Valuation model - Security Analysis and Investment Management

The value of a bond- or any asset, real or financial- is equal to the present value of the cash flows expected from it. Hence, determining the value of a bond requires:

  • An estimate of expected cash flows
  • An estimate of the required return. To simplify the analysis of bond valuation we will make the following assumptions:
  • The coupon interest rate is fixed for the term of the bond.
  • The coupon payments are made every year and the next coupon payment is receivable exactly a year from now.
  • The bond will be redeemed at par on maturity.

Given these assumptions, the cash flow for a non-callable bond comprises an annuity of a fixed coupon interest payable annually and the principal amount payable at maturity. Hence the value of a bond is:

P =nΣ (t=1) C/ (1 + r)t + M/(1 + r)n (10.1)

Where P = value (in rupees)
n = number of years
C = annual coupon payment (in rupees)
r = periodic required return
M = maturity value
t = time period when the payment is received.

Since the stream of semi-annual coupon payments is an ordinary annuity, we can apply the formula for the present value of an ordinary annuity. Hence the bond value is given by the formula:

P = C × PVIFAr, n + M × PVIFr, n (10.1 a)

To illustrate how to compute the value of a bond, consider a 10-year, 12 per cent coupon bond with a par value of Rs. 1000. Let us assume that the required yield on this bond is 13 per cent. The cash flows for this bond are as follows:

  • 10 annual coupon payments of Rs. 120.
  • Rs. 1000 principal repayment 10 years from now.

The value of the bond is:

P = 120×PVIFA13%, 10 yr + 1,000×PVIF13%, 10yr = 120 × 5.426 + 1000 × 0.295 = 651.1 + 295 = Rs. 946.1

Bond values with semi-annual interest

Most bonds pay interest semi-annually. To value such bonds, we have to work with a unit period of six months, and not one year. This means that the bond valuation equation has to be modified along the following lines:

  • The annual interest payment, C, must be divided by 2 to obtain the semi-annual interest payment.
  • The number of years to maturity must be multiplied by two to get the number of half-yearly periods.
  • The discount rate has to be divided by two to get the discount rate applicable to half-yearly periods.
  • With the above modifications, the basic bond valuation becomes:
P =nΣ (t=1) C/2/ (1 + r/2)t + M/ (1 + r/2)2n = C/2 (PVIFAr/2, 2n) + M (PVIFr/2, 2n) (10.2) Where P = value of the bond C/2 = semi-annual interest payment R/2 = discount rate applicable to a half-year period M = maturity value

2n = maturity period expressed in terms of half-yearly periods. Illustration Illustration 10.1: Consider a 8-year, 12 per cent coupon bond with a par value of Rs. 100 on which interest is payable semi-annually. The required return on this bond is 14 per cent.

Solution: Applying Eq. 10.2, the value of the bond is:

P =16 Σ (t=1) 6/(1.07)t + 100/ (1.07)+ 16 = 6 (PVIFA7%, 16 yr) + 100 (PVIF7%, 16 yr) = Rs. 6 (9.447) + Rs. 100 (0.388) = Rs. 95.5.

Illustration 10.2: At an annual rate of compounding of 9 per cent, how long does it take for a given sum to become double and triple its original value?

Solution: Pt = P0 (1 + r)n

When the n value is not given it can be solved by using log ln

n ln (1 + r) = ln Pt n ln (1 + 0.09) = ln 2 n. ln 0.0862 = ln 0.6931 n = 8.04 years To triple n ln (1 + 09) = ln 3 n. ln 0.0862 = ln 1.0986 = 12.74 years

Illustration 10.3: Of the following which amount is worth more at 16 per cent; Rs. 1000 today or Rs. 2100 after five years.

Solution: The present worth of Rs. 2100 = 2100 (1 + r) n = 2100 (1 + 0.16)5

= 2100 × 0.476 = 999.60

The present worth of Rs. 2100 is Rs. 999.60 which is less than Rs. 1,000. Hence Rs. 2100 after five years is not worthful.

Illustration 10.4: Determine the price of Rs. 1,000 zero coupon bond with yield to maturity of 18 per cent and 10 years to maturity. What is YTM of this bond if its price is Rs. 220?

Solution:

  1. Price = Face value/ (1 + YTM)n
    =1,000/ (1 + 0.18)10
    = 1,000/5.2338
    = Rs. 191.07
  2. Face value /Bond value)1 /T – 1 = YTM
    (Rs. 1000/ Rs. 200)1/w – 1 = YTM
    (4.55) 0.1 – 1 = YTM
    1.163 – 1 = 0.163
    YTM = 16.3

Illustration 10.5: Arvind considers Rs. 1000 par value bond bearing a coupon rate of 11% that matures after 5 years. He wants a minimum yield to maturity of 15%. The bond is currently sold at Rs. 870. Should he buy the bond?

Solution:

P0 =Coupon / (1+ Y) + … + Coupon + Face value/ (1+ Y)5 (Or) P0 = (Coupon) (PVIFA, n) + (Principal amount) (PVIF/k,n) P0 = 110 (PVIFA 15%, 5 years) + 1000 (PVIF/15%, 5 yrs) = 110 (3.352) + 1000 (0.497) = 368.7 + 497 = 865.7.

At Arvind’s anticipated minimum yield of 15% the price should be Rs. 865.70 but the market price is higher. Hence, he should not buy.

Illustration 10.6: Anand owns Rs. 1,000 face value bond with five years to maturity. The bond has an annual coupon of Rs. 75. The bond is currently priced at Rs. 970. Given an appropriate discount rate of 10%, should Anand hold or sell the bond?

Solution:

P0 = Coupon (PVIFA k, n) + Principal amount (PVIF k, n) = 75 (PVIFA 10%, 5 yrs) + 1000 (PVIF 10%, 5 yrs) = 75 × 3.7908 + 1000 (0.6209) = Rs. 284.31 + 620.9 = Rs. 905.21.

The market price Rs. 970 is higher than the estimated price Rs. 905.2. It is better for Anand to sell the bond.


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