How does interest rate affect the price of bonds?
By Vincent Kok There is a relationship between bond prices and interest rates， and the maturity of a bond has an impact on its price sensitivity to interest rates. This article examines the relationship. A dollar today is worth more than a dollar in the future， simply because a dol-lar today can be deposited into a bank ac-count to earn interest. If one-year interest rates are 5%， the $1 received today and deposited into the bank account would be worth $1.05 in a year's time. $1 today is also known as the present value of the $1.05 expected in a year， given the one-year interest rates of 5%. The relationship between present value， future value and interest rates is given by the simple discounting formula :- Present Value = sum of future cashflows / (1+interest rate) In our earlier example， the present val-ue of $1 was therefore obtained from:- =$1.05/ (1+5%) =$1.00 A bond holder receives a stream of interest， or coupons for owning the bond and gets back his principal on maturity of the bond. In order to receive these streams of future cashflows， the pros-pective bond holder pays a price to the issuer of the bond. The price paid upfront is the present value of all the future cash-flows of coupons and principal on matu-rity， discounted at the appropriate interest rate， which is also known as the yield to maturity (YTM) of th bond. The YTM is the current market interest rates， which could differ from the fixed interest or coupon rate paid by each bond. The YTM is determined by (among other factors) inflation， demand and supply of funds and Central Bank policy. On the other hand， the bonds coupon or fixed in-terest rate is determined at the launch of the bond and stays fixed during the bond's lifetime. The following examples will illustrate how changes in YT or market interest rates affect bond prices. Let's calculate the price of a two-year bond， which pays annual coupons of 8%， if the current interest rates or YTM is 5%. Note that this bond pays a higher coupon than the prevailing interest rates of 5%， which therefore makes the bond attractive to investors. Applying the present value formula to obtain the price of the bond:- Present Value = 〔$8/(1+5%) 〕 + 〔$8+$100/(1+5%)2.〕 = $105.58 A very important point to note is the inverse relationship between yield (deno-minator) and price. A rise in interest rates will reduce the price， or present value of all the future cashflows， of the bond. Conversely， a fall in interest rates will in-crease the price， or present value of all future cashflows of the bond. For in-stance， if the interest rates or yields rise to 6%， the new price of the bond will now only be:- Present Value = 〔$8/(1+6%)〕 + 〔$8+$100/(1+6%)2.〕 = $103.67
Shorter maturity bonds are typically less price-sensitive to interest rate changes than long maturity bonds. In ge-neral， the price sensitivity of a two-year fixed income bond is twice that of a one-year fixed income bond. Likewise， a 10-year fixed income bond will be about 10 times more sensitive to interest rates than a one-year fixed income bond. The longer the maturity， the higher the price sensitivity of the bond to interest rate changes. A fixed income bond investor has to understand these two concepts. For example， if an economy is undergoing a severe recession， there is a greater chance for the Central Bank to reduce interest rates. If interest rates fall， bond prices will rise， as shown by our earlier exam-ples. The prices of longer maturity fixed income bonds will rise more than shorter maturity bonds. Hence， a fixed income investor who expects market interest rates to fall should invest in longer maturity fixed income bonds to maximise price appreciation. On the other hand， if the economy has been booming and inflation is high， there is a greater chance that the Central Bank will raise interest rates. If interest rates do rise， bond prices will fall， according to the inverse relationship between bond price and interest rates. The fixed income investor should hold only shorter maturity bonds to avoid heavier price falls from ri-sing interest rates. When these shorter maturity bonds mature， the fixed income investor can reinvest the proceeds in new higher coupon bonds， assuming that in-terest rates do rise as expected. A good grasp of these two concepts enables a fixed income investor to tailor the maturity profile of his portfolio to his expectations of future interest rates. If the investor expects interest rates to fall， he should invest in longer maturity bonds which have higher price sensitivity to in-terest rates in order to maximise his re-turns. On the other hand， if the investor expects interest rates torise， he should keep his bond portfolio short in maturity to lessen price falls in his portfolio.
利率如何影響債券的價格?
債券價格與利率的關系，以及債券期限的長短在利率變動時對價 格構成了什么樣的影響呢?本文探討兩者的關系。
現在手頭上的1元比將來同樣的1元更值錢，那是因為這手頭上的 1元可存入銀行賺取利息。如果1年的利率是5%，那么一年后這1元將 值1.05元。
以年利率5%計算，手頭上的1元是一年后的1.05元的現值。現值 、未來值和利率間的關系可以用以下這個簡單的貼現方程式表達： 現值 = 未來會收現金/(1+利率) 根據上述例子，現值1元是根據這個方程式計算而得的： =$1.05/(1+5%) =$1.00 債券持有人會定期收到利息以及在到期時取回相等于債券面值的 金額。要在將來定期收到利息，債券投資者首先得付出一筆錢給債券 發行人。這筆錢就是投資者將來會收到的利息和到期時會取回的本金 的現值，而所用的貼現利率就叫做“到期利率”。
到期利率是市場現行的利率，與債券的票面利率可能有差距。影 響到期利率的
因素包括通貨膨脹率、資金需求與供應和中央銀行的貨 幣政策。而債券的票面利率在發行時已定下，并在有限期內通常維持 不變。
以下的例子將解釋到期利率或市場利率變動如何影響債券價格。
債券是兩年期，年利率是8%，到期利率是5%。這批債券所付的利 息高于市場，對投資者來說很具吸引力。根據以上方程式可算出債券 的價格： 〔$8/(1+5%)〕+〔$8+$100/(1+5%)2.〕 =$105.58 假設利率升高至6%，那么債券價格將是： 〔$8/(1+6%)〕+〔$8+$100/(1+6%)2.〕 =$103.67 利率與價格之間的反比關系是值得注意的一點。利率升高會使債 券價格下跌，相反的，利率下跌債券價格就會上升。
另外，期限較短的債券通常對利率變動的反應不會那么大，期限 較長的，在利率出現變化時，價格的變動會較大。
一般來說，兩年期債券的價格變動會相等于一年期的兩倍;同樣 的，10年期債券的反應會是一年期的10倍。
上述兩個概念對債券投資者來說是非常重要的。例如，經濟正處 于嚴重衰退時，中央銀行很可能會調低利率，在這種情形下，上述的 例子顯示債券價格將會升高。而長期債券的價格升幅會高過短期債券 ，因此預期利率會調低的債券投資者應該投資于較長期的債券，以便 取得較高的收益。
另一方面，如果經濟蓬勃發展，通貨膨脹率也高，中央銀行調高 利率的可能性很高。如果利率真的上升，債券價格就會往下落，這時 候投資者應該持有較短期的債券，減低債券價格滑落所帶來的沖擊。
在這批短期債券到期時，假設利率如預期般升高，投資者可將取回的 現金投資于票面利率較高的債券。
掌握這兩個概念，將能讓投資者根據自己對利率走勢的看法決定 投資組合的組成債券。如果認為利率會跌，那應投資于對利率較為敏 感的長期債券，以爭取更高的收益。相反地，預測利率會起的話，就 應確保投資組合中的債券是短期的，減少價格下跌所造成的沖擊。
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