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Fin 221: Chapter 5

BOND PRICES AND

INTEREST RATE RISK

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The Time Value of Money• Investing—in financial assets or in real assets—means giving

up consumption until later.• People have positive time preference for consumption,

indicating that money has different value in different time periods, (time value of money)

• The time value of money indicate that a given amount of money today (present value) is worth more than the same amount received at some future time (future value).

• 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

• Time value of money has nothing primarily to do with inflation.– Inflation expectations affect discount rate, but– Deferred consumption has opportunity cost by definition.

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Future Value and Present Value

The future value of an amount of money n periods in the future (FV) at the present time (PV) is given by (compounding):

FV = PV (1+i)n

where

i is the periodic interest rate and

n is the number of compounding periods.

The Present Value, the value now of a sum expected at a future time is given by (discounting):

PV = FV [1/ (1+i)n ]

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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 (coupon payments) to lender (investor or bondholder) over given number of years (maturity of the bond)

At maturity, holder receives principal, or the original amount of the loan (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.

For example, a $1,000 Par value, 5% Coupon Rate, maturing 30 years from today will generate payments of $50/year interest for 30 years + $1,000 par at end of year 30.

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• Bondholder thus owns right to a stream or a series of cash flows:– Ordinary annuity of interest payments; plus– Future lump sum in return of par value,– Discountable to a present value at any time while bond

is outstanding.

• Therefore, 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)

Bond Pricing: bond cash flows

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PV of bond cash flows

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

i) + (1C +

i) + (1C = PB N

NN2

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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.

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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 (par bond).

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

• If market rate exceeds coupon rate, bond trades below par or at a discount (discount bond).

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Zero coupon bonds• These are “pure discount” instruments. • Have no periodic coupon payments, but only a single payment

at maturity.• They eliminates reinvestment risk.• PB is simply PV of FV represented by par value (the single

payment), discounted at market rate.• The price is given by this formula:

PB zero coupon bond = Fmn/ [(1 +i/m)mn]Where:

PB = price of the zero coupon bondFn = amount of cash payments at maturity. i = interest rate (yield) for n periods

n = number of years until the payment is duem = number of times interest is compounded each year

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Bond yields: risks rewarded

Cash flows from a bond depends on coupon payments, income from reinvesting coupons, and gains or losses resulting from changes in market interest rates.

Bond 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.

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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 are:- Yield to maturity- Realized yield- Expected yield

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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.

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

• For example: an 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.

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Yield to maturity …. Cont’d.

) + (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.

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Realized yield

• Investor’s ex post or 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 reinvestment rate.

- sale of bond before maturity at premium or discount.

• It is the rate that equates the original price of the bond to the discounted actual cash flows from the bond.

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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).

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Computing realized yield … cont’d

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%.

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Expected yield

• Predicted yield for a given holding period, or the expected yield if the bond is to be sold before maturity.

• First, forecast the expected interest rate(s)• Second, calculate bond price at end of holding

period, using the forecasted interest rate• Third, plug forecast results into bond pricing

formula and calculate i which equates the original price to the discounted expected cash flows from the bond.

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Bond price volatility (price risk) Bond price and yield are inversely related. The response of PB to changes in interest rates is measured by BP volatility. BP volatility is the percentage change in price for given change in interest rates. It is calculated using the formula:

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

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Bond theorems • Bond prices are inversely related to bond yields.• BP volatility is affected by maturity and coupon

rate of the bond:• The price volatility of a long-term bond is greater

than that of a short-term bond, holding the coupon rate constant (volatility increases as maturity increase).

• The price volatility of a low-coupon bond is greater than that of a high-coupon, bond, holding maturity constant (volatility decreases as coupon rate increase).

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Interest rate risk and Duration

• 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.

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Interest rate risk and Duration

• Duration is a measure of interest rate risk that considers both coupon rate and term to maturity. It refers to the period necessary to offset price risk and reinvestment risk, and thus eliminate interest rate risk.

• It is measured as the ratio of the sum of the time weighted discounted cash flows divided by the current price of the bond. That is it is equal to the PV of all cash flows weighted according to length of time to receipt, divided by the price of the bond.

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Interest rate risk and 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)

• Duration is calculated using the formula:

D

CF ti

CFi

tt

t

n

tt

t

n

*( )

( )

1

1

1

1

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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.

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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

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Duration and bond price volatility

Duration is 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

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Using Duration to Manage Interest Rate Risk

• 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.

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

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