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