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Chapter 4 Understanding Interest Rates 1 Present Value One lira paid to you one year from now is less valuable than one lira paid to you today. Even if we assume inflation zero, people prefer immediate money. People are impatient. This is called time value of money. 2 Present Value If the government promises to pay you 100 liras one year later, or alternatively pay you 100 liras right now, which one would you prefer? What If the government promises to pay you 115 TL one year later, or alternatively pay you 100 TL right now, which one would you prefer? What is the present value of 115 TL one year later for you? In other words, how many liras you receive today would make you equally happy with 115 TL received one year later? (assume infl. is zero) 3 Present Value If your answer is 100 TL today, then your discount rate is 15%: 115 PresentValue 100TL (1 0.15) The present value of 115 liras one year later is equal to 100 liras now. Notice that the present value decreases as the discount (interest) rate increases. If discount rate was 20%, then PV of 115 liras next year would be _____liras now. 4 Present Value If we think of a longer time period such as two years, how much money two years later would be equivalent to 100 liras now? With 15% annual discount=interest rate, FutureCash Flow 100 (1 0.15) 2 Cash Flow is the two-years-later value (or the repayment amount) of 100 TL (loan) today. We find CF = _132.25___ liras. 5 Present Value Today Year 1 Year 2 Year n 100 x 1 0.15 n 100 115 132.25 6 Simple Present Value CF PV where n 1 i PV present value CF fut ure cash flow payment i annualint erestrat e n number of years 7 Simple Present Value İf there is more than one payment in the future, then we add the present value of each payment. For example, if there are two payments: 1000 TL one year later and 1500 TL three years later: PV=(1000/(1+i))+(1500/(1+i)3) 8 Four Types of Credit Market Instruments Simple Loan Lender provides the borrower the loan amount at the present. Borrower must repay the principal + interest payment when the maturity date comes. Ex: 100 TL loan at 50% simple interest rate, 1 yr. maturity. Principal: 100 TL, interest payment: 50 TL. 9 Four Types of Credit Market Instruments Fixed Payment Loan Repayment of a certain loan is divided into equal payments. Ex: You borrowed 100,000 TL now to buy a house. If the interest rate is 10%, you will repay this loan by paying a fixed amount of 11,017 TL every year for the maturity of 25 years. Ex: Mortgages, car loans. 11,017 11,017 11,017 100,000 .. 2 (1 i) (1 i) (1 i) 25 10 Four Types of Credit Market Instruments Coupon Bond Issuer of this bond makes fixed interest (coupon) payments to the holder every year until the maturity date. For example, assume you bought a coupon bond with 1000 TL face value, 50% coupon rate, annual coupon payments and 5 years maturity issued by the Treasury. The treasury will pay you 500 TL every year and will pay you the 1000 TL five years later. 500 500 500 500 1500 PV 2 3 4 (1 i) (1 i) (1 i) (1 i) (1 i)5 11 Four Types of Credit Market Instruments Discount Bond(zero-coupon bond) The issuer pays the holder the face value at the maturity date. No coupon payments. The current price of the bond is smaller than the face value. Ex: Treasury discount bond with 1000 TL face value and one year maturity is currently sold for 900 TL. then the interest rate is (1000-900) / 900 = 11.1 % 12 Yield to Maturity To compare different instruments, we need a standard measure of the interest rate. The interest rate that equates the present value of future cash flow payments with the value of the loan today is called yield to maturity (YTM). Assumption is that the security is held until maturity and payments are done on time. We know future cash flows and present value; we calculate the annual interest rate which is YTM. 13 Simple Loan—Yield to Maturity If one borrows 100 TL credit for one year, and repays 118 TL at the end of the year, YTM of this credit is = (118-100)/100 = 0.18, i.e. 18% 14 Fixed Payment Loan— Yield to Maturity The same cash flow payment every period throughout the life of the loan LV = loan value FP = fixed yearly payment n = number of years until maturity FP FP FP FP LV = ...+ 2 3 n 1 + i (1 + i) (1 + i) (1 + i) 15 Fixed Payment Loan— Yield to Maturity You take out a 100,000 TL house mortgage with an annual interest rate (YTM) of 20% to be paid in 20 years. What is your payment every year? Using a financial calculator, we find annual payments (FP) as 20,536 TL. Total amount paid in 20 years: 410,720 TL. If the interest rate (YTM) increases, annual payments also increase (keeping maturity and loan value constant). For example, if interest rate was 10%, then we would find annual payments as FP = 11,746 TL. 16 Coupon Bond—Yield to Maturity Using the same strategy used for the fixed-payment loan: P = price of coupon bond C = yearly coupon payment F = face value of the bond n = years to maturity date C C C C F P= . . . + 2 3 n n 1+i (1+i) (1+i) (1+i) (1+i) 17 Coupon Bond—Yield to Maturity For a treasury coupon bond with 1000 YTL face value, ten years maturity and 100 TL coupon payments every year: If the yield to maturity is 10%, what is the current price of the bond? The current price =present value is 1000 TL. 100 100 1100 1000 ... 2 10 (1 0.10) (1 0.10) (1 0.10) 18 If the current price of the coupon bond is equal to its nominal value, its yield to maturity equals the coupon rate. In other words, if the YTM is equal to the coupon rate, then its current price is equal to its nominal value. The price of a coupon bond and the yield to maturity are inversely related If yield to maturity is greater than the coupon rate then the current price is below its face value. If YTM=20%, then current price = $581. (sold at a discount) 19 Consol or Perpetuity A bond with no maturity date. A consol never repays the principal but pays fixed coupon payments forever. But you can sell it anytime. Pc C / ic Pc price of the consol C yearly interest payment ic yield to maturity of the consol Can rewrite above equation as ic C / Pc For coupon bonds, this equation gives current yield— an easy-to-calculate approximation of yield to maturity 20 Consols The same consol formula can be used to approximately calculate the YTM or FP of a long-term coupon bond or a fixed payment loan. For example, consider a coupon bond with 1000 TL nominal value, 10% coupon rate, 10 years maturity and present value of 1200 TL. Then YTM is ~ 8.3 % (exact 7.13%). Or a fixed payment credit of 100,000 YTL with 20 years maturity has annual payments equal to 20,536 YTL. YTM of this credit is approx. 20.5 % (exact 20%). 21 Discount Bond—Yield to Maturity For any one year discount bond F-P i= P F = Face value of the discount bond P = current price of the discount bond The yield to maturity equals the increase in price over the year divided by the initial price. As with a coupon bond, the yield to maturity is negatively related to the current bond price. 22 Discount Bond—Yield to Maturity If a discount bond issued by the treasury has a face value of 1000 TL, maturity of one year, and is sold at the price of 900 TL, what is its yield to maturity? Answer: 1000 900 100 i 0.111 11.1% 900 900 23 Rate of Return In case one does not hold an asset until maturity, rate of return differs from YTM. Consider a treasury coupon bond with 20% yield to maturity (interest rate), 1000 TL face value, 200 TL coupon payments and two years maturity (initial price=1000) . One year later, a crisis breaks out and the interest rate rises to 40%. If you need money and sell the bond, price of the bond will be 857 TL (=1200/(1.4)). (If the int. rate did not increase, price would be still 1200/1.2 = 1000 TL) Your rate of return is : 24 Rate of Return Your rate of return is : 200/1000 + (857 - 1000)/ 1000 = 20 - 14.3 = 5.7 % much less than 20%. The bondholder faces interest rate risk. 25 Rate of Return More generally, rate of return R for holding a bond from time t to time t+1 is C Pt 1 Pt R where Pt Pt Pt is theprice of thebond at t imet Pt 1 is theprice of thebond at t imet 1 C is coupon payment Pt 1 Pt C is current yield and is capitalgain Pt Pt 26 Rate of Return and Interest Rates Rate of return is equal to yield to maturity only if you hold the bond until the maturity date. A rise in interest rates causes a fall in bond prices, and results in a capital loss if the asset is sold before the time to maturity. If the bond has a longer term (maturity), there is a greater risk of capital loss because of a given interest-rate change. See Table 2. So, usually long term bonds have higher int. Rates to compensate for the higher interest rate risk. 27 28 Interest-Rate Risk Prices and returns for longer-term bonds are more volatile than those for shorter-term bonds. There is more interest rate risk for long term bonds. This is why Turkish Treasury had a hard time selling long term bonds. There is no interest-rate risk for any bond if the owner holds it until the end of maturity. In that case, YTM= rate of return and YTM is known when the bond is purchased. 29 Real and Nominal Interest Rates Nominal interest rate is the rate quoted by banks. It is not adjusted for inflation. Real interest rate is adjusted for inflation so it more accurately reflects the cost of borrowing “Ex ante” real interest rate is calculated from expected rate of inflation at the beginning of the year. “Ex post” real interest rate is calculated from realized inflation at the end of the year. 30 Fisher Equation i ir e i = nominal interest rate ir = real interest rate e = expected inflation rate When the real interest rate is low, there are greater incentives to borrow and fewer incentives to lend. The real interest rate is a better indicator of the incentives to borrow and lend. 31 Real Interest Rates in Turkey 110 100 90 80 70 60 50 40 30 20 10 0 -10 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 -20 -30 -40 NomIntRate RealIntRate 32 Real Interest Rates in USA 33