BOND VALUATION AND RISK

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■ Bonds are debt obligations with long-term maturities that are commonly issued by governments or corporations to obtain long-term funds.

■ The price of a bond is the present value of the cash flows that will be generated by the bond, namely periodic interest or coupon payments and the principal at maturity.

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Current price of a bond (PV)

Where:

C = coupon payment paid in each period

Par= par value

k= required rate of return

n = number of period to maturity

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nkparC

k

C

k

CPV

1

...11 21

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Impact of the Discount Rate on Bond

Valuation

■The appropriate discount rate for valuing any asset is the yield that could be earned on alternative investments with similar risk and maturities.

■High risk securities have higher discount rates.

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Impact of the Timing of Payments on Bond Valuation

■ Timing affects the market price of a bond

■ Funds received sooner can be reinvested to earn additional returns

Valuation of Bonds with Semiannual Payments■ First, divide the annual coupon by two■ Second, divide the annual discount rate by two■ Third, double the number of years

PV of bond withSemiannual payments

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nk

parC

k

C

k

C221 )]2/(1[

2/...

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2/

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1. Discount bonds: Bonds Selling below Par

If coupon rate is below required rate, the price of the bond is below par (PV < 1,000)

2. Par Bonds: Bonds Selling at Par

If coupon rate equals the required rate, the price of the bond is equal to par value (PV = 1,000)

3. Premium Bonds: Bonds Selling above Par

If the coupon rate is above the required rate, the price of the bond is above the par (PV > 1,000)

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Relationship between Required Return and Present Value for a 10 Percent

Coupon Bond with Various Maturities

1. Factors That Affect the Risk-Free Rate

a. Impact of Inflationary Expectations (INF)

If the level of inflation is expected to increase (decrease), there will be upward (downward) pressure on interest rates and hence on the required rate of return on bonds.

Inflationary expectations are partially dependent on oil prices and exchange rate movements.

b.Impact of Economic Growth (ECON)- Strong economic growth tends to generate upward pressure on interest rates, while weak economic conditions put downward pressure on rates.

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),,,( DEFMSECONINFfR f

1. Factors That Affect the Risk-Free Rate (Cont.)

c. Impact of Money Supply Growth (MS)

i. The increased money supply may result in an increased supply of loanable funds. If demand for loanable funds is not affected, the increased money supply should place downward pressure on interest rates, causing bond portfolio managers to expect an increase in bond prices and thus to purchase bonds based on such expectations.

ii. In a high-inflation environment, bond portfolio managers may expect a large increase in the demand for loanable, which would cause an increase in interest rates and lower bond prices.

d. Impact of Budget Deficit (DEF)- An increase in the budget deficit can put upward pressure on interest rates. An increase in borrowing by the federal government can indirectly affect the required rate of return on all types of bonds.

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The value of any debt claim is equal to the sum of the discounted future cash flows. The discount [interest] for future cash flows is a function of the level of riskiness for a particular bond and is termed the Yield to Maturity (YTM).

Bond valuation model; Vb = Coupon * PVIFA + Face Value * PVIF

Current Yield Coupon Yield

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Impact of Interest Rate Movements on Bond Prices

Bond Prices move in the Opposite Direction to Interest Rates.

Bonds will sell at discounts or premiums or equal to Face values

Interest Rates move in the Same Direction as Inflationary Expectations

Interest Rates move in the Same Direction as Perceived Riskiness

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Determination of Bond Yields Yield to Maturity (YTM) Current Yield Tax treatment of gains and loses

Premiums Discounts

Using Expectations to Manage Total Returns on Bond Portfolios

If you know were interest rates are going you know where bond prices are going

Riding the yield curve or betting on the movement of interest rates

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Factors Affecting Bond Price Interest Rate Sensitivity

The time remaining to maturity; direct relationship The size of the coupon payments The frequency of coupon payments; i.e., annual,

Semi-, quarterly, monthly Measuring Sensitivity to Interest Rate

Movements: Duration As duration increases, the greater the sensitivity

to interest rate fluctuations Importance of determining the investment horizon Key strategy for immunizing the yield on a bond

portfolio

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Bondholders know that the price of their bonds change when interest rates change. But,

How big is this change? How is this change in price estimated?

Macaulay Duration, or Duration, is the name of concept that helps bondholders measure the sensitivity of a bond price to changes in bond yields. That is:

Two bonds with the same duration, but not necessarily the same maturity, will have approximately the same price sensitivity to a (small) change in bond yields.

2YTM1

YTMin ChangeDurationPrice Bond in Change Pct.

Example: Suppose a bond has a Macaulay Duration of 11 years, and a current yield to maturity of 8%.

If the yield to maturity increases to 8.50%, what is the resulting percentage change in the price of the bond?

-5.29%.

20.081

0.080.08511- Price Bond in Change Pct.

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Some analysts prefer to use a variation of Macaulay’s Duration, known as Modified Duration.

The relationship between percentage changes in bond prices and changes in bond yields is approximately:

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YTM1

DurationMacaulayDurationModified

YTMin Change Duration Modified- Price Bond in Change Pct.

Macaulay’s duration values are stated in years, and are often described as a bond’s effective maturity.

For a zero-coupon bond, duration = maturity.

For a coupon bond, duration = a weighted average of individual maturities of all the bond’s separate cash flows, where the weights are proportionate to the present values of each cash flow.

In general, for a bond paying constant semiannual coupons, the formula for Macaulay’s Duration is:

In the formula, C is the annual coupon rate, M is the bond maturity (in years), and YTM is the yield to maturity, assuming semiannual coupons.

12YTM1CYTM

YTMCM2YTM1

YTM2

YTM1Duration

2M

Calculating Duration

C=8% 2yr maturity Y=10% -> y = 5%

t CF PV$ @ y PV of CF t x PV(CF)1 40 0.9524 38.10 38.10 2 40 0.9070 36.28 75.56 3 40 0.8638 34.55 103.66

4 1040 0.8227 855.61 3,422.44 964.53 3,636.744

Dmac = 3636.74/964.536 = 3.7704596

Dmod (1/2 yrs) = 3.7704596/1.05 = 3.590914 -> Dmac/(1+Y/2)

All else the same, the longer a bond’s maturity, the longer is its duration.

All else the same, a bond’s duration increases at a decreasing rate as maturity lengthens.

All else the same, the higher a bond’s coupon, the shorter is its duration.

All else the same, a higher yield to maturity implies a shorter duration, and a lower yield to maturity implies a longer duration.

Use of Duration as an Immunization Strategy

"A portfolio of bonds is immunized from interest rate risk if the duration of the portfolio equals the desired investment horizon" Fisher and Weil Requires a known investment horizon; when do you

need the cash? Involves periodic adjustment in portfolio composition; Increase in YTM after the position is set results in a

decrease in duration and vice-versa (Reinvestment of cash flows at higher rates than original YTM)

Duration affected by calls, serial redemptions, and sinking fund provisions

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Use of Derivative Securities as Hedges Interest rate futures, as well as options on futures May also involve currency hedges SWAP agreements may also be used

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What cash flows are associated with bond investments?

What effects do interest rate increases (decreases) have on; Market values of bonds? Current yields? Yields to maturity? Duration measures?

What does it mean when a bond sells at par, at a discount, at a premium?

What does it mean to immunize a bond portfolio?

What information is necessary in order to make good bond investments?

Q&A: 2, 5, 7, 8, 11, 14, 17 Interp: a, b, c

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