EAR/APRTime Value of Money

Returns and Compounding

Returns are often stated in annual terms Interest is paid (accrues) within the year

Example: Savings accounts: interest accrues every month

How do Banks report “interest rates”? Unfortunately, they do not report annual growth rates

APR

Banks report “Annual Percentage Yields” or “APRs”

Let N be the number of times a bank pays interest per year Example: if the bank pays interest annually N=12

Let rm be the effective monthly interest rate

Then the APR is simply rm*12

APR

Example: A bank pays 5% semi-annually. What is APR?

APR=0.05*2 = 10%

What is the annual growth rate on the account? 1.052 -1=10.25%

Effective Annual Rates

The effective annual rate or EAR, is simply the annual growth rate. For the example above, the effective

annual growth rate is 10.25%

NrNAPR /

1)1( Nr

1)1( /1 NEAR

You can’t move directly fromEAR to APR or vice-versawithout finding r first.

Effective Annual Rate Effective Annual Rate: The effective return on

an investment assuming all interest payments are reinvested at the same rate.

Example: Account pays 1% per month In 1 year, a $1 investment has grown to

Effective annual return is 13%

13.101.11 12

Effective Annual Rates

What is EAR on UCCU savings account if APR is 12%? Assume interest accrues monthly.

%67.12

1267.112

12.11

12

EAR

EAR

Effective Monthly Rates

The EAR is 15% What is effective 1-month rate? What

is the APR? Assume interest accrues monthly.

04.1412*17.1

%17.1

115.1

115.112/1

12

APR

EMR

EMR

EMR

APR and EAR

APRs are always defined for 1 year

But we could have Effective 6 month rates Effective 3 year rates Effective 2-month rates

Effective rates

If the effective annual rate is 12%, what is the effective 6-month rate? 3 year rate?

We just need to find the 6-month and 3-month growth rates.

Effective Rates

Effective 6-month rate is given by

Effective 3-year rates is given by

%8.5

)1()12.1(

6

26

m

m

r

r

%49.40

)1()12.1(

3

33

y

y

r

r

Pricing Bonds Zero-Coupon Bond

Assume an investment of similar risk pays 8% per year.

Face Value of Bond =1000 Bond Matures in 10 years

Price? Just find present value

19.46308.1/1000 10 Price

Pricing Bonds

How about a coupon paying bond? Assume an investment of similar risk pays 8%

per year. Face Value of Bond =1000 Bond Matures in 3 years Bond pays a coupon of 50 at end of each year

(5% of face value)

Pricing Bonds

Cash flows from bond 50 at end of year 1 50 at end of year 2 1050 at end of year 3

We can “split up” each of these cash flows and think of each of them as a separate zero-coupon bond.

Pricing Bonds

Cash Flow 1: 50 at end of year 1 Price of this cash flow: 50/1.08=46.30

Cash flow 2: 50 at end of year 2 Price of this cash flow: 50/1.082 = 42.87

Cash flow at end of year 3 Price of this cash flow: 1050/1.083=900.21

Pricing Bonds

The price of this bond is just the sum of the PV’s of each of the pieces

Price= 46.3 + 42.87 + 900.21=989.38

Pricing Semi-Annual Bonds

Most Bonds pay coupons every six months Example:

Face value: 1000 Coupon: 30 every six months Matures: 1.5 years Investment of similar risk: pays 2% every six

months

Pricing Semi-Annual Bonds

Cash flows 30 at end of first 6 months 30 at end of first year 1030 when bond matures

PV of cash flows 30/1.02 = 29.41 30/1.022 = 28.84 1030/1.023 = 970.59

Price of bond = 1028.84

Yield-to-Maturity Suppose we know bond price, and cash flows.

Yield-to-Maturity comes from the interest rate that makes the price equal to the sum of the PV of all cash flows.

Notation: y: the rate that makes the price equal to the sum of

discounted cash flows YTM: yield to maturity

Yield-to-Maturity

Example: Zero coupon bond Face value=1000 Price=600 Matures in 5 years

For a zero-coupon bond, YTM=y

%76.101600

1000

600

1000)1(

)1(

1000600

5/15

5

yy

y

Yield-to-Maturity

Economic Interpretation: For a zero-coupon bond, YTM is the

effective annual return. From example above:

5-year return: 1000/600-1=0.67

%76.10167.1

67.1)1(

5/1

5

y

y

y

r

r

r

return annual effective

Yield-to-Maturity

Coupon paying bonds Example:

Bond matures in 3 years Pays 4% coupon annually FV=1000 Price=$987

32 )1(

1040

)1(

40

1

40987

yyy

Yield-to-Maturity

In this case, it is too difficult to solve for y algebraically

Use Financial Calculator N=3 pmt=40 Price=-987 FV=1000 Rate=?

For an annual coupon paying bond, YTM = y

%47.4y

Yield-to-Maturity

Economic Interpretation For annual coupon paying bonds, YTM is

the effective annual return assuming all coupons are reinvested at the same rate.

From previous example Suppose we invest all coupons at 4.473% What do we get in 3 years?

Yield-to-Maturity

Future value of first coupon: 40(1.0473)2=43.66

Future value of second coupon 40(1.0473)=41.79

Future value of last cash flow 1040

Total future value =43.66+ 41.79+1040 = 1125.45

Total 3-yr return = 1125.45/987-1=14.027%

Yield-to-Maturity

Effective Annual Return:

yr

r

r

y

y

y

%47.4114027.1

14027.1)1(

3/1

3

return annual effective

Yield-to-Maturity

Semi-Annual Coupon paying bonds Example:

Bond matures in 1 year Pays $20 coupon semi-annually FV=1000 Price=$990

2)1(

1020

1

20990

yy

Yield-to-Maturity

Again, in this case, it is too difficult to solve for y algebraically

Use Financial Calculator N=2 pmt=20 Price=-990 FV=1000 Rate=? %519.2y

Yield-to-Maturity

Economic Interpretation For semiannual coupon paying bonds, y is

the effective six-month return assuming all coupons are reinvested at the same rate.

From previous example Suppose we invest all coupons at 2.519% What do we get in 1 year?

Yield-to-Maturity

Future value of first coupon: 20(1.02519)=20.50

Future value of last cashflow 1020

Total future value =20.50 + 1020 = 1040.50

Total 1-yr return = 1040.50/990-1=5.10%

Yield-to-Maturity

Effective six-month return

yr

r

r

m

m

m

%519.210510.1

0510.1)1(

2/16

6

26

return month-six effective

Yield-to-Maturity

For semi-annual bonds, the YTM is always quoted as an APR:

For semi-annual bonds, the coupon paid is always quoted as a percentage of face value times two.

2yYTM

Example

Semi annual bond Matures in 2 years Coupon rate=8% Face=1000 Price=955

What is y? What is YTM?

Example

Finding y: N=4 pmt =40 FV=1000 PV=-955 Rate?

%55.10

%277.5

YTM

y

Example

Find bond price Semi annual bond YTM=6% Coupon rate=9% Matures 3 years FV=1000

Example

Finding Price: N=6 FV=1000 r = 3% pmt= 45 Price=?

26.1081Price

Example

Find Bond Price Annual bond YTM=6% Coupon rate = 9% Matures 3 years FV=1000

Example

Finding Price: N=3 (this is an annual bond) FV=1000 r = 6% pmt= 90 Price=?

19.1080Price