Valuing Bonds:Bond Pricing and Bond Yield

Dr. Himanshu JoshiFORE School of Management

New Delhi

Major Classes of Financial Assets or Securities

• Money market• Bond market• Equity Securities• Indexes• Derivative markets

The Money Market (International)

• Treasury bills (initial maturities of 28,91, or 182 days.)– Bid and asked price(asked price is the price you would have to pay to buy T-bill

from a dealer, and bid price is slightly lower price you would receive if you want to sell a bill to dealer)

• Certificates of Deposits (Time deposit with bank usually for 3 months or less)

• Commercial Paper (issued by large well known Co. With upto • 270 days maturity)• Bankers Acceptances (an order to bank by a bank’s customer to

pay a sum of money at a future date, typically within 6 months. )

The Money Market (International) Continued

• Eurodollars :• (euro dollars are dollar denominated deposits at foreign

bank or foreign branches of US banks. Escape regulation of Federal Reserve. Less than 6 month maturity)

• Repurchase Agreements (RPs) and Reverse RPs:• (the dealer sells government securities to an investor on an

overnight basis, with an agreement to buyback next day at slightly higher price.)

• In Reverse repo dealer find an investor holding government securities and buys them, agreeing to sell them at slightly higher price on a future date.

The Money Market (International) Continued

• Brokers’ Calls (individuals who buys stocks on margin borrow part of the funds to pay for

the stock from their broker. The broker in turn may borrow the funds from a bank, agreeing to repay the bank immediately on call if bank requires it. Rate is normally short term T-bill rate + 100 BPS )

• Federal Funds (SLR in India) bank’s fund maintained with fed.• LIBOR Market• London inter bank offer rate is the rate at which large banks in London are

willing to lend money among themselves. • LIBOR interest rate may be tied to currencies other than the US dollar.

LIBOR rates are widely quoted for transactions denominated in British pounds, yen, euro, and so on.

The Bond Market

• Treasury Notes and Bonds• Inflation-Protected Treasury Bonds• Federal Agency Debt• International Bonds• Municipal Bonds• Corporate Bonds• Mortgages and Mortgage-Backed Securities

Risk in Fixed Income Securities

Risk of Debt Securities

• Interest Rate Risk: debt securities, which pay fixed coupon rates, suffer a price decline when interest rates go up unexpectedly, because the stated coupon is inadequate to compensate for the prevailing higher level of interest rates.

Fixed Income Security Prices

PrevailingInterest Rate In the Market

Risk of Debt Securities

Likewise reinvestment of fixed contractual coupons becomes risky when market interest rate decline.

Re-investment Risk

PrevailingInterest Rate In the Market

Bond Price

Bond Price

• This bond was issued near par value of 100 in the middle of January 2007. price quoted here is for the year 2009 (January to December).

• Fluctuation in bond price may be due to: (a) An increase in interest rate in the market. (b) An increase in unanticipated inflation rate. (c) A fall in risk premium that causes investors to

prefer riskier securities than treasury securities.

Credit Risk

• Treasury securities do not carry credit risk. However there are corporate bonds that carry significant amount of credit risk: that the issuer may be unable to service all or some of the promised obligations due to financial distress, reorganization, workouts, or bankruptcy.

Liquidity Risk

• Some debt securities may trade in illiquid markets (few dealers, wide bid-offer spreads, low depth, and so on).

• Emerging market debt and some high yield debt fall into this category.

• Liquidity refers to the ease with which a reasonable size of a security can be transacted in the market within a short notice, without adverse price reaction.

Liquidity Risk• The seller or the buyer will face following:1. High Transaction costs such as fees and

commissions,2. Bid-offer spreads3. Market impact costs, which refer to the

possibility that following the placement of a buy (Sell) order the market makers may increase (Decrease) the prices at which they are willing to trade.

Contractual Risk• Debt securities may be callable by the issuer at the issuer’s

option.• Holders of mortgage loans have the right to prepay their old

mortgages if they can refinance them at a cheaper rate.• This implies that prepayment should increase when mortgage

rates in market drop. • The lender will want to charge a higher interest rate to account

for the fact that he or she is giving the borrower a valuation option to call away the loans when interest rate fall in the market.

• This is “call risk” in the mortgages.• Hence mortgages must trade at a yield higher than similar non

callable treasury debt securities.

Inflation Risk• Inflation risk is the risk that money obtained in the future will be worth

less than when it is invested, which is almost always the case.• The real risk is how much this risk will be. On the other hand, it is possible,

in some cases, to take advantage of deflation that occurs when interest rates rise.

• A good example is when interest rates are rising, newly issued fixed-income securities start to pay more, while prices of things that generally require borrowing, such as real estate, start declining.

• Thus, for instance, one could buy 4 week T-bills as a way to save for a house or for a down payment. As the T-bills expire, they can be re-invested at progressively higher rates (while rates are rising).

• In the meantime, real estate prices are falling because it is becoming more expensive to borrow the money to pay for it. So the money earned on the T-bills becomes even more valuable than the interest rate itself suggests when used to purchase real estate.

Event Risk

• Some debt securities may be sensitive to events such as hostile reorganizations or leveraged buyouts (LBOs). Such events can lead to a significant price loss.

• In October 1988 RJR Nabisco was taken over through an LBO. The resulting company took on heavy debt to finance the takeover. As a result Moody’s rating for RJR Nabisco’s debt from A1 to B3.

• The prices of RJR Nabisco dropped about 15%, and yield spread went from about 100 BPS above treasury to 350 BPS above treasury.

• In corporate debt market this risk is called event risk.

Event Risk (Protection)

• Investors often require protection against this type of risk by requiring a right from the sellers of bonds that allows investors to sell (Put) the bonds back to the seller at par value.

• Waga and Weltch (1993) examined the bondholder losses for 16 firms experiencing LBO were nearly 7% within 20 day window surrounding the even date.

Tax Risk

• If debt securities were originally issued with certain tax exemption features and subsequently there developed an uncertainty regarding their tax status, it could to lead to a price loss.

Foreign Exchange Risk

• Concept of ‘carry trade’• Depending upon the currencies in which the

investor is domiciled, debt securities may pose FX risk as well.

• Central bank of China and Japan hold significant amount of U.S government debt as investments, and consequently they are subject to the risk that the dollar could depreciate.

Cost of Debt..

• Debt may be in the form of Debentures, Bonds, Term Loans from financial institutions and Banks etc.

• Debt carries a fixed rate of interest payable, to them irrespective of the profitability of the firm.

• Payment of interest will reduce profit and will result into tax saving.

• Use of debt keep EPS high.• If company makes loss, the tax shield goes down and

cost of borrowing increases.

Cost of Debt

• Cost of perpetual Debt:

• KDt= I * (1-T) / D• Where: I = Annual interest payable.• D = net proceed of issue of debentures.• T= Tax rate.

Example:

• Aries limited has issued 30,000 irredeemable 14% debentures of Rs.150 each. The cost of floatation of debentures is 5% of the total issued amount. The company’s Tax rate is 40%. Calculate Cost of Debt?

Cost of redeemable Debt..

• KD = [ I + {Rv –Sv/N}]/ (Rv + Sv/2)

• KDt = KD * (1-T)• Where: Rv = Redemption Value, Sv = Sale value (Issue price)• N = Term of maturity, T = Company’s tax rate

Example:

• Surya limited has raised funds through issue of Rs.10,000 debentures of Rs.150 each at a discount of Rs. 10 per debenture with 10year maturity. The coupon rate is 16%. The floatation cost is Rs. 5 per debenture. The debentures are redeemable with 10% premium. Applicable tax rate is 40%. Calculate cost of debt.

Cost of deep discount Bond or Zero Coupon bond..

• Example: Express Cargo Ltd has issued 5 years zero coupon bonds of Rs.1000 each at a price of Rs.540. calculate the cost of debt.

• 540 = 1000/ (1+Kd)5

•Bond Pricing

• Face or par value• Coupon rate– Zero coupon bond

• Compounding and payments– Accrued Interest

• Indenture

Bond Characteristics

Bond Indenture: Illustration

• A bond with par value of $1000 and coupon rate of 8% might be sold for $1000. the bondholder is then entitled to a payment of 8% of $1000 = $80 per year, for the stated life of a bond say, 30 years.

• The $80 payment typically comes in two semiannual installments of $40 each. At the end of 30 year life of the bond issuer also pays the $1000 par value to the bondholder.

Bond Pricing

• The price of any financial instrument is equal to the present value of the expected cash flows from financial instrument.

• Determining the price require:1. An estimate of the expected cash flows.2. An estimate of the appropriate required

yield.

Bond Pricing

• The required yield refers to the yield for financial instruments with comparable risk, or alternative (or substitute) investments.

• The first step in determining the price of a bond is to determine its cash flows.

• The cash flows of a bond that the issuer can not retire prior to its stated maturity date. (a non callable bond) consist of:

1. Periodic coupon payments to the maturity date.2. The Par (or maturity) value at maturity.

1 (1 )(1 )

T

TB tt

ParValueCPrr

PB = Price of the bond

Ct = interest or coupon payments

T = number of periods to maturityy = semi-annual discount rate or the semi-annual yield to

maturity

Bond Pricing

Bond Pricing

• You may recall that PV of an annuity was: • PV = c/y [ 1 – 1/(1+y)N ]• Where 1/y [ 1 – 1/(1+y)N ] is called an annuity

factor.• And also PV of Terminal Value is:• Par Value * 1/(1+r)N • Where 1/(1+r)N is called PV factor.• So Price = Coupon* Annuity factor (r, T) + Par

Value* PV factor (r, T)

Bond Pricing bond price.xlsx

8% coupon, 30-year maturity bond with par value of $1,000 paying 60 semiannual coupons of $40 each. Suppose that interest rate is 8% annually or 4% per six months period. Then

Price = $40* Annuity factor (4%,60) + $1000* PV factor (4%,60)

Price = $909.94 + $95.06 = $ 1000

The Inverse Relationship Between Bond Prices and Yields

Bond Prices at Different Interest Rates (8% Coupon Bond, Coupons Paid Semiannually)

Coupon Rate, Required Yield and Price

Coupon Rate < Yield Price < Par Discount

Coupon Rate = Yield Price = Par Par

Coupon Rate > Yield Price > Par Premium

Relationship Between Bond Price and Time if Interest Rates are Unchanged

• If the required yield does not change between the time the bond is purchased and the maturity date, what will happen to the price of the bond?

• For a Bond Selling at Par: as the bond moves towards maturity it will continue to sell at par value. Its price will remain constant as the bond moves towards the maturity date.

• Bond Selling at Discount: ?• Bond Selling at Premium: ?

Reasons for the change in the Bond Price

1. There is a change in the required yield owing to changes in the credit quality of the issuer.

2. There is a change in the price of the bond selling at a premium or a discount, without any change in the required yield, simply because the bond is moving towards the maturity.

3. There is a change in the required yield owing to a change in the yield on comparable bonds. (i.e., change in the required yield by the market)

Measuring Yields

• Current Yield• Yield to Maturity• Yield to Call • Yield to Put• Yield to Worst

Current Yield of a Bond

• Current yield relates the annual coupon interest to the market price.

• Current Yield = Annual $ Coupon Payment/ Price • Example: 8%, 30 year bond currently selling at

$1276.76.• Current Yield = $80/$1276.76 = 0.0627 or 6.27%.• YTM = 6.09%.• Coupon Rate (8%) > Current Yield (6.27%)>YTM

(6.09%) ?

Current Yield

• Limitations?

Current Yield Limitation..

• The current yield calculation takes into account only the coupon interest and no other source of return that will affect an investors’ yield.

• No consideration is given to the capital gain/loss that the investor will realize when bond is purchased at a discount/premium.

• No consideration on reinvestment of coupon interests.

Yield to Maturity

• In practice, an investor considering the purchase of a bond is not quoted a promised rate of return.

• Instead the investor must use Bond Price, Maturity Date, Coupon Payments, to infer the return offered by the bond over its life.

• YTM is often interpreted as a measure of true average rate of return that will be earned if it is bought now and held until maturity.

• Bond price used in the function should be the reported flat price, without accrued interest.

Yield to Maturity

• Interest rate that makes the present value of the bond’s payments equal to its price

Solve the bond formula for r

1 (1 )(1 )

T

Ttt

BParValueCP

rr

Yield to Maturity

• 15 – year 7%- Semiannual pay bond with par value of $1000 selling at $769.42.

• Coupon = 7% of $1000 = $70 annual• Cash flow 1. $35 semiannual payments for 30

periods.• Cash flow 2. $1000 principal amount to be

received 30 periods from now.

YTM for a Zero Coupon Bond

• Y = {CFn/P}1/n - 1 or

• Y = {M/P}1/n - 1• Where M = Maturity value and P = Price of the

bond.Q. 10 year zero coupon bond with a maturity

value of $1000, selling at $439.18.

Current Yield vs. YTM..

1. Yield to maturity calculation takes into account not only the current coupon income but also any capital gain or loss that the investor will realize by holding the bond to maturity.

2. YTM consider timing of the cash flows.3. It also consider re-investment of the Coupon

interests, however, assumes that reinvestment is made on the YTM only.

Relationship among the Coupon rate, Current Yield, and YTM

Bond Selling at: Relationship

Par Coupon rate = Current Yield = YTM

Discount Coupon Rate < Current Yield < YTM

Premium Coupon Rate > Current Yield >YTM

• Prices and Yields (required rates of return) have an inverse relationship

• When yields get very high the value of the bond will be very low

• When yields approach zero, the value of the bond approaches the sum of the cash flows

Bond Prices and Yields

The Inverse Relationship Between Bond Prices and Yields

Yield to Call

• What if the bond is callable, and may be retired prior to the maturity? (YTM is not Relevant).

• The price at which a bond may be called back is referred to as the call price.

• For some issues, the call price is the same regardless of when the issue is called.

• For other callable issues, there is a call schedule that specifies a call price for each call date.

Figure 14.4 Bond Prices: Callable and Straight Debt

Example 14.4 Yield to Call

Yield to Put

• When bondholders can force the issuer to buy the issue at a specified price. As with callable issue, putable issue can have a put schedule.

• The schedule specifies when the issue can be put and the price, called the put price.

• The YTP (yield to put) is the interest rate that makes the present value of cash flows to the assumed put date plus the put price on that date equal to the bond price.

• PV of Cash Flows to put date + PV of Put Price = Bond Price

Yield to Worst..

• A Practice in the industry is for an investor to calculate the yield to maturity, yield to every possible call date, and the yield to every possible put date.

• The minimum of all of these is called Yield to Worst.