TOPIC 5 – VALUATION OF BONDS
5.1 PRESENT VALUE METHOD
Definition – Long term debt instrument representing the issuer’s obligation to pay interest and repay principal.
Bonds are fixed income securities i.e securities with specified payment dates and amounts.
The par value of most bonds is RM 1000 i.e the value assigned to a security when it is issued. Coupon bonds . Most of the bonds are coupon bonds. Coupon is the periodic interest that the issuer pays to the holder of the bonds.
Intrinsic value of a security (estimated value) is the present value of the expected cash flows from the asset.
The future cash flows must be discounted at an appropriate rate to determine the present value.
n
Vo = ∑ cash flows /(1 + k )t
t=1
Where
Vo = the value of the asset now.
Cash flows = the future cash flows resulting from ownership of the asset.
k = the appropriate discount rate or required rate of return.
n= number of periods over which the cash flows are expected.
The investor has to determine :-
1. The expected cash flows.
2. The timing of the expected cash flows.
3. The discount rate demanded by the investor.
Formula :-
2n
V =∑ Ct/2 /(1 + r/2)t + MV / (1 + r/2)2n
t=1
Where
V = the present value of the bond today.
C = the annual coupons or interest payments
MV = the maturity value (or par value) of the bond.
n = the number of years to maturity of the bond.
r = the appropriate discount rate for the bond.
Ct/2 refers to the semiannual payment of interest/coupon.
This calculation can be solved easily by using the present value tables.
V = C/2 (PVIFA r/2% , 2n ) + MV(PVIF r/2% , 2n)
Example 1
Consider newly issued bond A with a three year ,maturity, sold at par with a 10% coupon rate. Assumed interest is paid semiannually and discount rate is 10%.
6
VA = ∑ 50/(1 + 0.05)t + 1,000 /(1+ 0.05)6
t=1
= 50(PVIFA 5%, 6) + 1,000(PVIF 5%, 6)
= 50(5.0757) + 1000 (0.7462)
= 999.99 @ 1000.
The price is expected to be RM 1000 since it has just been sold at par.
Example 2
Consider bon B issued 5 years ago. The coupon bond was 7%. The current discount rate is 10%. The bond has 3 years left to maturity.
VB = 35 (5.0757) + 1000 (0.7462)
= RM 923.85
5.2 YIELD TO MATURITY AND CURRENT YIELD
i) Current Yield
The ratio of the coupon interest to the current market price.
Current Yield = interest received / market price.
Example
A bond with face value of RM 1000 has coupon rate at 10% and the market price is RM 920. Calculate its current yield.
Current Yield = 100/920 = 10.9%
ii) Yield To Maturity
The promised compounded rate of return on a bond purchased at the current market price and held to maturity.
Formula:-
2n
P = ∑ Ct/2 /(1 + YTM/2)t + MV / (1 + YTM/2)2n
t=1
or
V = C/2 (PVIFA YTM/2% , 2n ) + MV(PVIF YTM/2% , 2n)
Where:-
P= the market price of the bond which is known.
N= the number of years to maturity.
YTM = the yield to maturity to be solved for.
C = the coupon in dollars
MV = the face or par value or maturity value.
Example 1:-
A 10% coupon bond has 10 years remaining to maturity. Current market oprice is RM 885.30. Face value of the bond is RM 1000.
Use the formula :-
20
885.30 = ∑ 50 /(1+ r/2)t + 1000/(1 + r/2)20
t=1
Use trial and error process.
i) Try r = 10%
P = 50(PVIFA 5%, 20) + 1000 (PVIF 5%,20)
= 50 (12.462) + 1000 (0.377)
= 623.1 + 377
= 1000.1
ii) Try r = 12%
P = 50 (11.470) + 1000 ( 0.312)
= 573.5 + 312
= 885.50
iii) Try r = 14%
P = 50 (10.594) + 1000 (0.258)
= 529.7 + 258
= 787.7
r between 10% and 14 %
Use interpolation
r = 10% + (1000 – 885.3) / (1000 – 787.7) x 4%
= 10% + (114.7 / 212.3 x 4%)
= 10% + 2.16
= 12.16 %
Formula for a zero coupon bond:-
r/2 = (MV/P) 1/2n – 1
Example 2
A zero coupon bond has 12 years to maturity and is sold for RM 300. Face value is RM 1000. Calculate its yield to maturity.
r/2 = (1000/300) 1/24 – 1
= 24√ 3.3333 – 1
= 1.0514 -1
= 0.0514
r = 0.0514 x 2
= 10.29 %
5.3 Specific Characteristics
i. The call provision
Gives the issuer the right to ‘call in’ the bonds by paying off the obligation.
ii. The sinking fund
Provides for the orderly retirement of the bond issue during its life.
iii. The collateral
Refers to the security behind a bond. Bonds are senior securities. i.e senior to any preferred stock and to the common stock of a corporation in terms of priority of payment and in case of bankruptcy and liquidation.
iv. The conversion feature
Allows a bond or preferred stock to be converted into shares of common stock at the option of the holder.
v. Zero-coupon bond
A bond sold with no coupon at a discount and redeemed for face value at maturity.
5.4 Bond Risks
i. Interest Rate Risk (IRR)
The prices of outstanding bonds must change inversely with changes in current market interest rates. When the market interest rate declines (rises), the prices of bonds outstanding rise (fall).
IRR is the change in the price of a security resulting from a change in interest rates.
ii. Default Risk
The failure to pay the specified interest payments or repay the principal at the time specified in the indenture (contract).
iii. Reinvestment Rate Risk
The uncertainty about the rate at which future cash flows can be reinvested. The expected return does not occur. The actual return does not the same as expected.
iv. Inflation Risk
The risk of the real return being less than the nominal return. The risk of unanticipated inflation. Bonds are fixed income securities. i.e since the payment in dollars is fixed; the value of the payment in real terms declines as the price level rises.
v. Maturity Risk
Indicates that the further into the future an investor goes in purchasing a long-term security, the more risk there is in the investment.
vi. Call Risk
Involve with the callable bonds. The option of the issuer to redeem a bond before maturity date. Bonds will not called unless it is to the issuer’s advantage. e.g when interest rates decline, bonds with higher coupons are likely to be called.
vii. Liquidity (Marketability) Risk
Concerned with the secondary market where a security is traded. A security is liquid if it can be sold easily and quickly.
5.5 Relationship between bond price and interest rates
Bond prices change because interest rates and required yields change.
Bond prices move inversely to interest rates.
Table 1:Bond prices and market yields (10% coupon bond)
Time to maturity Bond prices at different market yields and maturities
6% 8% 10% 12% 14%
1 1038.27 1018.86 1000 981.67 963.84
5 1170.60 1081.11 1000 926.40 859.53
10 1297.55 1135.90 1000 885.30 788.12
15 1392.01 1172.92 1000 862.35 751.82
20 1462.30 1197.93 1000 849.54 733.37
25 1514.60 1214.82 1000 842.38 723.99
30 1553.51 1226.23 1000 838.39 719.22
The table shows ; for any given maturity, the price of bond will decline as the required yield increases.
A decrease in rates will raise bond prices more than an increase in rates will lower bond prices. (As example use data in the table).
From the table; for 15-year 10% coupon bond, the price would be RM 1172.90 if market rate move to decline from 10% to 8 % which resulting in a price appreciation of 17.29%. A rise in market rates from 10% to 12% will result in a change price to RM 862.35 or price decline of 13.77%.
Additional information needed:-
1. The effects of maturity
For a given change in market yields, changes in bond prices ore directly related to time to maturity. As interest rates change, the prices of longer term bonds will change more than the prices of shorter term bonds.
Example:-
There are two 10% coupon bonds and a drop in market yields from 10% to 8%; 15-year bond and 30-year bond.
The price of 15-year bond will be RM 1172.92 while the price of 30-year bond will be RM 1226.23.
Long-term bond prices fluctuate more than do short-term bond prices.
2. The effects of coupon
Bond price fluctuations (volatility) and bond coupon rates are inversely related.
Conclusions:-
A decline (rise) in interest rates will cause a rise (decline) in bond prices, with the most volatility in bond prices occurring in longer maturity bonds and bonds with low coupons.
5.6 Bond Ratings
Definition – Current opinions on the quality of most large corporate and municipal bonds as well as commercial paper. Also can be defined as letters assigned to bonds to indicate their relative probability of default.
Examples of credit-rating agencies are standard & Poor’s (S&P) Corporation, Moody’s Investor Service Incorp., RAM (Rating Agency of Malaysia) etc. Rating firms perform the credit analysis for the investor.
Ratings are made by committees within the rating organization. They are assigned to specific issues of bonds. It reflects the rating companies’ judgements of the issuer’s ability and determination to fulfill its contractual obligation.
The emphasis is on the issuer’s likely prosperity, not on the resources the investor can expect in the event of financial difficulties. It tied closely to the issuer’s financial statements and will change if the issuer’s financial conditions change enough to warrant it.
For S&P corp. consists of letters ranging from AAA, AA, A, BBB and so on to D while Moody’s uses letters from Aaa, Aa, A, Baa etc to D.
Categories AAA through BBB represent investment grade securities. AAA have very strong capacity to meet all obligations. BBB have adequate capacity.
Bond ratings and bond coupon rates are inversely related.
BB, B, CCC and CC represent speculative securities in terms of the issuer’s ability to meet its contractual obligations . These categories carry significant uncertainties.
C reflects that the issuer currently not paying interest. D refers to bonds are in default.
However bond ratings are reflection of the relative probability of default and not the absolute probability of default.
Yield to maturity (YTM) is the yield promised by the bondholder on the assumption that the bond will be held to maturity, that all coupon and principal payments will be made and coupon payments are reinvested at the bond's promised yield at the same rate as invested. It is a measurement of the return of the bond. This technique in theory allows investors to calculate the fair value of different financial instruments.
The calculation of YTM is identical to the calculation of internal rate of return.
• If a bond's current yield is less than its YTM, then the bond is selling at a discount.
• If a bond's current yield is more than its YTM, then the bond is selling at a premium.
• If a bond's current yield is equal to its YTM, then the bond is selling at par.
Given that many bonds have different characteristics, there are some variants of YTM:
• Yield to Call: when a bond is callable (can be repurchased by the issuer before the maturity), the market looks also to the Yield to Call, which is the same calculation of the YTM, but assumes that the bond will be called, so the cashflow is shortened.
• Yield to Put: same as Yield to Call, but when the bond holder has the option to sell the bond back to the issuer at a fixed price on specified date.
• Yield to Worst: when a bond is callable, "puttable" or has other features, the yield to worst is the lowest yield of Yield to Maturity, Yield to Call, Yield to Put, and others.
