## Discount Factors and Present Value

Once the spot rates have been determined, it is natural to define the corresponding discount factors d, for each time point. These are the factors by which future cash flows

Spot Years Rate

10 11 12

5.571 6.088 6 555

6 978 7.361

7 707

8 020 8 304 8 561

8 793 9.003 9.193

9 365 9 520 9 661

9 789 9 904

10 008 10 103 10 188

FIGURE 4.2 Spot rate curve, The yearly rate of interest depends on the length of time funds are held

must be multiplied to obtain an equivalent present value. For the various compounding conventions, they are defined as follows:

(a) Yearly For yearly compounding, dk -

(b) in periods per year For compounding m periods per year,

(1 +sk/m)mk ' (c) Continuous For continuous compounding, d,

The discount factors transform future cash flows directly into an equivalent present value Hence given any cash flow stream (..vo, , .vt, . ., ,x„), the present value, relative to the prevailing spot rates, is

The discount factor dk acts like a price for cash received at time k. We determine the value of a stream by adding up "price times quantity" for all the cash components of the stream.

 Year 1 2 3 4 5 6 7 8 9 10 Total PV Discount 947 889 827 764 701 641 583 528 477 431 Cash flow 8 8 8 8 8 8 8 8 8 108 PV 7 58 7 11 661 6 11 5 61 5 12 4.66 4 22 3 82 46 50 97,34

Each cash flow is discounted by the discount factor for its time

Each cash flow is discounted by the discount factor for its time

Example 4.1 (Price of a 10-year bond) Using the spot rate curve of Figure 4.2, let us find the value of an 8% bond maturing in 10 years

Normally, for bonds we would use the rates and formulas for 6-month compounding; but for this example let us assume that coupons are paid only at the end of each year, starting a year from now, and that I-year compounding is consistent with our general evaluation method. We write the cash flows together with the discount factors, take their products, and then sum, as shown in Table 4 1 The value of the bond is found to be 97.34.,

Example 4.2 (Simplico gold mine) Consider the lease of the Simplico gold mine discussed in Chapter1 2, Example 2 6, but now let us assume that interest rates follow the term structure pattern of Figure 4,2 We shall find the present value of the lease.

The cash flow stream is identical to that of the earlier example; namely, \$2M each year for 10 years. The present value is therefore just the sum of the first 10 discount figures multiplied by \$2M, for a total of \$13 58M,