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, LLC distributors of the North Carolina Capital Management Trust

Email: [email protected]

UNC School of Government: Cash Management & Investment of Public Funds

UNC 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 n

n n

Purchase Price of the bond Interest or coupon received

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

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

UNC 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

UNC School of Government: Cash Management & Investment of Public Funds

Determinants of Interest Rates where:

i = RFR + I + RP RFR = real risk-free rate of interest I = expected rate of inflation RP = risk premium

UNC 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!

UNC 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 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

UNC School of Government: Cash Management & Investment of Public Funds n

Bond Pricing n

n

n

n

Bond Prices move inversely to interest rates. 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 to interest rate changes. Zero coupon bonds are the most price sensitive.

UNC School of Government: Cash Management & Investment of Public Funds Treasury Rates Example of Dramatic Rise in Longer Term Rates

4.0

3.5 3.0

2.50%

Yield (%)

2.5

2.0

1.41%

1.5

1.67%

4/30/2013

6/30/2013

1.0

.65%

0.5

0.0

0

2

5

10

Years

30

UNC 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)

UNC School of Government: Cash Management & Investment of Public Funds

UNC School of Government: Cash Management & Investment of Public Funds

UNC School of Government: Cash Management & Investment of Public Funds

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)

UNC School of Government: Cash Management & Investment of Public Funds I. II.

ANNUALIZING INTEREST AND YIELD

Typically invest for a portion of the year and must convert interest earned to equivalent annual yield 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%

UNC 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

UNC 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%

UNC 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.

UNC 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.

UNC 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

UNC 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

UNC 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 $975,000 = .02564 X 2 = .05128 = 5.128%

X

360 180

UNC 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 Amount Invested

_____365_____ Days to Maturity

*For securities with maturity of 182 days or less B.

Calculation $ 25,000 $975,000

X

365 180

= .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%

UNC 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%. (Simple Interest)

UNC School of Government: Cash Management & Investment of Public Funds VI. Yield to Maturity (YTM) A.

Calculations for YTM : 1. 2.

Interest earned on interest 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. 2.

$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 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) = = = =

c.

1,000,000 (.09875/2) (55/182) 1,000,000 (.0494) (.3022) 1,000,000 (.014929) $14,929

Amount invested: $1,019,929

UNC 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 Amount Invested

X 360 127

= 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%

UNC School of Government: Cash Management & Investment of Public Funds CALCULATION OF AVERAGE YIELD FOR INVESTMENTS DURING QUARTER

(1)

A. Investments $500,000 100,000 100,000 250,000 100,000 60,000

(2)

(3)

(4)

Annual Coupon Portion of Year Invested Rate Days Days/365 6.05% 91 .2493 5.50 91 .2493 5.50 76 .2082 5.30 60 .1644 5.10 30 .0822 5.00 91 .2493

(5) Interest Earned (1)x(2)x(4) $ 7,542 1,371 1,145 2,178 419 750 $13,405

UNC School of Government: Cash Management & Investment of Public Funds Calculation of average daily amount invested during quarters 1. Dollar days calculation Amount Invested $500,000 100,000 100,000 250,000 100,000 60,000

Days Invested 91 91 76 60 30 91 439

2. Average daily amount invested: Dollar days Days in quarter = $85,700,000 91 = $941,758

$ Days $45,500,000 7,100,000 7,600,000 15,000,000 3,000,000 5,500,000 $85,700,000

UNC 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%

UNC 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 $100,000

.012

x

360 90

=

x

4

=

4.80%

UNC 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

360 365

=

7.742%

UNC 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 Price = .98190111 Principal = $981,901.11 Discount = $18,098.88

MM Yield = 7.29% BEY = 7.39%

UNC 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

981,901.11

5.

x

360

91

=

7.29%

Convert the MMY calculated in # 4 to a bond equivalent yield. 18,098.88 981,901.11

x

365 91

=

7.39%

UNC 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: Investments $300,000 $200,000 $100,000

Annual 6.0% 5.0% 5.5%

Days Invested Rate in Quarter 30 days 91 days 60 days Total interest earned:

Quarterly Interest $1,480 $2,493 $ 904 $4,877

Dollar Day Calculation Amount Invested $300,000 $200,000 $100,000

Days Invested 30 days 91 days 60 days

__$ Days__ 9,000,000 18,200,000 6,000,000 33,200,000

Avg. Daily Amount Invested = $ Days / Days in Qtr. = 33,200,000/91 = $364,835

UNC 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

UNC School of Government: Cash Management & Investment of Public Funds

How to Calculate the Price of a Bond

UNC School of Government: Cash Management & Investment of Public Funds

School of Government: Cash Management & Investment of Public Funds

School of Government: Cash Management & Investment of Public Funds

Bond is trading at a discount

UNC 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

UNC School of Government: Cash Management & Investment of Public Funds

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