chapter 5 bond prices and interest rate risk. copyright© 2006 john wiley & sons, inc.2 the time...

CHAPTER 5

BOND PRICES AND

INTEREST RATE RISK

Copyright© 2006 John Wiley & Sons, Inc. 2

The Time Value of Money:

Investing—in financial assets or in real assets—means giving up consumption until later.

Positive time preference for consumption must be offset by adequate return. Opportunity cost of deferring consumption determines minimum rate of return required on a risk-free investment—

Present sums are theoretically invested at not less than this rate;

Future cash flows are discounted by at least this rate.

Time value of money has nothing primarily to do with inflation.

Inflation expectations affect discount rate, butDeferred consumption has opportunity cost by definition.


Future Value or Compound Value

The future value (FV) of a sum (PV) is

FV = PV (1+i)n

where

i is the periodic interest rate and

n is the number of compounding periods.


Present Value

The value now of a sum expected at a future time is given by

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.


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 $1,000

Coupon Rate 5%

Issued Today

Matures 30 years from today

ScheduledPayments: $50/year interest for 30 years

$1,000 par at end of year 30


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: bond cash flows


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.


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.


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.


Bond yields: set by market

Discount rate at which bond price equals discounted PV of expected payments.

Measure of return ideally capturing impact of

Coupon payments

Income from reinvestment of coupons

Any capital gain or loss


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.

Normally determined by iteration—try different discount rates until PB=present value of future payments. Analagous to IRR of a capital project.

The longer until maturity, the less valid the reinvestment assumption.


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.


Expected yield

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

Must forecast—Expected interest rate(s)Bond price at end of holding period

Plug forecast results into bond pricing formula


Realized yield

Investor’s ex post or “hindsight” actual rate of return, given the cash flows actually received and their timing. 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; sells bond 3 years later for $902.63.

Solve for i such that $1,000 (the original investment) equals PV of 2 annual payments of $80 followed by a 3rd annual payment of $982.63 (the actual cash flows this investor received).


Computing realized yield, cont.

i) + (1 ... +

i) + (1 +

i) + (1 = 21 3

63.98280801000

Solving either by trial and error or with a financial calculator results in a realized yield of 4.91%.


Total Return

Yield metric that considers capital gains or losses as well as changes in reinvestment rate.

Find interest rate that compounds initial purchase price to sum of terminal value of bond plus future value of all coupon payments received (based on known or assumed reinvestment rate).


Computing Total Return

Terminal Value of Bond

+Future Value of Reinvested Interest Coupons

Total Accumulated

Purchase Price

of Bond = Total Accumlated

(1+ i)n

Solve for i.


Total Return--Example

Investor buys bond for $985. Bond pays $80 annual interest coupon. Investor holds bond for 5 years, reinvesting each interest coupon at 7%. After 5 year investor sells bond for $1020.

Investor will accumulate $1480.06=terminal value of bond + $460.06 FV of interest payments.

985 = 1480.06/(1+i)5 i = 8.25% (to 2 decimal places).


Bond price volatility (price risk)

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

100P

PPP%

1t

1ttB


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

Price Risk

Reinvestment Risk

Duration as a risk management tool


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.

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

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 - a measure of interest rate risk

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)

Macaulay’s Duration

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 can be calculated for an entire portfolio

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

i

i

ni Dw Duration Portfolio

1


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 5.6—If yield increases to 12%:

%..

..

)i(

i DPB%

245100101

020882

1001


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

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

chapter 5 bond prices and interest rate risk. copyright© 2006 john wiley & sons, inc.2 the time...

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