Chapter 3

UnderstandingInterest Rates

Four Types of Credit Instruments

1. Simple (Interest) Loan2. Fixed Payment Loan (Amortizing)3. Coupon Bond

• Face or Par Value ($1,000 increments)• Maturity• Coupon Rate (% of the Face Value)

4. Discount Bond (Zero Coupon)• Purchased at a Discount (Below Face Value)• Matures to Face Value

Present ValueConcept of Present ValueSimple loan of $1 at 10% interestYear 1 2 3 n

$1.10 $1.21 $1.33 $1 (1 + i)n

$1PV of $1 = ———

(1 + i)n

Calculating Present Value is Referred to as Discounting

Yield to Maturity: LoansYield to maturity = interest rate that equates today’s value with present value of all future payments1. Simple Loan (i = 10%)

$100 = $110/(1 + i)

$110 – $100 $10i = ————— = —— = .10 = 10%$100 $100

2. Fixed Payment Loan (i = 12%) $126 $126 $126 $126

$1000 = ——— + ——— + ——— + ... + ———(1 + i) (1 + i)2 (1 + i)3 (1 + i)25

FP FP FP FPLOAN = ——— + ——— + ——— + ... + ———

(1 + i) (1 + i)2 (1 + i)3 (1 + i)25

Mortgage Payments Table

Bond Table

Yield to Maturity: Bonds

3. Coupon Bond (Coupon rate = 10% = C/F)

$100 $100 $100 $100 $1000PB = ——— + ——— + ——— + ... + ——— + ————

(1 + i) (1 + i)2 (1 + i)3 (1 + i)10 (1 + i)10

C C C C FPB = ——— + ——— + ——— + ... + ——— + ————

(1 + i) (1 + i)2 (1 + i)3 (1 + i)N (1 + i)N

Perpetuity: Fixed coupon payments of $C forever (No Payback)

C CPc = —— i = ——

i Pc

Yield to Maturity: Bonds

4. Discount Bond (Pd = $900, Face = $1000)

$1000$900 = ———

(1 + i)

$1000 – $900i = —————— = .111 = 11.1%

$900

F – Pdi = ———Pd

Relationship Between Price and Yield to Maturity

Three Interesting Facts in Table 11. When bond is at par, yield equals coupon rate2. Price and yield are inversely related3. Yield is greater than the coupon rate when the bond price is below par

value

Current Yield

Cic = ——

PB

Two Characteristics1. Is better approximation of yield to maturity, the nearer the

bond price is to par and the longer the maturity of bond2. Change in current yield always signals change in same

direction as yield to maturity

Yield on a Discount Basis

(F – Pd) 360idb = ———— ————————————

F (number of days to maturity)

One year bill, Pd = $900, F = $1000$1000 – $900 360

idb = ————————— = .099 = 9.9%$1000 365

Two Characteristics1. Understates yield to maturity; longer the maturity, greater is

understatement2. Change in discount yield always signals change in same

direction as yield to maturity

Bond Page of the Newspaper

Distinction Between Interest Rates and Returns

Rate of Return

C + Pt+1 – PtRET = —————— = ic + g

Pt

Cwhere: ic = —— = current yield

Pt

Pt+1 – Ptg = ——— = capital gainPt

Key Facts about RelationshipBetween Interest Rates and Returns

Maturity and the Volatility of Bond Returns

Key Findings from Table 21. Only bond whose return = yield is one with maturity =

holding period2. For bonds with maturity > holding period, i PB

implying capital loss3. Longer is maturity, greater is price change associated

with interest rate change4. Longer is maturity, more return changes with change in

interest rate5. Bond with high initial interest rate can still have

negative return if i

Maturity and the Volatility of Bond Returns

Conclusion from Table 2 Analysis1. Prices and returns more volatile for long-term

bonds because they have higher interest-rate risk2. No interest-rate risk for any bond whose maturity

equals holding period

Reinvestment Risk

Reinvestment Risk

1. Occurs if an investor holds a series of short term bonds over long term holding period

2. i at reinvestment is uncertain

3. gain from an i , lose when i

Calculating Duration, i = 10% 10-yr 10% Coupon Bond

Calculating Duration, i = 20% 10-yr 10% Coupon Bond

Formula for Duration

Key facts about durationEverything else equal,1. when the maturity of a bond lengthens, the duration rises as well.2. when interest rates rise, the duration of a coupon bond falls.3. the higher the coupon rate on the bond, the shorter the duration of

the bond.4. duration is additive: the duration of a portfolio of securities is the

weighted-average of the durations of the individual securities, with the weights equaling the proportion of the portfolio invested in each.

n

tt

tn

tt

t

iCP

iCPtDUR

11 )1()1(

Duration and Interest Rate Risk%P – DUR i/(1 + i)

i 10% to 11%: Table 3—10% coupon bond

%P = 6.76 .01/(1 + .10)= –.0615 = –6.15%.

Actual decline = 6.23%

20% coupon bond, DUR = 5.72 years

%P = – 5.72 .01/(1 + .10) = –.0520 = –5.20%

The greater the duration of a security, the greater the percentage change in the market value of the security for a given change in interest rates. Therefore, the greater the duration of a security, the greater its interest-rate risk.

Distinction Between Real and Nominal Interest Rates

Real interest rateInterest rate that is adjusted for expected

changes in the price level

ir = i – e

1. Real interest rate more accurately reflects true cost of borrowing

2. When real rate is low, greater incentives to borrow and less to lend

Distinction Between Real and Nominal Interest Rates

Real interest rates an Example

if i = 5% and e = 0% then

ir = 5% – 0% = 5%

if i = 10% and e = 20% then

ir = 10% – 20% = –10%

U.S. Real and Nominal Interest Rates