3 4 LEGISLACIÓN COMPARADA.

It is relatively easy to determine the present value of a bond since its cash flows and the discount rate can be determined. If there is no risk of default, then there will be no difficulty in estimating the cash flows associated with a bond. The expected cash flows consist of annual interest payments plus repayment of principle. The appropriate capitalization or discount rate would depend upon the risk of the bond.

MBE 839 QUANTITATIVE TECHNIQUES FOR BANKING AND FINANCE

Bonds are classified into three, namely:

 B o nd w it h ma t u r it y;

 Pure D isco u nt Bo nds, a nd

 P e r p e t u a l B o n d s . (a) Bonds with Maturity:

The government and companies issue bonds that specify the interest rate (coupon) and the maturit y period. The present value o f bo nd (debenture) is the discount value of its cash flows, that is, the annual interest payments plus bond's terminal, or maturity value. The discount rate is the interest rate that investors could earn on bonds with similar characteristics. By comparing the present value of a bond with its current market value, it can be determined whether the bond is over-valued or under-over-valued.

Example 1 — Value of Bond with Maturity

Suppose an investor is considering the purchase of a five-year NI,000 per value bond, bearing a nominal rate of interest at 7% per annum. The investor's required rate of return is 8%. What will he be willing to pay now to purchase the bond if it matures at par?

The investor will receive cash of N70 as interest each year for 5 years and N1,000 on maturity (i.e. at the end of the fifth year).

The present value can be determined as follows:

Bo = 70, +70, + 70, + 70,+

(1.08)⁴

70, (1.08)⁵

+ 1,000

(1.08)¹ (1.08)² (1.08)³ (1.08)⁵

Observation:

N70 is an annuity for 5 years and N1,000 is received as a lump sum at the end of the fifth year.

Using the present value tables in appendix, given at the end of this book, the present value of bond is:

Bo = 70 x 3.993 + 1,000 x 0.681

 279.51 + 681

61

MBE 839 QUANTITATIVE TECIINIQUES FOR BANKING AND FINANCE

This implies that N1,000 bond is worth N960.51 today if the required rate of return is 8 percent. The investor would not be willing to pay more than N960.51 for bond today. Note that N960.51 is a composite of the present value of interest payments, N279.51 and the present value of the maturity value N681.00.

Since most bonds will involve payment of an annu9ity (equal interest payments each year) and principal at maturity, we can use the following formula to determine the value of a bond.

Bond value Bo

Present value of interest + Present value of maturity value



INT, + INT? + INT, + 13,

L(1)-Kd)¹ (1+Kd)² (11-1c)_ (1+1Q)"

INT, + B.

=I (1 ^-f^-Kd)' (¹+¹⁽d)ⁿ

Note: Bo is the present value of a bond (debenture),

INT, is the amount of interest in period t (from year 1 to n),

Kd is the market interest rate or the bond's required rate of return, Bn is bond's terminal or maturity value in period in n and

n is the number of years to maturity.

In equation (1), the right-hand side consists of an annuity of interest payments that are constant (i.e. INT, = INT2 = INT) over the bond's life and a final payment on maturity. Thus, an annuity formula can be used to value interest payments as shown below:

Bo TNT X INT, + _________ + 130 ... (2) (1+Kd) ^I (Kd) Kd(l+Kd):, (1 +1(d)ⁿ

Yield to Maturity (YTM) — the measure of a bond's rate of return that consists both the interest income and any capital gain or loss. YTM is bond's internal rate of return. The yield-to-maturity of 5 year bond, paying 6 percent interest on the face value of N1,000 and currently selling for N883.40 is 10 percent as shown below:

883A = 6 0 , + 6 0 , + 60, + 6 0 , + 60+ 1, 0 00 (1+YTM)^I (1+YTM)² (1+YTM)³ (1+YTM)⁴ (1+YTM)⁵ It is, however, simpler to calculate a perpetual bond's yield-to-maturity.

It is equal to interest income divided by the bond's price.

(1)

MBE 839 QUANTITATIVE TECHNIQUES FOR BANKING AND FINANCE

For Example:

If the rate of interest on N1,000 per value perpetual bond is 8 percent, and its price is N800, its YTM will be:

n=co

130

INT =

-1 (

ICV (

Cd)

K_d INT

80 0.10 = 10% (

)

800 Bonds Value and Semi - Annual

In practice, some companies pay interest on bonds (or debentures) semi-annually. The formula for bond valuation can be modified in terms of half-yearly interest payments and compounding periods as given below

— ref. equation (1) giving it semi-annual approach.

BO

= li

=1 (1+L)t +

ix o

2 2

Bo

60 1

000

=I

(1.06)

(1.06)

60 x Annuity factor (6%, 20) + 1000 x PV factor (6%, 20) 60x 11.4699+ 1,000 x 0.3118

688.20 + 311.80 N1,000.00

Pure Discount Bonds do not carry an explicit rate of interest. It provides for the payment of a lump sum amount at a future date in exchange for the current price of the bond. The difference between the face value of the bond and its purchase price gives the return or YTM to the investor.

For Example:

A company may issue a pure discount bond of N1,000 face value for

N520 today for a period of five years. Thus, the debenture has:

MBE 839 QUANTITATIVE Turf INIQUES FOR BANKING AND FINANCE

( a ) p u r c h a s e p r i c e o f N 5 2 0 ;

( b ) maturity value (equal to the face value) of N1,000 of five years

The rate of interest can be calculated as follows:

520 = 1,000

(1+YTM)⁵

(1 + YTM)⁵ = 1,000 = 1.9231 520

1.923l"-1 = 0.14 or 14%

( c )P e r p e t u a l B o n d s :

Perpetual bonds also known as consols, has an indefinite life and therefore, it has no maturity value. Perpetual bonds or debentures are rarely found in practice. In the case of the perpetual bonds, as there is no maturity, or terminal value, the value of the bonds would simply be the discounted value of the infinite stream of interest flows.

Example:

Suppose that a 10 percent NI,000 bond will pay N100 annual interest into perpetuity? What will be the value of the bond if the market yield or interest rate were 15 percent?

The value of bond is determined as follows:

Bo = INT = 100 = N667

Kd 0.15

SELF ASSESSMENT EXERCISE 1

From the above example, calculate the value of the bond if the yield or interest rate were 5%, 10%, 20%, 25% and 30%.