1
School of Government: Cash Management & Investment of Public Funds
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: [email protected]
School of Government: Cash Management & Investment of Public Funds
2
School of Government: Cash Management & Investment of Public Funds
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
School of Government: Cash Management & Investment of Public Funds
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
3
School of Government: Cash Management & Investment of Public Funds
Determinants of Interest Rates
i = RFR + I + RP where:
RFR = real risk-free rate of interest I = expected rate of inflation RP = risk premium
School of Government: Cash Management & Investment of Public Funds
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!
4
School of Government: Cash Management & Investment of Public Funds
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
School of Government: Cash Management & Investment of Public Funds
n Bond Pricing
n Bond Prices move inversely to interest rates
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.)
n The lower the coupon of the bond, the greater the price sensitivity
5
School of Government: Cash Management & Investment of Public Funds
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 2 5 10 30
Yie
ld (
%)
Years
Treasury RatesExample of Dramatic Rise in Longer Term Rates
4/30/2013 6/30/2013
2.50%
1.67%
.65%
1.41%
School of Government: Cash Management & Investment of Public Funds
* 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)
6
School of Government: Cash Management & Investment of Public Funds
School of Government: Cash Management & Investment of Public Funds
7
School of Government: Cash Management & Investment of Public Funds
Back to Yields
Alternative Methods to Calculate Yields
n Discount yield (DY) n Money market yield (MMY) n Bond equivalent yield (BEY) n Yield to maturity (YTM)
School of Government: Cash Management & Investment of Public Funds
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%
8
School of Government: Cash Management & Investment of Public Funds
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
School of Government: Cash Management & Investment of Public Funds
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%
9
School of Government: Cash Management & Investment of Public Funds
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.
School of Government: Cash Management & Investment of Public Funds
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.
10
School of Government: Cash Management & Investment of Public Funds
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
School of Government: Cash Management & Investment of Public Funds
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
11
School of Government: Cash Management & Investment of Public Funds
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%
School of Government: Cash Management & Investment of Public Funds
C. Calculation of BEY for the $1 million, 180-day maturity T-bill
A. BEY* = Interest X 365 Amount Invested Days to Maturity
*For securities with maturity of 182 days or less
B. Calculation
$ 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%
12
School of Government: Cash Management & Investment of Public Funds
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%.
School of Government: Cash Management & Investment of Public Funds
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
13
School of Government: Cash Management & Investment of Public Funds
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%
School of Government: Cash Management & Investment of Public Funds
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
14
School of Government: Cash Management & Investment of Public Funds
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
School of Government: Cash Management & Investment of Public Funds
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%
15
School of Government: Cash Management & Investment of Public Funds
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%
School of Government: Cash Management & Investment of Public Funds
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
16
School of Government: Cash Management & Investment of Public Funds
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
School of Government: Cash Management & Investment of Public Funds
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
17
School of Government: Cash Management & Investment of Public Funds
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
School of Government: Cash Management & Investment of Public Funds
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
18
School of Government: Cash Management & Investment of Public Funds
How to Calculate the Price of a Bond
School of Government: Cash Management & Investment of Public Funds
19
School of Government: Cash Management & Investment of Public Funds
School of Government: Cash Management & Investment of Public Funds
Bond is trading at a discount
20
School of Government: Cash Management & Investment of Public Funds
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
21
School of Government: Cash Management & Investment of Public Funds
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 www.marketwatch.com/news/markets/ www.bloomberg.com/news/
Charts www.economagic.com/popular.htm
Bank Info www.bauerfinancial.com www.bankrate.com www.fdic.gov
Top Related