Investment Yield Formulas and Yield Case Studies Presented by : Gary Porter, C.F.A. Vice President Capital Management of the Carolinas, LLC distributors of the North Carolina Capital Management Trust Email: gporter@capitalmc.com 1
What is Yield? n The annual return on an investment n Yield on a bond is based on: n Purchase Price of the bond n Interest or coupon received n Coupon payments are generally fixed n Price of the bond after purchase fluctuates n Bond prices change due to changing interest rates n Supply & demand, time to maturity, credit quality Computing Bond Yields Yield Measure Purpose Nominal Yield Measures the coupon rate Current Yield Measures current income rate Yield to Maturity* Measures expected rate of return for bond held to maturity Yield to Call Measures expected rate of return for bond held to a call date * Probably the most widely used 2
Determinants of Interest Rates where: i = RFR + I + RP RFR = real risk-free rate of interest I = expected rate of inflation RP = risk premium Bond Pricing n Bond prices move inversely to interest rates Interest Rates go Bond Prices go Interest Rates go Bond Prices go If you learn nothing else about bond prices, learn this! 3
Bond Pricing Ex: - Bond purchased at par (price = 100%) 2 year maturity, 5% coupon bond Cost = Principal Amt x Price ($1,000 x 100% = $1,000) Interest rate moves to 4% New Price = 101.9039% New Market Value = $1,019.04 Interest rate moves to 6% New Price = 98.1415% New Market Value = $ 981.42 n Bond Pricing n Bond Prices move inversely to interest rates n n The longer the maturity of the bond, the more sensitive (variable) its price is to changes in rates (10 yr. security price will move more than 2 yr.) The lower the coupon of the bond, the greater the price sensitivity 4
4.0 Treasury Rates Example of Dramatic Rise in Longer Term Rates 3.5 3.0 2.5 2.50% Yield (%) 2.0 1.5 1.41% 1.67% 4/30/2013 6/30/2013 1.0 0.5.65% 0.0 0 2 5 10 30 Years * Modified Duration is used to approximate the percent change in bond value for a given percent change in yield, using the following formula: Percent change in bond value = (-DM * change in yield) 5
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Back to Yields Alternative Methods to Calculate Yields n n n n Discount yield (DY) Money market yield (MMY) Bond equivalent yield (BEY) Yield to maturity (YTM) ANNUALIZING INTEREST AND YIELD I. Typically invest for a portion of the year and must convert interest earned to equivalent annual yield II. What is the annual yield on $100,000 invested for 91 days and interest earned of $1,500? A. $ 1,500 = 1.5% or.015 $100,000 B. Annualize.015 x 365 91 =.015 x 4.011 =.06016 = 6.016% 7
III. Given $100,000 invested for 91 days at an annual yield of 6.016%, what interest would be earned? A. Convert annual yield to yield for period invested.06016 x 91 365 =.06016 x.2493 =.0150 = 1.5% B. Calculation of interest $100,000 x 1.5% = $1,500 CALCULATION OF YIELD DEPENDS ON NUMBER OF DAYS USED FOR YEAR I. Some securities for which yield is quoted using a 360-day year Treasury bills Commercial paper Bankers acceptances Some others II. Securities for which yield is calculated using a 365-day year Treasury notes and bonds NC Capital Management Trust CDs sometimes Others occasionally III. To re-state the yield of a security quoted on a 360-day basis to a yield on a 365-day basis, multiply the 360-day yield by the ratio of 365/360. 6.7% x (365/360) = 6.7% x 1.0139 = 6.79% 8
IV. To re-state the yield of a security quoted on a 365-day basis to a yield on a 360-day basis, multiply the 365-day yield by the ratio of 360/365 6.79% x (360/365) = 6.79% x.9863 = 6.7% V. Always verify number of days used in investment quotes given to you. ALTERNATIVE METHODS FOR CALCULATING YIELD I. Discount yield (DY) A. Used for securities that are issued at discount from face value, e.g., T- bills. The WSJ provides discount and bond equivalent yield quotes for T Bills. B. Discount yield or percentage multiplied by the face value of a security calculates the dollar amount of the discount. DY is calculated by dividing the discount (face value less the amount invested) by the face value. C. DY uses a 360-day year D. Example: Invest $950,000 in discount instrument with face value of $1,000,000 for 270-day term $1,000,000 - $950,000 = $50,000 =.05 x (360/270) =.0667 = 6.67% annualized $1,000,000 $1,000,000 discount yield. 9
II. Money market or (MMY) A. May be used to provide yield quotes for CDs and other money market instruments. B. MMY is calculated by dividing interest earned or discount by the amount invested. MMY is higher than the discount yield but less than bond equivalent yield C. MMY uses a 360-day year D. Example: Invest $950,000 in CD for 270 days. Interest earned is $50,000 $50,000 =.05263 x (360/270) =.0699 = 6.99% $950,000 III. Bond equivalent yield (BEY) required by LGC on 203 s A. Used for Treasury notes and bonds, NC Capital Management Trust, etc. B. Yield is calculated by dividing the interest earned or discount by the amount invested. BEY is higher than MMY and DY C. BEY uses a 365-day year. To convert MMY to bond equivalent yield, multiply MMY by 365/360. Example: 6.99 % x 365/360 = 6.99 x 1.0139 = 7.1% D. For securities that have maturities longer than 182 days and for which interest is earned or paid semi-annually, BEY is calculated using a 365-day year and considering compound interest, i.e., interest earned on interest 10
IV. Conversions among DY, MMY, and BEY Discount A. MMY = Interest X 360 Amount Invested Days to Maturity B. Conversion of DY to MMY for $1 million T-bill, 180-day maturity at DY of 5% 1. Discount (interest earned) on T-bill = $1,000,000 X.05 x 180 360 = $25,000 2. Amount invested to buy bill $1,000,000 - $25,000 = $975,000 3. Conversion MMY = $ 25,000 X 360 $975,000 180 =.02564 X 2 =.05128 = 5.128% C. Calculation of BEY for the $1 million, 180-day maturity T-bill A. BEY* = Interest X 365 Amount Invested Days to Maturity B. Calculation *For securities with maturity of 182 days or less $ 25,000 X 365 $975,000 180 =.02564 x 2.0278 =.05199 = 5.199% D. Conversion of MMY to BEY for 180-day maturity T-bill 5.128% X 365 360 5.128% x 1.0138 = 5.199% 11
V. Effect of more frequent compounding or earning of interest A. Security for which quoted annual interest rate is 5%, but for which interest is compounded daily, will earn an effective annual yield of 5.26%. Due to daily compounding. B. Security for quoted annual interest rate is 5%, and for which interest is compounded just once a year, will earn an effective annual yield of just 5%. VI. Yield to Maturity (YTM) A. Calculations for YTM : 1. Interest earned on interest 2. Capital gains or losses B. Actual realized yield over the life of an investment equals YTM only if all coupon interest payments received before maturity are reinvested at the current YTM C. Conversion of YTM to MMY for short maturity T-Note 1. $1 million T-note with a 9 7/8% coupon maturing on 5/31/03 purchased at 1/24/03 for 100.16 with a YTM of 8.28%. Coupon period 11/30/02-5/31/03 2. Amount Invested = [Face Value x Price] + Accrued Interest a. Face Value x Price = Principal 1,000,000 x 1.005 = $1,005,000 b. Accrued interest = Face Value x (Coupon Rate/2) x (Number of days from end of last coupon date to date of purchase/number of days from last coupon date to next coupon date or maturity) = 1,000,000 (.09875/2) (55/182) = 1,000,000 (.0494) (.3022) = 1,000,000 (.014929) = $14,929 c. Amount invested: $1,019,929 12
3. Value at Maturity = Face Value x (1 + Coupon Rate/2) = 1,000,000 (1.0494) = $1,049,400 4. MMY = Value at Maturity - Amount Invested X 360 Amount Invested 127 = 1,049,400-1,019,929 x 2.835 1,019,929 = 29,471 x 2.835 1,019,929 =.0288952 x 2.835 = 8.19% CALCULATION OF AVERAGE YIELD FOR INVESTMENTS DURING QUARTER (1) (2) (3) (4) (5) Annual Coupon Portion of Year Invested Interest Earned A. Investments Rate Days Days/365 (1)x(2)x(4) $500,000 6.05% 91.2493 $ 7,542 100,000 5.50 91.2493 1,371 100,000 5.50 76.2082 1,145 250,000 5.30 60.1644 2,178 100,000 5.10 30.0822 419 60,000 5.00 91.2493 750 $13,405 13
Calculation of average daily amount invested during quarters 1. Dollar days calculation Days Amount Invested Invested $ Days $500,000 91 $45,500,000 100,000 91 7,100,000 100,000 76 7,600,000 250,000 60 15,000,000 100,000 30 3,000,000 60,000 91 5,500,000 439 $85,700,000 2. Average daily amount invested: Dollar days Days in quarter = $85,700,000 91 = $941,758 C. Calculation of average yield Interest earned Average daily amount invested = $13,405 $941,758 = 1.42% D. Annualized yield 1.42 x 365 91 = 1.42 x 4.011 = 5.69% 14
INVESTMENT YIELD CASES 1. If $1,200 is earned on $100,000 invested in a CD for 90 days, what is the MMY for this investment? $1,200 x 360 = $100,000 90.012 x 4 = 4.80% 2. Convert a 7.85 % interest rate, quoted on a 365-day basis, to an equivalent rate using a 360-day basis..0785 x 360 = 7.742% 365 15
3. Calculate the principal amount that should be paid for a T-bill with a $1 million face value maturing on 12/18/x3 and settled (purchased) on 9/18/x3. The T-bill has a quoted discount rate of 7.16 % Par x ((1-(DR/100) x (DTM/360))) 1,000,000 x ((1-(7.16/100) x 91/360))) 1,000,000 x (1 (.01809888)) 1,000,000 x.98190111 = $981,901.11 Par = $1,000,000 MM Yield = 7.29% Price =.98190111 BEY = 7.39% Principal = $981,901.11 Discount = $18,098.88 4. Convert the discount yield quoted in # 3 (7.16 %) to an equivalent money market yield. Discount Yield 7.16% to MMY MMY = 18,098.88 x 360 = 7.29% 981,901.11 91 5. Convert the MMY calculated in # 4 to a bond equivalent yield. 18,098.88 x 365 = 7.39% 981,901.11 91 16
6. Calculate the average annualized yield earned from the following three investments made during a quarter, using a 365 day year: Days Invested Quarterly Investments Annual Rate in Quarter Interest $300,000 6.0% 30 days $1,480 $200,000 5.0% 91 days $2,493 $100,000 5.5% 60 days $ 904 Total interest earned: $4,877 Dollar Day Calculation Amount Invested Days Invested $ Days $300,000 30 days 9,000,000 $200,000 91 days 18,200,000 $100,000 60 days 6,000,000 33,200,000 Avg. Daily Amount Invested = $ Days / Days in Qtr = 33,200,000/91 = $364,835 Average Yield = Interest Earned Avg Daily Amount Invested = $4,877 $364,835 =.0134 or 1.34% Annualized Yield = 1.34% x 365 = 5.37% 91 17
How to Calculate the Price of a Bond 18
Bond is trading at a discount 19
Pricing Bonds with a PV Table (May be on the Exam) Add the PV of all coupons and maturity amount using PV s provided in table format 20
Websites Financial Calculator www.ficalc.com/ Interest Rates www.bloomberg.com/markets/rates/index.html Economic Info www.research.stlouisfed.org/fred2/ www.nber.org/releases/ News Charts www.marketwatch.com/news/markets/ www.bloomberg.com/news/ www.economagic.com/popular.htm Bank Info www.bauerfinancial.com www.bankrate.com www.fdic.gov 21