11 SOG PRINTYield Calculations Oct 2016 2.pdf

UNC 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, LLCdistributors of the North Carolina Capital Management Trust

Email: [email protected]


What is Yield?n The annual return on an investment

n Yield on a bond is based on:n Purchase Price of the bondn Interest or coupon received

n Coupon payments are generally fixedn Price of the bond after purchase fluctuates

n Bond prices change due to changing interest ratesn Supply & demand, time to maturity, credit quality


Computing Bond YieldsYield 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


Determinants of Interest Rates

i = RFR + I + RPwhere:

RFR = real risk-free rate of interestI = expected rate of inflation

RP = risk premium


Bond Pricingn 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!


Bond PricingEx: - Bond purchased at par (price = 100%)

2 year maturity, 5% coupon bond

Cost = Principal Amt x Price ($1,000 x 100% = $1,000)

Interest rates in the market move to 4%

New Price = 101.9039% New Market Value = $1,019.04

Interest rates in the market move to 6%

New Price = 98.1415% New Market Value = $ 981.42


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 to interest rate changes.

n Zero coupon bonds are the most price sensitive.


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 2 5 10 30

Yiel

d (%

)

Years

TreasuryRatesExampleofDramaticRiseinLongerTermRates

4/30/2013 6/30/2013

2.50%

1.67%

.65%

1.41%


* 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)


Back to YieldsAlternative Methods to Calculate Yieldsn Discount yield (DY)n Money market yield (MMY)n Bond equivalent yield (BEY)n Yield to maturity (YTM)


ANNUALIZING INTEREST AND YIELDI. Typically invest for a portion of the year and must convert interest earned

to equivalent annual yieldII. 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%


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 YEARI. Some securities for which yield is quoted using a 360-day year

Treasury billsCommercial paperBankers acceptancesSome others

II. Securities for which yield is calculated using a 365-day yearTreasury notes and bondsNC Capital Management TrustCDs sometimesOthers 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%


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 YIELDI. 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.


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


IV. Conversions among DY, MMY, and BEYDiscount

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 180360

= $25,000

2. Amount invested to buy bill

$1,000,000 - $25,000= $975,000

3. ConversionMMY = $ 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

*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 365360

5.128% x 1.0138

= 5.199%


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%. (Simple Interest)


VI. Yield to Maturity (YTM)A. Calculations for YTM :

1. Interest earned on interest2. 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/x3 purchased at 1/24/x3 for 100.16 with a YTM of 8.28%. Coupon period 11/30/x2 - 5/31/x3

2. Amount Invested = [Face Value x Price] + Accrued Interest

a. Face Value x Price = Principal

1,000,000 x 1.005 = $1,005,000 Price=(100 + 16/32)

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


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 360Amount Invested 127

= 1,049,400 - 1,019,929 x 2.8351,019,929

= 29,471 x 2.8351,019,929

= .0288952 x 2.835

= 8.19%


CALCULATION OF AVERAGE YIELD FOR INVESTMENTS DURING QUARTER

(1) (2) (3) (4) (5)AnnualCoupon Portion of Year Invested Interest Earned

A. Investments Rate Days Days/365 (1)x(2)x(4)$500,000 6.05% 91 .2493 $ 7,542100,000 5.50 91 .2493 1,371100,000 5.50 76 .2082 1,145250,000 5.30 60 .1644 2,178100,000 5.10 30 .0822 41960,000 5.00 91 .2493 750

$13,405


Calculation of average daily amount invested during quarters1. Dollar days calculation

DaysAmount Invested Invested $ Days

$500,000 91 $45,500,000100,000 91 7,100,000100,000 76 7,600,000250,000 60 15,000,000100,000 30 3,000,000 60,000 91 5,500,000

439 $85,700,0002. Average daily amount invested:

Dollar days Days in quarter

= $85,700,00091

= $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 36591

= 1.42 x 4.011

= 5.69%


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


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 = $981,901.11Discount = $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


6. Calculate the average annualized yield earned from the following three investments made during a quarter, using a 365 day year:

Days Invested QuarterlyInvestments 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

UNC School of Government:Cash Management & Investment of Public FundsHow to Calculate the Price of a Bond

School of Government:Cash Management & Investment of Public Funds

School of Government:Cash Management & Investment of Public Funds

Bond is trading at a discount


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


WebsitesFinancial Calculator www.ficalc.com/

Interest Rates www.bloomberg.com/markets/rates/index.html

Economic Info www.research.stlouisfed.org/fred2/www.nber.org/releases/www.federalreserve.gov/econresdata/default.htm

News www.marketwatch.com/news/markets/www.bloomberg.com/news/

Charts www.economagic.com/popular.htm

Bank Info www.bauerfinancial.comwww.bankrate.comwww.fdic.gov

11 SOG PRINTYield Calculations Oct 2016 2.pdf

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