The document explains bond valuation, defining a bond as a long-term contract where a borrower makes interest and principal payments to bondholders. It discusses various types of bonds, such as treasury, corporate, municipal, and foreign bonds, along with their associated risks and characteristics. Additionally, it details bond valuation methods, types of returns like coupon rate, current yield, yield to maturity, and bond duration, which measures interest rate risk.

1. Bond Valuation
Prepared by: Roshan Ali

2. What is a Bond?
 A bond is a long-term contract under which a borrower agrees to make payments of
interest and principal, on specific dates, to the holders of the bond.
 A bond is a long-term promissory note issued by a business or governmental unit. The
issuer receives money in exchange for promising to make interest payments and to
repay the principal on a specified future date.

3. Types of Bonds
 Treasury bonds: sometimes referred to as government bonds, are issued by the U.S.
federal government. It is reasonable to assume that the federal government will make
good on its promised payments, so these bonds have almost no default risk. However,
Treasury bond prices decline when interest rates rise, so they are not free of all risks.
 Corporate bonds: as the name implies, are issued by corporations. Unlike Treasury
bonds, corporate bonds are exposed to default risk—if the issuing company gets into a
trouble, it may be unable to make the promised interest and principal payments. Default
risk is often referred to as “credit risk,” and the larger the credit risk, the higher the
interest rate the issuer must pay.

4. Types of Bonds
 Municipal bonds: or “munis,” are issued by state and local governments. Like corporate
bonds, munis have default risk. However, munis offer one major advantage: The
interest earned on most municipal bonds is exempt from federal taxes and also from
state taxes if the holder is a resident of the issuing state.
 Foreign bonds: are issued by foreign governments or foreign corporations. Foreign
corporate bonds are, of course, exposed to default risk, and so are some foreign
government bonds. An additional risk exists if the bonds are denominated in a currency
other than that of the investor’s home currency.

5. Features of Bonds
 Face or Par value: The par value is the stated face value of the bond. The par value generally
represents the amount of money the firm borrows and promises to repay on the maturity date.
 Coupon interest rate: The rate of return that is calculated on the bases of par value.
 Maturity date: The future date where the bond will be returned to company and investor will get
back the principal amount.
 Frequency of coupon payment: Number of times the coupon payment will be made during the year.
 Callable bond: Most corporate bonds contain a call provision, which gives the issuing corporation
the right to call the bonds for redemption.

6. Features of Bonds
 Convertible bonds: Owners of convertible bonds have the option to convert the bonds
into a fixed number of shares of common stock.
 Bond with warrants: Warrants are options that permit the holder to buy stock at a fixed
price, thereby providing a gain if the price of the stock rises.
 Indexed bonds: also called purchasing power bonds, first became popular in Brazil,
Israel, and a few other countries plagued by high inflation rates. The interest payments
and maturity payment rise automatically when the inflation rate rises, thus protecting
the bondholders against inflation.

7. Types of Returns
 Bonds returns can be calculated in various ways:
 Coupon Rate
 Current Yield
 Spot Interest rate
 Yield to Maturity (YTM)
 Yield to Call (YTC)

8. Types of Returns: Coupon rate
 A nominal rate of interest that is fixed and printed on the bond certificate.
 Coupon rate is calculated on bond’s face value, irrespective of its current market price.
 Coupon rate is used to calculate coupon payments payable to bondholders.
 Coupon payments are made periodically, mostly every six month.

9. Types of Returns: Coupon rate
 For Example: JK company issued 10%, 20 year bond having face value of Rs.10,000.
Calculate annual coupon payment.
 In case of Semi-annual coupon payments:

10. Types of Returns: Current yield
 The price of the bond that prevails in the market may differ from its par/face value. thus, it
may be sold at premium or discount.
 Current yield relates to annual interest receivable on a bond to its current market price.
× 100
= Coupon Payment
= Market value
Current yield thus measures the annual return accruing to a
bondholder who purchases the bond from secondary market and sells it
before maturity presumably at a price he bought the bond.

11. Types of Returns: Current yield
 For example: : JK company issued 10%, 20 year bond having face value of Rs.10,000. Bond
is currently selling at premium of Rs.1,500. what is the current yield on bond?
× 100
Current yield = 8.70%
 What is the current yield if bond sells at discount of Rs.1,000?
× 100
Current yield = 11.11%

12. Types of Returns: Spot interest rate
 Spot interest rate is the yield that is earned on zero coupon bond or deep discount bond.
 Zero coupon bonds do not pay any coupon payment. However, they are issued at
discount and redeemed at par. The difference between the two values represents spot
interest rate.
 Example of this includes Treasury bills (T-bills) issued by central banks. In case of
Pakistan, T-bills are issued by State Bank of Pakistan (SBP).
above formula can be used to determine spot interest rate on discount bond. We
have to calculate “r” which represents the rate of return.

13. Types of Returns: Spot interest rate
 For example: A Zero coupon bond has a face value of Rs.1,000 and maturity period of
5 year. If the issue price of the bond is Rs.519.37. What is the spot interest rate?
FV = Rs.1,000
PV = Rs. 519.37
n = 5 year
r = ?
1,000
1,000/519.37 =
=
1.140 = 1+r r = 1.140 – 1 r =
14%

14. Types of returns: Yield to Maturity (YTM)
 It is the rate of return that an investor is expected to earn on an annualized basis from a bond
purchased at current market price and held till maturity.
 It is the internal rate of return earned on bond if held till maturity.
 YTM is the rate of discount (r) on a bond which makes the present values of cash inflows,
coupon payment and redemption value, equal to the cash outflows on purchasing of a bond.
+
If the above equation is solved for “r”, given the other things, then “r” which equates
both sides of the equation will be known as Yield To Maturity (YTM).
Hence, the equation can be rewritten as:

15. Types of returns: Yield to Maturity (YTM)
 In order to find YTM, we need to use “trial & error” to determine the rate which equates both sides.
 Approximate YTM can be calculated through following formula:
N= number of years to maturity.

16. Types of returns: Yield to Maturity (YTM)
 For example: Suppose you were offered a 14-year, 10% annual coupon, Rs.1,000 par
value bond at a price of Rs.1,494.93. What rate of interest would you earn on your
investment if you bought the bond and held it to maturity?
𝑌𝑇𝑀 = 0.052 or 5.2%
This rate is called the bond’s yield to maturity (YTM), and
it is the interest rate generally discussed by investors when they
talk about rates of return.

17. Types of returns: Yield to Maturity (YTM)
 Yield To Maturity (YTM) is equal to coupon rate if bond was purchased at its par
value.
 YTM is less than coupon rate if bond was purchased at above par value.
 YTM is greater than coupon rate if bond was purchased at below the par value.

18. Types of returns: Yield to Call (YTC)
 If you purchased a bond that was callable and the company called it, you would not
have the option of holding the bond until it matures. Therefore, the yield to maturity
would not be earned. Investor will estimate its expected rate of return as the Yield to
Call (YTC) rather than as the YTM.
 YTC is computed on the assumption that the bond’s cash flows are terminated at the
call date with the redemption of the bond at the specific call price. To calculate the
YTC, solve this equation for r:
+
 Thus, YTC is the rate of discount which makes the present value of cash inflows till
call equal to current market price of the bond.
 Above formula demands the use of trial & error approach to obtain YTC.

19. Types of returns: Yield to Call (YTC)
 To calculate yield to call (YTC) manually following formula provides the approximate
answer:

20. Types of returns
 For example: An investor has 14% debenture with face value of Rs.100 that matures at
par in 15 years. The debenture is callable in 5 years at Rs.114. It is currently selling for
Rs.105. calculate:
 YTM
 YTC
 Current Yield

21. Bond valuation
 Bond Price: Intrinsic value of the bond is equal to the present value of all future cash
flows discounted at required rate of return.
 The value of any financial asset—a stock, a bond, a lease, or even a physical asset such
as an apartment building or a piece of machinery—is simply the present value of the
cash flows the asset is expected to produce.
 Hence, value of bond is equal to the present value of its expected cash flows.

22. Bond Valuation: Semiannual coupon
 Although some bonds pay interest annually, the vast majority actually pay interest
semiannually. To evaluate semiannual payment bonds, we must modify the valuation model
as follows:
1. Divide the annual coupon interest payment by 2 to determine the dollars of interest paid every
6 months.
2. Multiply the years to maturity, N, by 2 to determine the number of semiannual periods.
3. Divide the nominal (quoted) interest rate, r, by 2 to determine the periodic (semiannual)
interest rate.

23. Bond Valuation
 Example: Jackson Corporation’s bonds have 12 years remaining to maturity. Interest is
paid annually, the bonds have a $1,000 par value, and the coupon interest rate is 8%.
The bonds have a yield to maturity of 9%. What is the current market price of these
bonds?
Solution:
INT= $1000×8%= $80

24. Bond Valuation
 Example: Renfro Rentals has issued bonds that have a 10% coupon rate, payable
semiannually. The bonds mature in 8 years, have a face value of $1,000, and a yield to
maturity of 8.5%. What is the price of the bonds?

25. THANKS

26. Bond Duration
 Bond duration: is a way of measuring how much bond prices are likely to change if and when
interest rates move. In more technical terms, bond duration is measurement of interest rate risk.
 Understanding bond duration can help investors determine how bonds fit in to a broader
investment portfolio.
 Duration is measured in years. Generally, the higher the duration of a bond or a bond fund
(meaning the longer you need to wait for the payment of coupons and return of principal), the
more its price will drop as interest rates rise.
 A bond's duration is easily confused with its term or time to maturity because certain types of
duration measurements are also calculated in years.

27. Bond Duration in action
 As a general rule, for every 1% increase or decrease in interest rates, a bond's price will
change approximately 1% in the opposite direction for every year of duration.
 For example, if a bond has a duration of five years and interest rates increase by 1%,
the bond's price will decline by approximately 5%. Conversely, if a bond has a duration
of five years and interest rates fall by 1%, the bond's price will increase by
approximately 5%.
 Understanding duration is particularly important for those who are planning on selling
their bonds prior to maturity. If you sell that bond before maturity (or if you are
invested in a fund that buys and sells bonds while you own it) then the price of your
bonds will be affected by changes in rates.

28. Bond Duration in action
 Because every bond and bond fund has a duration, those numbers can be a useful tool
that you and your financial professional can use to compare bonds and bond funds as
you construct and adjust your investment portfolio.
 If, for example, you expect rates to rise, it may make sense to focus on shorter-duration
investments (in other words, those that have less interest-rate risk).
 It's also important to remember that duration is only one of many factors that could
affect the price of your bonds.

29. Types of Duration
 The duration of a bond in practice can refer to two different things. Time required to recover
the cash flows and the expected change in a bond's price to a 1% change in interest rates.
 Macaulay duration: is the weighted average time until all the bond's cash flows are paid. By
accounting for the present value of future bond payments, the Macaulay duration helps an
investor evaluate and compare bonds independent of their term or time to maturity.
 Modified duration: Unlike Macaulay's duration, modified duration is not measured in years.
Modified duration measures the expected change in a bond's price to a 1% change in
interest rates.

30. Macaulay duration
 Macaulay duration finds the present value of a bond's future coupon payments and
maturity value.
 Because Macaulay duration is a partial function of the time to maturity, the greater the
duration, the greater the interest-rate risk or reward for bond prices.
 Macaulay duration can be calculated manually as follows:
 Where

31. Macaulay duration
 The first part is used to find the present value of all future bond cash flows.
 The second part finds the weighted average time until those cash flows are paid.
 When these sections are put together, they tell an investor the weighted average amount of
time to receive the bond's cash flows.
 For Example: Imagine a three-year bond with a face value of Rs.1000 that pays a 10%
coupon semi-annually (Rs.50 every six months) and has a yield to maturity (YTM) of 6%.
find the Macaulay duration.

32. Macaulay duration
 In order to find the Macaulay duration, the first step will be to use this information to find
the present value of all the future cash flows as shown in the following table:
Years Cash flows
Discounted Cash flows
(DCF) (DCF/VB)(Years)
0.5 Rs. 50 Rs. 48.56 0.0219
1 50 47.17 0.0425
1.5 50 45.82 0.0619
2 50 44.5 0.0801
2.5 50 43.22 0.0973
3 1050 881.6 2.381
VB= 1,110.87 ∑=2.6847

33. Modified Duration
 The modified duration of a bond helps investors understand how much a bond's price
will rise or fall if the YTM rises or falls by 1%. This is an important number if an
investor is worried that interest rates will be changing in the short term.
 The modified duration of a bond with semi-annual coupon payments can be found with
the following formula:

34. Modified Duration
 Following the same example, the modified duration can be found as under:
 Hence, one percentage change in interest will cause 2.61 percentage change in the
value of the bond.
 To convert into change in terms of currency, we use modified duration