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Chapter 3

UnderstandingInterest Rates

Four Types of Credit Instruments

1. Simple (Interest) Loan

2. Fixed Payment Loan (Amortizing)

- Coupon Bond
- Face or Par Value ($1,000 increments)
- Maturity
- Coupon Rate (% of the Face Value)
- Discount Bond (Zero Coupon)
- Purchased at a Discount (Below Face Value)
- Matures to Face Value

Present Value

Concept of Present Value

Simple loan of $1 at 10% interest

Year 1 2 3 n

$1.10 $1.21 $1.33 $1´(1 + i)n

$1

PV of $1 = ———

(1 + i)n

Calculating Present Value is Referred to as Discounting

Yield to Maturity: Loans

Yield to maturity = interest rate that equates today’s value with present value of all future payments

1. Simple Loan (i = 10%)

$100 = $110/(1 + i) Þ

$110 – $100 $10

i = ————— = —— = .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

FPFPFPFP

LOAN = ——— + ——— + ——— + ... + ———

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

Yield to Maturity: Bonds

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

$100 $100 $100 $100 $1000

PB = ——— + ——— + ——— + ... + ——— + ————

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

CCCCF

PB = ——— + ——— + ——— + ... + ——— + ————

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

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

CC

Pc = —— i = ——

i Pc

Yield to Maturity: Bonds

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

$1000

$900 = ——— Þ

(1 + i)

$1000 – $900

i = —————— = .111 = 11.1%

$900

F – Pd

i = ———

Pd

Relationship Between Price and Yield to Maturity

Three Interesting Facts in Table 1

1. When bond is at par, yield equals coupon rate

2. Price and yield are inversely related

3. Yield is greater than the coupon rate when the bond price is below par value

Current Yield

- C
- ic = ——
- PB
- Two Characteristics
- 1. Is better approximation of yield to maturity, the nearer the bond price is to par and the longer the maturity of bond
- 2. Change in current yield always signals change in same direction as yield to maturity

Yield on a Discount Basis

(F – Pd) 360

idb = ———— ´ ————————————

F (number of days to maturity)

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

$1000 – $900 360

idb = ——————— ´ —— = .099 = 9.9%

$1000 365

Two Characteristics

1. Understates yield to maturity; longer the maturity, greater is understatement

2. Change in discount yield always signals change in same direction as yield to maturity

Distinction Between Interest Rates and Returns

Rate of Return

C + Pt+1 – Pt

RET = —————— = ic + g

Pt

C

where: ic = —— = current yield

Pt

Pt+1 – Pt

g = ——— = capital gain

Pt

Maturity and the Volatility of Bond Returns

Key Findings from Table 2

1. Only bond whose return = yield is one with maturity = holding period

2. For bonds with maturity > holding period, i PB ¯ implying capital loss

3. Longer is maturity, greater is price change associated with interest rate change

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

5. Bond with high initial interest rate can still have negative return if i

Maturity and the Volatility of Bond Returns

Conclusion from Table 2 Analysis

1. Prices and returns more volatile for long-term bonds because they have higher interest-rate risk

2. 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¯

Formula for Duration

Key facts about duration

Everything 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.

Duration and Interest Rate Risk

%DP – DUR´Di/(1 + i)

i 10% to 11%:

Table 3—10% coupon bond

%DP = 6.76 ´ .01/(1 + .10)

= –.0615 = –6.15%.

Actual decline = 6.23%

20% coupon bond, DUR = 5.72 years

%DP = – 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 rate

Interest rate that is adjusted for expected changes in the price level

ir = i – pe

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 pe = 0% then

ir = 5% – 0% = 5%

if i = 10% and pe = 20% then

ir = 10% – 20% = –10%

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