Financial Management MGT201
BONDS VALUATION AND YIELD ON BONDS
After going through this lecture, you would be able to have an understanding of the following topics
· Bond Valuation
· Yield of Bonds
In previous lecture, we studied are bonds are long term debt instruments. Like Stock bonds are also
direct claim securities which means that the value of these bonds is determined by the future cash flows
that bond holders will receive. These cash flows are of two basic types
1. Cash inflow: in the form of coupon receipt with regular interval over the life of the bond
2. Other cash flow is the par value of the bond which you will receive at the maturity date of the
Present Value formula for the bond:
PV= ∑ CFt / (1+rD)t =CF1/(1+rD)+CFn/(1+rD)2 +..+CFn/ (1+rD) n +PAR/ (1+rD) n
NPV = Intrinsic Value of Bond or Fair Price (in rupees) paid to invest in the bond. It is the Expected or
Theoretical Value and needs to be compared to the Market Price. It is different from the Par (or Face)
Value which is printed on the Bond paper.
rD = Bondholder's (or Investor's) Required Rate of Return for investing in Bond (Debt). DIFFERENT
from the Coupon Rate and the Market / Macroeconomic Interest Rate!
There are basically 2 kinds of Cash Flows:
(1) Annuity from Fixed Regular Coupon Receipts (CF= Coupon Rate x Par Value) and
(2) Single Cash Flow from Par Value (or Initial Investment) Returned to the Investor on maturity.
In this equation
rD: It represents the required rate of return. It is the return which is required by the investor based on
his opportunity cost. In case of Pakistan, the investor required a higher return on bond then the rate of
markup offered by the PLS account in bank. It is different from the Coupon Rate and the Market /
Macroeconomic Interest Rate.
Coupon or CF:
It is a fixed rate and it is equal to
(CF= Coupon Rate x Par Value)
Par value of the bond is fixed but the market price varies with the change in the supply and demand,
perception of investor for that bond.
Defense Savings Certificates: Suppose that you invest in a Defense Savings Certificate whose Par
Value is Rs 100,000. The Bond Issuer is the Government of Pakistan. The Certificate has small
detachable coupons. You (as the Bondholder or Investor) can present one Coupon at the end of
every month and receive Rs 1,000 cash. After 1 year, you will be repaid your Principal Investment
(or Par Value) of Rs 100,000. Assume your Required Return (rD) is 10% pa. What is the Present
Value of this Investment to you?
In the previous lectures, we have solved simple version of similar example we solved a problem
similar to this where we had to calculate the NPV of the Defense Savings Certificate with 1 Annual
Coupon payment after 1 year.
We arrived at the following approximate answer:
CF1 / (1+ i) +
CFI1 / (1+ i)
= -100,000 + 12,000/(1+0.10) + 100,000/(1+0.10)
= -100,000 +
(NOTE: PV = NPV + Io = 10,909 + 90,909 = 101,818)
But this is not the correct exact answer to our present example because it ignores monthly compounding.
Accurate Solution - Monthly Compounding:
The Accurate solution to the Savings Certificate Example with Monthly Coupons requires us to
use a monthly cash flow diagram and do monthly discounting. There is an Annuity Stream of 12
Financial Management MGT201
Coupons (Cash Inflows) of Rs 1,000 each at the end of every month. There is a final Cash Inflow worth
the Par Value of Rs 100,000 at the end of the 12th month.
The Cash Flow Diagram for Bonds is a Combination of 2 Flows: (1) an Annuity Stream (of
Coupon Receipts) every month for 12 months and (2) One Par Receipt at the end of the 12th month.
You can draw their individual Cash Flow Diagrams and then add them up later. You can compute their
PV's separately and then add them up later.
Cash flows from coupons represents by the upward pointing arrows which represents cash
In combine diagram, at the end of the year there are two upward pointing arrows. One for
coupon rate and the other is for the payment of par value of bond.
Bond Cash Flow Diagram
Savings Certificate Example
Par Receipt :
2 Cash Flow
SAME point in
time can be
Calculate the PV of Coupons from the FV Formula for Annuities (with multiple compounding
within 1 year):
FV = CCF (1 + rD/m )nxm - 1/rD/m
Use Monthly Basis for this example. n = 1 year m = 12 months
CCF = Constant Cash Flow = Rs 1,000 = Monthly Coupon
rD = Annual Nominal Required Rate of Return for investment in Bond (Debt) = 10% pa.
Periodic Monthly Required Rate of Return is rD/m = 10/12 = 0.833 % = 0.00833 p.m.
m = 12 months
Coupon Annuity Cash Flow Receipts
FV = 1,000 x [(1.00833)12- 1] /0.00833 = +Rs 12,566 (at the end of 1 year)
PV (Coupons Annuity) = FV / (1 + rD/m) nxm
= 12,566 / (1.00833)12 = +Rs 11,374
Final Par Value Cash Flow Receipt
FV = 100,000 (at the end of 1 year)
PV (Par) = 100,000 / (1.00833)12 = +Rs 90,522
PV = PV (Coupons Annuity) + PV (Par) = 11,374 + 90,522
= + Rs 101,896 (Final Answer)
So this Certificate is worth Rs 101,896 to you today. It is worth more than the Market Price (Rs
100,000). So it is a good investment.
NOTE: Our answer is slightly higher than what we got when we used Annual compounding (Rs
101,818).when we consider multiple compounding the present value of the bond increases. Its NPV is
greater than zero so on the basis of our capital budgeting techniques you should invest in that project.
Now, we consider over all rate of return on a bond. We have studied expected price of the bond. These
two are complimentary. When bonds trader talk about he overall return on a particular bond they
referred to yield to maturity.
Financial Management MGT201
Bond Yield to Maturity (YTM):
We can calculate the Value of our Investment in Bonds. But how can we compute its Rate of
Return? Both are important whether you are talking about Investment in Real Assets or Securities.
The most common way to compare the Overall Rate of Return of different Bonds is to compare
In capital budgeting, you can calculate IRR using the NPV equation. Similarly, you can calculate it
by setting the PV Equation for Bond Valuation equal to the Present Market Price and solve for "rD".
Use Trial and Error or Iteration. The value of "rD" that gives PV = Market Price is the YTM for that
= Bond Market Price =
CFt / (1+rD)t
CF1/ (1+rD) +CF2/(1+rD)2+...+CFn/(1+rD)n+ PAR/(1+rD)n
All variables are known (ie. CF, PAR, and n) EXCEPT rD .Set PV equal to the Actual Present Market
Price of Bond and solve for rD
YTM = rD
Bond YTM Example:
Term Finance Certificate (TFC): The TFC (a kind of Bond) of Company ABC is traded in the
Karachi Stock Exchange for Rs 900. The Par Value of the TFC is Rs 1,000. The Coupon Rate is fixed
at 15% pa. Coupons are paid annually. The TFC will Mature after exactly 2 Years (it is a 5 Year Bond
issued 3 Years ago). What is the Overall Expected Rate of Return (ie. YTM) offered by this TFC?
Market Price (Rs 900) is LESS than its Par Value (Rs 1,000). This Bond is selling at a Discount. Why?
Possibly Interest Rate Risk. Market Interest Rate rises above TFC's Fixed Coupon Rate so Market Price
of the TFC falls below Par. Note: when Market Interest Rates rise, Required Rate of Return (rD) for
Investors rises. But, Coupon Rate fixed by Bond Issuer at time of issue.
The Expected (or Promised) Rate of Return for Investors is the Yield to Maturity (or YTM).
Compute the Overall Return (or YTM) for the TFC using the Old IRR-like Approach:
PV = Market Price = Rs 900
Par Value =Rs 1,000. Receive this after 2 Years (remaining life)
Annual Coupons =Coupon Rate x Par =15%x1, 000 = Rs 150
rD = Minimum Return Required by the Investors investing in
The Bond Market = YTM.
This is unknown in the equation.
PV = 900 = 150 / (1+ rD) + 150 / (1+rD)2 + 1,000 / (1+rD)2
900 = 150 / (1+ rD) + 1,150 / (1+rD)2 . Use Trial & Error
rD > 15%:
Try rD = 20%: PV = 924 (close)
Try rD = 21%: PV = 909 (closer)
YTM = 21.7%: (Gives PV=Rs 900)
YTM: YTM is the expected rate of return for which the bond holder holds the bond until maturity but if
the bond holder before maturity is called by the issuer or if the holder of the bond decides to sell the
bond before maturity then your answer will change .all the calculation will remain the same only par
value is replaced as
PV=∑ CFt / (1+rD)t =CF1/(1+rD)+CF2/(1+rD)2+...+CFn/(1+rD)n+CALL/(1+rD)n
Where CALL = PAR Value + 1 Year's Worth of Coupon Receipts
YTM =Total or Overall Yield = Interest Yield + Capital Gains Yield
TFC Example Total Yield = YTM = +21.7%
Interest Yield or Current Yield = Coupon / Market Price
TFC Example Interest Yield = Rs 150 / Rs 900 = +16.7% pa
Capital Gains Yield = YTM - Interest Yield
TFC Example Capital Gains Yield = 21.7% - 16.7% = +5 %
n = Maturity or Life of Bond (in years)
N=1 year ,m= no. of intervals in a year =12
CCF=constant cash flow =1000=monthly copoun .we can plug the values in this formula to know what
the future value of annuity is going to be ?take a look at the copoun annuity :
Financial Management MGT201
FV=1000[(1-0.00833)12-1]/0.0083=+12566 at the end of one year what is the present value of this
This is the present value of cash flow from coupon. Now we need to calculate the present value of face
value at maturity suppose face value =100,000 then
Now, we combine the present value of coupon interest and present value of par both
When we compare the answer with annual cash flows where coupon was not compounded
monthly .it is grater because monthly compounding increase future cash flows as well as the present
value .2nd thing is that this NPV is grater than the initial investment which is Rs.100,000so, we should
under take this project because the NPV is grater .now, the next area is the rate of return so, the
important thing in this regard is yield to maturity .this is abbreviated as YTM.it is easy to understand
because we have discussed IRR in capital budgeting .where we set NPV=0 and calculated for r .here
market price is the YTM of the bond and then solving for the variable rD=required rate of return .so,
let's try to understand YTM using a very simple example ,the example that we will pick out is that of
term finance certificate or TFC which is by the stock exchanges of Pakistan for Rs.900.let's assume that
its par value is Rs.1000fixed or coupon interest rate is 15 p.a. and it is paid annually ,total life of the
TFC is 5 years 3 years have already passed and it will mature 2 years from now what will be over all
expected rate of return .So, let's see the equation if we compute the over all yield here we can equate
PV=market value is the YTM for the bond the PV=900 which is market price PAR=1000.
Annual coupon rate =coupon rate *par =15/100*1000=150
rD=minimum return required by the investor in the bond market =YTM it is unknown ?
PV= 900=150/(1+rD )+150/ (1+rD)2+1000/(1+rD)2
we also know that the value of rD should be more than 15% you will try different values for example if
you try 20% you will come up with PV=24 (close),try rD=21.7% PV=900 so,
YTM =21.7% =900
Therefore 21.7% is the yield to maturity for this TFC because rD=YTM .YTM is the expected
rate of return for which the bond holder holds the bond until maturity but if the bond holder before
maturity is called by the issuer or if the holder of the bond decides to sell the bond before maturity then
your answer will change .all the calculation will remain the same only par value is replaced as call value
Call=par value +I, year copoun receipts
Another thing to keep in mind is that YTM has two components first is
YTM=interest yield on bond +capital gain yield on bond from his example
YTM= 21.7% so,let's calculate the interest yield
INTEREST YIELD =annual copoun interest /market price
=150/900 =16.7% so,
CAPITAL YIELD =YTM INTEREST YIELD
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