Chapter 4 Understanding Interest Rates 現值 present value a dollar paid to you one year from now is less valuable than a dollar paid to you today(存款) 例：simple loan, principal 本金 $100, 一年後還 $110 ，年利率 $100 $110 $121 $133

現值 現在的$100，N年後價值 N年後的$ 1，現在的價值多少？ the present value of future(N年後) $1 when the discount factor is PV of future $1 =

到期殖利率yield to maturity The YTM of a debt instrument is the annual discount factor that equates 1 and 2 1. The debt instrument’s value today 2. the present value of all its future payments 觀念：YTM是在「買入後即持有至到期日為止」的假定下計算的年(利息)投資報酬率 例 simple loan, principal 本金 $100, 一年後還 $110 1=$100 ；2= ，則YTM即 r，r=10% simple loan的票面利率即其YTM

Debt Instruments 借貸工具 Simple loan 簡單貸款—到期一次償還(F包含本利) - P +F 放款日 maturity date到期日 Fixed-payment loan 固定金額償還貸款 (P為本金) - P +FP +FP +FP ……………….+FP 每期固定收到FP的金額，包括本金與利息在內，直到到期日。常見於房貸。 注意這裡考慮的是站在貸放者(lender)的立場，思考應投資在什麼工具上。故計算的是貸放者的報酬率。 F是到期值或面值。Par value or face value

Debt Instruments 借貸工具 Coupon bond 息票債券 - P +C +C +C ………………….+(C+F) 購買日 票面必載：1. 發行者與到期日 2. 其票面利率即 coupon rate，息票利率 3. 到期值 F coupon rate is the dollar amount of the yearly coupon payment expressed as a percentage of the face value of the bond(= C/ F) 常見的息票債券有：公債Treasury notes, bonds

Debt Instruments 借貸工具 以及公司債corporate bond等 Discount bond 折價債券(貼現債券)或 零息債券 zero-coupon bond 票面記載：1. 發行者與到期日 2. 到期值 F(包含本息) 債券存續期間無收入發生 -P +F 購買日 常見的是國庫券 T-bills Perpetual bond 永久債券 無到期日、存續期間無窮久的息票債券

到期殖利率的計算 Fixed-payment loan 例 本金$1000, yearly payment $126, 共25年。則 - $1000 +$126 +$126 +$126…… +$126 1) value today=loan value $1000, 2) PV= 使1)=2) 的 r即此借貸的YTM。

YTM of fixed-payment loan 公式：本金LV，共攤還 N次本息，每次固定為 FP (每年攤還1次)使下式等號成立之 r即為YTM LV=

YTM of coupon bond 例 $1000-face-value bond with ten years to maturity and a 10% coupon rate, if its current price is P ： 1. Calculate coupon payments C=$1000*10%=$100 2. Its cash flow is - $P +$100 +$100 +$100……+$1100 YTM= r, P=

YTM of coupon bond 公式 其中 , C= yearly coupon payments coupon rate

YTM of coupon bond Table 1 coupon rate 10%, face value $1000, N=10 years price $P YTM% 1200 7.13 1100 8.48 1000 10.00 900 11.75 800 13.81 Facts: 1. When the coupon bond is priced at its face value, YTM = coupon rate

YTM of coupon bond 2. The price of a coupon bond and the YTM are negatively related. 3. The YTM is greater than the coupon rate when the bond price is below its face value.

YTM of discount bond 例 one year US Treasury bill, face value $1000 if the current price is $900, r=YTM -$900 + $1000 公式 for one year discount bond YTM=

YTM of consol Consol is a perpetual bond with no maturity date and no repayment of principal that makes fixed coupon payments $C forever. - P +C +C +C +C………… 公式

Other Measures Current yield 當期殖利率 is an approximation of the YTM on coupon bonds The current yield better approximates the YTM when the bond’s price is nearer to the bond’s par value and the maturity of the bond is longer A change in the current yield always signals a change in the same direction of the YTM

Yield on a discount basis Discount yield 折價率 The discount yield understates the more accurate measure of YTM The longer the maturity of the discount bond, the greater this understatement becomes A change in the discount yield always indicates a change in the same direction for YTM

The distinction between interest rates and returns Interest rates只考慮持有期間的收入，但是 rate of return 還考慮了買賣價差收入。尤其是沒有持有至到期日為止 例 face value $1000 coupon bond, coupon rate 10%, bought at $1000, held for one year and then sold for $1200.則YTM=10%，但並非實際報酬率 - $1000 +$100 +$1200 RET= rate of return Pt= purchase price Pt+1= sold price

Rate of return RET= current yield + rate of capital gain The only bond whose return equals the initial YTM is one whose time to maturity is the same as the holding period

債券願付價格的決定 債券願付價格的決定 當債券的成交價格已知時，可以使用下列公式計算出債券的投報率 例 $1000-face-value bond with ten years to maturity and a 10% coupon rate, if its current price is $2100 式中唯一的未知數是r，即此一投資行為的YTM 換言之，若投資人花$2100投資此一債券，持有至到期為止，平均每年的投報率是 r

債券願付價格的決定 但是，若投資人尚未決定出價多少購買本債券，也就是若債券的成交價格尚未決定，投資人如何評估出價多少 買入這張債券？ 投資人首先須決定希望這張債券的投報率是多少，通常投資人會參考當時市場利率水準，例 3%，則投資人應出價多少(P*)買下這張債券、持有至到期日為止，能獲得持有期間平均每年3%的投報率？