Understanding Interest Rates Chapter 4 Understanding Interest Rates

Present Value （現值） A dollar paid to you one year from now is less valuable than a dollar paid to you today Why? A dollar deposited today can earn interest and become $1 x (1+i) one year from today.

Discounting the Future (未來值折現)

Simple Present Value (簡單現值)

Time Line Cannot directly compare payments scheduled in different points in the time line $100 $100 $100 $100 Year 1 2 n PV 100 100/(1+i) 100/(1+i)2 100/(1+i)n

Four Types of Credit Market Instruments Simple Loan(簡單放款) Fixed Payment Loan(固定償還放款) Coupon Bond(附息債券) Discount Bond(貼現債券)(零息債券)

Yield to Maturity (到期收(殖)益率) The interest rate that equates the present value of cash flow payments received from a debt instrument with its value today

Simple Loan

Fixed Payment Loan

Coupon Bond

Table 1 Yields to Maturity on a 10%-Coupon-Rate Bond Maturing in Ten Years (Face Value = $1,000) When the coupon bond (附息債券) is priced at its face value (面額), the yield to maturity equals the coupon rate (票面利率) The price of a coupon bond and the yield to maturity are negatively related The yield to maturity（市場殖利率） is greater than the coupon rate (票面利率) when the bond price (債券價格) is below its face value (面值)

Consol or Perpetuity (永續債券) A bond with no maturity date(無到期日) that does not repay principal(本金) but pays fixed coupon(固定票息) payments forever For coupon bonds, this equation gives the current yield, an easy to calculate approximation to the yield to maturity

Discount Bond (貼現債券)

Following the Financial News: Bond Prices and Interest Rates

由利率換算成價格 公式：(台灣票券市場) 應用： 國庫券初級市場標售價格 國庫券、商業本票、銀行承兌匯票等工具的次級市場買斷或賣斷的價格(注意利息所得稅的計算)

票券利率另一常用公式 公式： (台灣票券市場)

由利率換算成價格 公式：

應用在商業本票的發行成本 貼現息=發行金額×市場利率×(天數/365) 保證費=發行金額×保證費率×(天數/365) 簽證費=發行金額×簽證費率×(天數/365) 承銷費=發行金額×承銷費率×(天數/365) 發行成本=貼現息+保證費+簽證費+承銷費

應用在附買回協定 附買回價格 =原約定價格 ×〔1+利率×(天數/365) ×(1-20%)〕

Rate of Return

Rate of Return and Interest Rates The return equals the yield to maturity only if the holding period(持有期間) equals the time to maturity(到期期間) A rise in interest rates(利率上升) is associated with a fall in bond prices(債券價格下降), resulting in a capital loss(資本損失) if time to maturity is longer than the holding period The more distant a bond’s maturity(距到期日越長), the greater the size of the percentage price change(價格變動百分比) associated with an interest-rate change(利率變量)

Rate of Return and Interest Rates (cont’d) The more distant a bond’s maturity(距到期日越遠), the lower the rate of return(報酬率下降越多) the occurs as a result of an increase in the interest rate(利率上升會導致) Even if a bond has a substantial initial interest rate, its return can be negative if interest rates rise (利率與報酬率成反比)

Table 2 One-Year Returns on Different-Maturity 10%-Coupon-Rate Bonds When Interest Rates Rise from 10% to 20%

Interest-Rate Risk Prices and returns for long-term bonds are more volatile than those for shorter-term bonds There is no interest-rate risk for any bond whose time to maturity matches the holding period

Real and Nominal Interest Rates Nominal interest rate makes no allowance for inflation Real interest rate is adjusted for changes in price level so it more accurately reflects the cost of borrowing Ex ante real interest rate is adjusted for expected changes in the price level Ex post real interest rate is adjusted for actual changes in the price level

Fisher Equation

FIGURE 1 Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953–2008 Sources: Nominal rates from www.federalreserve.gov/releases/H15. The real rate is constructed using the procedure outlined in Frederic S. Mishkin, “The Real Interest Rate: An Empirical Investigation,” Carnegie-Rochester Conference Series on Public Policy 15 (1981): 151–200. This procedure involves estimating expected inflation as a function of past interest rates, inflation, and time trends and then subtracting the expected inflation measure from the nominal interest rate.