Debenture Valuation
Debenture Valuation
A bond is an instrument of debt issued by a business house or a
government unit. The bonds may be issued at par, premium or discount. The par
value is the amount stated on the face of the bond. It states the amount the
firm borrows and promises to repay at the time of maturity.
The bonds carry a fixed rate of interest payable at fixed
intervals of time. The interest is calculated by multiplying the value of bonds
with the rate of interest.
Bond valuation is, generally, called debt valuation because the
features that distinguish bonds from other debts are primarily non-financial in
nature. Since bonds have a promised payment stream, they are less risky as
compared to the shares. But it does not mean that they are totally risk free.
Therefore, the required rate of return on a firm’s bond will
exceed the risk free interest rate but will be less than the required rate of
return on shares. The differences in required rates of return among bonds of
different companies are caused by differences in ‘default risk’. The value of
the bond depends upon the discount rate. It will decrease with every increase
in the discount rate.
For the purpose of valuation,
bonds may be classified into two categories:
(i) Bonds with a maturity period, and
(ii) Bonds in perpetuity.
ADVERTISEMENTS:
(i) Bonds with a Maturing
Period:
When the bonds have a definite maturity period, its valuation is
determined by considering the annual interest payments plus its maturity value.
The following formula can be
used to determine the value of a bond:
where, Vd = Value of bond or debt
R1, R2……. =Annual
interest (Rs.) in period 1, 2, …, and so on
Kd = Required rate of return
M = Maturity value of bond
n = Number of years to maturity.
It must be observed
from the above equation that as n becomes large, it becomes difficult to
calculate (1 + kd)n.
Symbolically:
Vd =(R)(ADFi, n) + (M)(DFi, n)
Illustration 1:
An investor is considering the purchase of a 8% Rs. 1,000 bond
redeemable after 5 years at par. The investor’s required rate of return is 10%.
What should he be willing to pay now to purchase the bond?
Solution:
Bonds Redeemable in
Installments:
A company may issue a bond or debenture to be redeemed
periodically. In such a case, principal amount is repaid partially each period
instead of a lump sum at maturity and hence cash outflows each period include
interest and principal. The amount of interest goes on decreasing each period
as it is calculated on the outstanding amount of bond/debenture.
The value of such a bond can be
calculated as below:
Illustration 2:
A company is proposing to issue a 5 year debenture of? 1,000
redeemable in equal installments at 14 percent rate of interest per annum. If
an investor has a minimum required rate of return of 12 per cent, calculate the
debenture’s present value for him. What should he be willing to pay now to
purchase the debenture?
Solution:
(ii) Bonds in Perpetuity:
Perpetuity bonds are the bonds which never mature or have
infinitive maturity period. Value of such bonds is simply the discounted value
of infinite streams of interest (cash) flows.
where, Vd = Value or bond or debt
Kd = Required rate of return
R1 = Interest at period 1
R2 = Interest at period 2
R = Annual Interest (as interest is constant)
Illustration 3:
Mr. A has a perpetual bond of the face value of Rs. 1,000. He
receives an interest of Rs. 60 annually. What would be its value if the
required rate of return is 10%?
Solution:
Vd = R/Kd
= 60/10
= Rs. 600
Relationship between the
Required Rate of Return and Coupon Interest Rate:
We have observed earlier that the value of a bond or debenture
is influenced by the coupon or fixed rate of interest payable on the bond and
the investor’s required or desired rate of return.
The relationship between the
required rate of return and the coupon interest rate can, thus, be summarised
as below:
(i) If the investor’s required rate of return and the coupon
interest rate are the same, the value of the debt (bond or debenture) shall be
equal to its face value or paid-up value, as the case may be.
(ii) If the required rate of return is higher than the interest
rate payable on bond or debenture, the value of the bond shall be lower than
its face or paid-up value.
(iii) If the required rate of return is lower than the interest
rate payable on bond or debenture, the value of the bond shall be higher than
its face or paid-up value.
The above relationship can be explained with the help of
following illustration.
Illustration 4:
Face value of a Debenture = Rs. 1,000
Annual Interest Rate of Debenture = 12%
Maturity Period = 5 years
What is the value of the
debenture, if:
(a) Required rate of return is 12%
(b) Required rate of return is 15%
(c) Required rate of return is 10%
Solution:
Vd = (R) (ADFi,n) + (M)(DFi,n)
Vd = 120(3.605) + 1000 (.567)
Or, Vd = 432.60 + 567
= Rs. 999.60 or say Rs. 1,000.
(b) Vd = 120 (3.352) + 1,000 (.497)
= 402.24 + 497
= Rs. 899.24
(c) Vd = 120 (3.791) + 1,000 (.621)
= 453.92 + 621
= Rs. 1075.92 or say Rs. 1076
Bond Values with Semi-Annual
Interest Rates:
We have so far determined the valuation of debentures
considering the annual interest payments for the sake of simplicity. However,
in most of the cases, interest is payable on semi-annual or half yearly basis.
To determine the value of such
bonds/debentures, the bond valuation equation has to be modified on the
following lines:
(1) The annual interest amount, R, should be divided by 2 to
find out the amount of half-yearly interest.
(2) The maturity period, n, should be multiplied with 2 to get
the number of half yearly periods.
(3) The required rate
of return, Kd, should be divided 2 to get an appropriate discount
rate applicable to half-yearly periods.
Thus, the basic bond valuation
equation as modified would be:
Illustration 5:
An investor holds a debenture of Rs. 100 carrying a coupon rate
of 12% p.a. The interest is payable half-yearly on 30th June and 31st December
every year. The maturity period of the debenture is 6 years and it is to be
redeemed at a premium of 10%. The investor’s required rate of return is 14%
p.a. Compute the value of the debenture.
Solution:
Vd = (R/2)(ADFi/2,2n) + M (DFi/2,2n)
12/2(7.943) + 110(.444)
47.658 + 48.840
Rs. 96.498 or say Rs. 96.50
Yield to Maturity or Bond’s
Internal/Rate of Return:
We have so far assumed that the investor’s required rate of
return, also called the discount rate, is given for calculating the value of
the bond/debenture. However, in many cases, we may be required to calculate the
required rate of return when the cash inflows and the current value/price of
the bond are given.
This rate also known as ‘yield
to maturity’ or ‘the internal rate of return’ for the bond can be calculated by
solving the following basic equation:
Vd = R1/(1 + kd)1 + R2/(1 + kd)2 + – – – Rn/(1 + kd)n
For example, suppose that the current value of a 8% debenture,
of Rs. 1,000 redeemable after 5 years at par, is Rs. 924.28.
The yield to maturity or the
internal rate of return can be calculated as below:
924.28 = 80/(1 + kd)1 + 80/(1 + kd)2 + 80/(1 + kd)3 + 80/(1 +kd)4 + 80/(1 + kd)5 + 1000/(1 +
kd)5
We can find the value
of kd equal to 10 percent from the above equation by
trying several values of Kd by hit and
trial method. At 10% the equation becomes:
= 80(3.791) + 1000(0.621)
= 303.28 + 621
= 924.28
However, the approximate value
of yield to maturity can also be found by using the following simple formula:
Ydm = I + (F-V)/n/0.4F + 0.6V
where, I = Annual interest payment
F = Face value of bond/debenture
V = Current value/price of bond
n = Number of years to maturity
Thus, in the above example, the
yield to maturity can be calculated as:
Ydm = 80 + (1000-924.28)/5)/(4/10 × 1000) + (6/10
× 924.28)
= 95.14/954.57
= 10% (appx.)
In case of perpetual or
irredeemable bonds/debentures, the yield to maturity can be calculated by using
the following simple equation:
Vd = R/kd or kd = R/Vd
where Vd = Value of debenture
R = Annual interest payment
kd = Required rate of return or yield to
maturity.
Illustration 6:
Mr. A has a perpetual bond of the face value of Rs. 1000. He
receives an interest of Rs. 60 annually. Its current value is Rs. 600. What is
the yield to maturity?
Solution:
Vd = R/kd
or kd = R/Vd
or kd = 60/600 = .10
Thus, the yield to maturity is 10%.
Valuation of Zero Coupon/Deep
Discount Bonds (DDBs/ZCBs):
The deep discount bond does not carry any interest but it is
sold by the issuer company at deep discount from its eventual maturity
(nominal) value. The Industrial Development Bank of India (IDBI) issued such
DDBs for the first time in the Indian capital market at a price of Rs. 2700
against the nominal value of Rs. 1,00,000 payable after 25 years.
Since there is no intermediate payment of interest between the
date of issue and the maturity date, these DDBs may also be called zero coupon
bonds (ZBBs).
The valuation of a deep discount bond can also be made in the
same manner as that of the ordinary bond or debenture. The only point to
remember is that there shall be only one cashflow at the time of maturity in
case of a deep discount bond.
Thus, the value of a DDB may be taken as equal to the present
value of this future cashflow discounted at the required rate of return of the
investor for number of years equal to the life of the bond.
The following formula can be
used to determine the value of a DDB:
Vddb = FV/(1+rn)
where Vddb = Value of a deep discount bond
FV = Face value of DDB payable at maturity
r = Required rate of return
n = Number of years to maturity/Life of DDB.
We can also make use of the present value tables to simplify our
calculations.
Symbolically:
Vddb = (FV) x (DFi,n)
Illustration 7:
A deep discount bond (DDB) is issued for a maturity period of 20
years and having a face value of Rs. 1,00,000. Find out the value of the DDB if
the required rate of return is 10%.
Solution:
Vddb = FV/(1+r)n = (FV) ×
(DFi,n)
= 1,00,000/(1 + .10)20
= (1,00,000) × (.14864)
= Rs. 14,864.
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