CHAPTER 5 BOND PRICES AND INTEREST RATE RISK. Learning Objectives Explain the time value of money...

CHAPTER 5

BOND PRICES AND INTEREST RATE RISK

Learning Objectives• Explain the time value of money and its application

to bonds pricing.

• Explain the difference measures of yield.

• Explain how changes in interest rates cause bond prices to change.

• Describe interest rate risk an how it can be measured.

• Describe how interest rate risk can be managed.

Copyright© 2006 John Wiley & Sons, Inc. 2

The Time Value of Money

• Time value of money is based on the positive time preference for consumption.

• Which means people prefer to consume goods today rather than consume similar goods in the future.

• Time value of money means that a dollar today worth more than a dollar received at some future date. Because if you have a dollar today, you can invest it and earn interest.


Future Value or Compound Value

The future value is the value of a given amount of money invested today (PV) at a given point in the future (FV) at a given rate of interest.

FV = PV (1+i)n where

i is the periodic interest rate andn is the number of compounding

periods.Copyright© 2006 John Wiley &

Sons, Inc. 4

Future Value or Compound Value

• Example:Suppose you have deposited $100 into a savings

account for 5 years and the bank pays 4% interest annually. What is the future value of this deposit?

If the bank decides to pay interest semiannually instead of annually, what will be the future value of the deposit?


Present Value

The present value is the value today of a given sum of money to be received at a given point in the future.


i) + (1

1 FV = PV n

With risk present, a premium may be added to the risk-free rate. The higher the discount rate, the lower the present value.

Present Value

• Example:Suppose you could buy a financial claim that

would be worth $121.67 in 5 years. How much would you pay for this financial claim today if the interest rate is 4%?


Bond Pricing: What is a bond?

• A form of loan—a debt security obligating a borrower to pay a lender principal and interest.

• Borrower (issuer) promises contractually to make periodic payments to lender (investor or bondholder) over given number of years

• At maturity, holder receives principal (or face value or par value).

• Periodically before maturity, holder receives interest (coupon) payments determined by coupon rate, original interest rate promised as percentage of par on face of bond.


What is a bond? Example

Par value $1,000Coupon Rate 5%Issued TodayMatures 3years from today

ScheduledPayments: $50/year interest for 3 years

$1,000 par at end of year 3


Examples

• Example 1:A bond pays BD 100 of coupon interest annually and the

face value is BD 1000. what is coupon rate?

• Example 2:A coupon rate of a bond is 20% annually, the face value is BD

1000. What is the coupon payment?

• Example 3:What is the face value of a bond which pays BD 100

annually, and the coupon rate is 10%?


Bond Pricing: bond cash flows

• Bondholder thus owns right to a stream of cash flows:

– Ordinary annuity of interest payments and

– Future lump sum in return of par value,

• Discountable to a present value at any time while bond is outstanding.


Bond Pricing: Present Value

• The value (price) of a bond is the present value of the future cash flows promised,discounted at the market rate of interest (the required rate of return on this risk class in today’s market)


PV of bond cash flows


i) + (1F + C ... +

i) + (1C +

i) + (1C = PB N

NN2

21

1

Where PB = price of bond or present value of promised payments;Ct = coupon payment in period t, where t = 1, 2, 3,…, n;Fn = par value (principal amount) due at maturity;i = market interest rate (discount rate or market yield); andn = number of periods to maturity.

Not annual Compounding


1 2/ / /

/ / /mnmn

1 2 mn

+ C m C m C m FPB = + + ... (1 + i m) (1 + i m) (1 + i m)

Where m is the number of times coupon payments are made each year.

e.g. semiannual coupon payments, m = 2, quarterly coupon payments, m = 4.

Bond pricing -Notes

• The coupon rate, face value and term to maturity are fixed throughout the life of a bond.

• The coupon rate and the market interest rate may differ.


ExamplesExample 1:If a bond has an 8% coupon rate (annually), a 3 –year

maturity and similar bonds are selling for an 10 % yield, what is the price of the bond if the face value is $ 1000?

Example 2:In example 1, assume annual coupon rate is 5% and the

similar bonds are selling for an 5% yield, what is the price of the bond?


Examples-continued

Example 3:In example 2, assume the similar bonds are selling for

an 2% yield, what is the price of the bond?

There are 3 types of bonds:1.Par bond, if the price of the bond = the par value.2.Premium, if the price of the bond > the par value. 3.Discount, if the price of the bond < the par value.


Bond pricing: principles

• Cash flows are assumed to flow at end of the period and to be reinvested at i. Bonds typically pay interest semiannually.

• Increasing i decreases price (PB); decreasing i increases price; thus bond prices and interest rates move inversely.

• If market rate equals coupon rate, bond trades at par.

• If coupon rate exceeds market rate, the bond trades above par—at a premium.

• If market rate exceeds coupon rate, bond trades below par—at a discount.


Zero coupon bonds are “pure discount” instruments.

• No periodic coupon payments.

• Issued at discount from par.

• Single payment of par value at maturity.

• PB is simply PV of FV represented by par value, discounted at market rate.


Zero coupon bonds pricing

• Where:PB = price of the zero coupon bond.Fn = amount of cash payments at maturity. i = interest rate (yield) for n periods.n = number of years until the payment is due.m = number of times interest is compounded each year


) + (1

Fmn = PB

mi mn

Examples

• Example 1:A zero coupon bond with a BD 2,000 face value, 10 –

year maturity, annually compounding. What is the price if the market interest rate is 12 %?

• Example 2:In example 1, assume it is a semi-annually

compounding. Calculate the price.


Bond yields: risks rewarded

Yield rewards investor for at least 3 risks:

• Credit or default risk: chance that issuer may be unable or unwilling to pay as agreed.

• Reinvestment risk: potential effect of variability of market interest rates on return at which payments can be reinvested when received.

• Price risk: Inverse relationship between bond prices and interest rates.


Common yield measures

• Yield to Maturity

• Expected Yield

• Realized Yield

• Total Return


Yield to maturity

• Investor's expected yield if bond is held to maturity and all payments are reinvested at same yield.

• If the bond’s purchase price is known, the bond – pricing formulas can be used to find the bond’s yield. This yield called yield to maturity.


Yield to maturity- continued

Two methods to calculate yield to maturity:1.The PB formula but we will find “i” that let the

present value of the cash flows on the right hand side of the equation equal to the PB. This can be done by “trial and error” method—try different discount rates until PB=present value of future coupon payments.


Yield to maturity- continued

2- the following formula:

WherePB = price of bond or present value of promised payments; C = coupon payment, Fc = Face value (principal amount) due at maturity; n = number of periods to maturity.(Fc + PB) / 2 is called average investment value

Dr. Hisham Abdelbaki FIn221-CH5

26

2/

/)(

PB) + (Fc

nPBFcC = Yield

Computing yield to maturity

Investor buys 5% percent coupon (semiannual payments) bond for $951.90; bond matures in 3 years. Solve the bond pricing equation for the interest rate (i) such that price paid for the bond equals PV of remaining payments due under the bond.


Computing yield to maturity, cont.

) + (1 ... +

) + (1 +

) + (1 =

iii21

)2/)2/)2/ (

025,1

(

25

(

2590.951 6


Solving either by trial and error or with a financial calculator results in yield to maturity of 3.4% semiannually, or 6.8% annually.

Computing yield to maturity, cont.

2/

/)(

PB) + (Fc

nPBFcC = Yield


2/9.9511000

3/)9.9511000(50

)+ ( = Yield

by using the formula:

= 66.03 / 975.95 = 6.8% annually.

Computing yield to maturityExample 2:A BD 1000 bond with a coupon rate of 10%, interest paid

annually, matures in 8 years and sells for BD1120. what is the yield to maturity on the bond?

= 8% annually


2/

/)(

PB) + (Fc

nPBFcC = Yield

2/11201000

8/)11201000(100

)+ ( = Yield

Summary

The yield to maturity is what return the investor earn on a bond if:

1. The borrower makes all cash payments as promised,

2. Interest rates do not change over the bond’s maturity, and

3. The investor holds the bond to maturity.

• Not occur of one or more of the above will lead to the actual return on a bond will be different from the promised yield.

• The actual return on a bond called realized yield.


Realized yield

• Is the return earned on a bond given the cash flows actually received by investor and assuming that coupon payments are reinvested at realized yield.

• May differ from YTM due to—

– change in the amount or timing of promised payments (e.g. default).

– change in market interest rates affecting premium or discount.


Computing Realized Yield

Investor pays $1,000 for 10-year 8% coupon bond and then sells the bond 3 years later. At the time he sells the bond, 7-years bonds with similar characteristics sell at yield of 10%.What will be the realized rate of return?


Computing realized yield, cont.


Steps for solution:

1- calculate the PB for the bond BUT till 7 years.

2- calculate the yield to maturity BUT for 3 years (period to maturity) and by using the PB from (1) as principal (face value) will be received at maturity. (Use the YTM formula)

Computing realized yield, cont.

1. calculate PB

= 902.63

2. Calculate the realized rate.

i=4.91%


)10.1(

100080

10.

80

10.

807

+ ... +

) + (1 +

) + (1 = PB

21

)1(

63.9028080801000 3i

+ ... +

i) + (1 +

i) + (1 = 21

Computing realized yield

• ExampleIf a 7% coupon bond, semiannual payments, purchased

at par (BD 1000) is sold 2 years later for BD 990, what is realized rate of return?


Expected yieldInvestors usually plan to sell the bonds before

maturity, therefore the actual return on the bond will be different from the promised yield.

Predicted yield for a given holding period (same procedure as YTM, but for some holding period shorter than maturity)

To calculate the expected yield you should forecast the bond price at the end of the holding period, then apply the bond pricing formula to find the interest rate .


Expected yield

• Example:You buy a 10-year, 8% coupon bond at face

value of $1000, and the expected yield to maturity on similar bonds with 8 years to maturity will be 6% after 2 years. What is the expected yield if you will sell the bond after 2 years?


Expected yield-continued

Steps:1- calculate the price of the bond after 8 years by using the expected

yield (6%). Note: the question may be include the PB.

= 1124.202- calculate the yield after 2 years using the price from the first step.

i= 13.81%


) + (1

+ ... +

) + (1 +

) + (1 = PB 21 06.0

100080

06.0

80

06.0

808

i) + (1

+ i) + (1

= 21

20.112480801000

Total Return

• It is the rate of return that considers capital gains or losses as well as changes in reinvestment rate.


Bond price volatility (price risk)

100P

PPP%

1t

1ttB


Price risk: Bond prices and yields vary inversely. As the market interest rate rise (fall), bond price declines (rise).

The bond price volatility (price risk) can be measured as the percentage change in price for given change in interest rates:

where %∆PB = percentage change in price Pt = new price in period t

P t – 1 = bond’s price one period earlier

Bond theorems

• Bond prices are inversely related to bond yields.

• The price volatility of a long-term bond is greater than that of a short-term bond, holding the coupon rate constant.

• The price volatility of a low-coupon bond is greater than that of a high-coupon, bond, holding maturity constant


Interest Rate Risk and Duration

• Interest rate risk is the risk related to changes in interest rates that cause a bond’s total return to differ from the promised yield or the yield to maturity. Interest rate risk comprises:

• Price Risk• Reinvestment Risk


Interest rate risk comprises price risk and reinvestment risk.• Price risk is the variability in bond prices caused by their

inverse relationship with interest rates.

- increases in interest rates lead to capital losses. - decreases in interest rates lead to capital gains.

• Reinvestment risk is the variability in realized yield caused by changing market rates at which coupons can be reinvested.

- if interest rates increase, coupons will be reinvested at higher yields, thus lead to higher reinvestment income.

- if interest rates decrease, coupons will be reinvested at lower yields, thus lead to lower reinvestment income.


Price risk and Reinvestment risk.

• Price risk and reinvestment risk work against each other. – As interest rates fall —• Bond prices rise but• Coupons are reinvested at lower return.

– As interest rates rise—• Bond prices fall but• Coupons are reinvested at higher return.


Duration

How can we measure the interest rate risk?Problem: bond prices varies directly with maturity and inversely

with coupon rate.Therefore, we should account for both effects.

A good measure of interest rate risk (bond price volatility) that considers both effects is duration.

Duration: is a weighted average of the number of years until each of the bond’s cash flows is received. It refers to the period necessary to offset price risk and reinvestment risk, and thus eliminate interest rate risk.


Duration - a measure of interest rate risk

• Macaulay’s Duration


where: D = duration of the bondCFt = interest or principal payment at time tt = time period in which payment is maden = number of periods to maturityi = the yield to maturity (interest rate)

D

CF ti

CFi

tt

t

n

tt

t

n

*( )

( )

1

1

1

1

Duration Example #1

Suppose we have a bond with a 3-year term to maturity, an 8% coupon paid annually, and a market yield of 10%. Duration is:


Duration Example #2

If the yield increases to 15%:


Duration concepts (all else equal):

Higher coupon rates mean shorter duration and less price volatility.

Duration equals term to maturity for zero coupon securities.

Longer maturities mean longer durations and greater price volatility.

The higher the market rate of interest, the shorter the duration.


Duration is good measure of interest rate risk

• Bond price volatility and duration have a positive relationship.

- The greater the duration, the greater the percentage change in a bond’s price for a given change in interest rates.

• Thus, duration is good measure of interest rate risk because duration varies directly with maturity and inversely with coupon rate.


Duration is directly related to bond price volatility….

…and is thus used as a measure of price risk:


Using the 3-year, 4% coupon bond in Exhibit 5.4—If yield increases to 12%:

%..

..

)i(

i DPB%

245100101

020882

1001


The previous equation works well for small changes in interest rate because based on duration the price-yield relationship is a linear straight line which is not true in reality.

Duration can be calculated for an entire portfolio

i

i

ni Dw Duration Portfolio

1


Duration can provides a rank ordering of portfolios by their exposure to changes in interest rates, just as it does for individual bonds.

where: wi = proportion of bond i in portfolio and Di = duration of bond i.

Using Duration to Measure & Manage Interest Rate Risk

• Duration is the holding period for which reinvestment risk just offsets price risk: the holder obtains the original, promised yield to maturity.

• Financial institutions use duration to manage interest rate risk and actually achieve the desired yield for the desired holding period.

• Zero-coupon approach: zero-coupon bonds have no reinvestment risk. The duration of a “zero” equals its term to maturity. Buy a “zero” with the desired holding period and lock in the YTM.Must hold to maturity to evade price risk.

• Maturity matching approach : Selecting a term to maturity equal to the desired holding period eliminates price risk, but not reinvestment risk.

• Duration matching approach : To realize yield to maturity, investors select bonds with durations matching their desired holding periods.


• Example: Suppose you wish to plan for a vacation 2 years from

now. So, you have a 2-year investment horizon over which you must accumulate enough funds to take your vacation. You consider investing in bonds, but you are concerned about the potential effect of interest rate risk on the returns to your investment and hence your ability to afford the vacation.


Using Duration to Measure & Manage Interest Rate Risk

CHAPTER 5 BOND PRICES AND INTEREST RATE RISK. Learning Objectives Explain the time value of money...

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