Week 4 Tuesday, January 31, 2017 12:03 PM 1/31/17 Need to adjust discount rate to different time periods A discount rate of r for one period can be converted to an equivalent discount rate for n periods Equivalent n-Period Discount rate = (1+r)n-1 n can never be greater than 1 Quarterly: n = 1/4 Monthly: n = 1/12 Semiannually: n = 1/2 SWe also discuss several other topics like jorge iber ttu
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uppose a bank account pays interest monthly with an effective annual interest rate of 6%. Want to have 100,000 at the end of 10 years. How much should we save every month? Convert it to a monthly rate: (1+.06)^(1/12) -1 = 0.0049 .49% FV annuity = 100,000 ((1+r)n-1) = FV C/.0049*((1+.0049)^120 -1) = 100,000 C = 615.49 Annual Percentage Rates (APR) -indicates amount of simple interest earned in once year, without compounding -typically less than actual amount of interest you will earn Bank advertises 6% APR with monthly compounding. They're offering a monthly interest of .5% each month. Actual interest rate is (1.005)^12 -1 = 0.0617 , 6.17% APR doesn't reflect true amount and cannot be used as a discount rate. APR quotes actual interest rate earned each compounding period. Interest rate per compounding period = APR/m m = number of compounding periods per year Convert APR to EAR (Effective annual rate) 1+EAR = (1+ APR/m)m m = number of compounding periods per year Week 4 Pae 1 30,000 car loan with 6.75% APR for 60 months .0675/12 = 0.0056 = r PV = 30,000 n = 60 C = ? C = $590.50 You are 3 years into the loan and want to sell the car. When you sell the car, you will pay the balance on the car loan. After 36 payments, how much do you still owe? 24 payments. n =24 PV formula 590.50*1/.005625*(1-(1/(1.005625^24))) = 13,222.321219 Inflation and Real Versus Nominal Rates -Nominal interest rates: rate at which money will grow if invested for a certain period, unadjusted for inflation -real interest rate: rate of growth of your purchasing power, after adjusting for inflation Growth in purchasing power = 1+ real rate = Real rate = At the start of 2008, 1 year US bond rates were 3.3%, while inflation was .1%. IN 2011, interest rates were .3% and inflation was 3%. What were the real rates of interest? 2008: (3.3%-.1%)/(1.001) = 0.032 3.2% 2011: (.3%-3%)/(1.03) = -0.0262 -2.62% Professor Hartwig bought bonds issued by MIT with a yield 4.67%. The expected inflation rate in 2017 is 2.40% What is the expected real rate of return this year? (4.675%-2.4%)/(1.024) = 0.0222 2.22% Yield Curve and Discount Rates -Term Structure: Relationship between investment term and interest rate Week 4 Pae 2 -Term Structure: Relationship between investment term and interest rate -Yield Curve: plot of bond yields as a function of bond maturity date -usually upward sloping but can change with economic downturn etc. -an inverted curve could happen if you thought a recession will occur in the future -associated with lower interest rates and inflation -Risk Free Interest Rate: interest rate at which money can be borrowed or lent without risk over a period. Assumed to be a US treasury security because default risk is 0. -Risk Premium: pay more for riskier terms The longer the loan, the higher the interest rate risk. The rates could increase and you lose out on potential interest earnings. -a risk-free cash flow of Cn received in n years has the present value: PV = Cn/(1+rn)n Compute the PV of a risk free 5 year annuity of $1000 per year given the yield curve in 2008. 1.0091, 1.0098, 1.026, 1.0169, 1.0201 are the risk free rates for each year PV = 1000/1.0091 + 1000/1.0098^2 + 1000/1.0126^3 + 1000/1.0169 + 1000/1.0201 = 4898.4763 Yield Curve and the Economy -Interest Rate Determination -Federal Funds Rate: overnight loan rate charged by banks with excess reserves at a Federal Reserve bank to banks that need additional funds to meet reserve requirements -The Fed determines very short-term interest rates through its influence in the market -Term Structure of Interest rates: longer the debt, higher the yield. -if interest rates are expected to rise, long term interest rates will be higher than short term to attract investors -if interest rates are expected to fall, long term rates will tend to be lower than short term Interest rates tend to fall during recessions because demand for credit falls and because the Fed lowers rates 2/2/17 Week 4 Pae 3 Bond certificate includes terms of the bond, amounts and dates of payment, etc. Bond is a security sold by governments and corporations to raise money from investors in promise for future payments Treasury Bill (T bill): maturity of up to 52 weeks Treasury Note: 2,3,5,7 and 10 years Treasury Bond: 20, 30 years Sovereigns: terms that refers to bonds issues by a federal government Municipal bond: bond issued by a state or local government, tax exempt, lower yields Maturity date: date of final payment and principal returned Term: time remaining until bond matures 1. Face Value: par or principal amount a. Notional amount used to compute interest amount b. Standard increments such as $1000 c. Repaid at maturity 2. Coupons: periodic payment made to holder of bond a. Bond certificate specifies the frequency of the payments and the % yield (rate set by issuer) b. Coupon is an old term but it persists Coupon rate: expressed as an APR, so amount of each coupon payment, CPN, is: CPN = (coupon rate * face value)/payments per year A $1000 bond with a 10% coupon and semi-annual payment: (.1*1000)/2 = 50 Zero Coupon Bonds: no coupon paid -only 2 cash flows -the bond's market price at time of purchase -bond's face value at maturity -purchased at discount and face value repaid at maturity -a one year, risk free, zero coupon bond with a $100,000 face value has an initial price of $96,618.36 Yield to Maturity of a Zero Coupon Bond -discount rate that sets present value of promised payments equal to current market price of bond 1+YTM = (FV/Price)1/n Maturity 1 yr 2 yr 3 yr 4 yr Week 4 Pae 4 Maturity 1 yr 2 yr 3 yr 4 yr Price 96.62 92.45 87.63 83.06 YTM = (100/96.62)^1 -1 = 0.035 YTM = (100/92.45)^.5 -1 = 0.04 YTM = (100/87.63)^(1/3) -1 = 0.045 YTM = (100/83.06)^.25 -1 = 0.0475 What is the price of a 5 year risk free, zero coupon bond with face value of 100? P = 100/1.05^5 = 78.3526 As time goes on, willing to pay less and less for the bond because money is "tied up" for longer Maturity 1 yr 2 yr 3 yr 4 yr Price 97.59 95.23 93.48 92.28 YTM = (100/97.59) ^ 1 -1 = 0.0247 YTM = (100/95.23)^.5 -1 = 0.0247 YTM = (100/93.48)^(1/3) -1 = 0.0227 YTM = (100/92.28)^.25 -1 = 0.0203 Example of an inverted yield curve because YTM is decreasing over time Coupon Bonds -pay face value at maturity -also make regular coupon interest payments -two types of US Treasury coupon securities: -Treasury Notes: 2, 3, 5, and 10 years -Treasury Bonds: 20 and 30 years Return on coupon bond comes from: -difference between purchase price and principal value -periodic coupon payments To compute the yield to maturity of a coupon bond, we need to know the coupon interest payments, and when they are paid P = CPN * 1/ytm(1- 1/(1+ytm)n) + FV/(1+ytm)n 5 year, 1000$ bond with a 2.2% coupon rate and semiannual coupons. Week 4 Pae 5 5 year, 1000$ bond with a 2.2% coupon rate and semiannual coupons. Price is 963.11. What is the YTM? CPN = (.022*1000)/2=11.0 Calculator: N = 10, -963.11 = PV, PMT = 11, FV = 1000, CPT: I/Y YTM = 1.5 semiannual rate; annual rate = 3% Bond prices move inversely with interest rates/YTM When interest rates rise, the value of bonds may fall Lower coupons have lower prices. Why Bond Prices Change Zero coupon bonds always trade for a discount Coupon bonds may trade at discount or premium When the bond price is greater than face value, it is premium or above par -coupon rate > YTM When price = face value, it is at par -Coupon rate = YTM When price is less than face value, it is below par or at a discount -coupon rate < YTM Three 30-year bonds with annual coupon payments One bond has a 10% coupon rate, one has a 5% coupon rate, and one has a 3% coupon rate If the YTM for each is 5%, determine price of each bond per 100$ of face value? 10% coupon: 176.86 5% coupon: 100 3% coupon: 69.26 Over time, the price of all bonds converges to its face value. Interest rate risk diminishes over time. Week 4 Pae 6