1. Interest Rates and Returns: Some Definitions and Formulas

2. Yield to Maturity
Yield to maturity is the interest rate that equates the present value of payments received from a debt instrument with its value today.

3. Current Yield
In some cases, yield to maturity can be difficult to calculate.
Current yield is a simple formula that approximates yield to maturity.
Current yield = Coupon/Bond Price

4. Yield on a Discount Basis
A discount bond is one that is sold at a discount from its face value.
The yield or interest received is determined by the difference between the price paid and its face value.

5. Discount Bonds Yield formula: Corporate Bond
i = (Face - Price Paid)/Price Paid
U.S. Treasury bills are sold on a discount basis. The formula used to calculate that yield is:
i = (Face - Price Paid)/Face * (360 # Days to Maturity)

6. Discount Bonds: Characteristics
The formula used to calculate yield on a Treasury discount bond understates the yield to maturity associated with a corporate bond.
The longer the maturity, the greater the understatement.
A change in discount yield always signals a change in the same direction as yield to maturity.

7. Coupon Bond
A bond is a debt instrument. A coupon bond is a bond that pays its owner a fixed coupon payment every year until maturity, at which time a specified final amount (face value) is repaid
We expect to get:
Coupon payments each year
Principal at maturity.
How much should we be willing to pay today for a stream of income?

8. Coupon Bond
PB = C/(1 + i) + C/(1 + i)2 + C/(1 + i)3 + ……..C/(1 + i)n + P/(1 + i)n
where
C is a fixed coupon
i is the rate of interest
PB is the price or present value of the bond
P is the principal
n is years to maturity

9. Coupon Bond Example
Let the coupon payment be $100, the rate of interest 10%, and the principal equal to $ If n is 4, how much should we pay today for this bond?
PB = 100/( ) + 100/( ) /( ) /( ) /( )4
PB = 100/(1.10) + 100/(1.21) + 100/(1.331) + 100/(1.4641) /(1.4641)
PB = = $1,000.

10. Things to Notice
When a coupon bond is priced at its face value, the yield to maturity equals the coupon rate.

11. More Things to Notice
The yield to maturity and the coupon rate do not have to be the same.
If the bond price is less than the face value, the yield to maturity is greater than the coupon rate.
In this case, the difference between the bond price and the face value adds to the total return.
If the bond price is greater than the face value, the yield to maturity is less than the coupon rate.
In this case, the difference subtracts from the total return.

12. Even More Things to Notice
The price of a coupon bond and the yield to maturity are inversely related.
An increase in the interest rate decreases the bond price.
A decrease in the interest rate increases the bond price.
This is the reason the market participants are so interested in the actions of the Federal Reserve.

13. Coupon Period of Less than One Year
Bonds generally have coupons due at intervals shorter than one year.
In this case the market value of the bond is still the sum of the present values of all the payments due, but some adjustments to the formula must be made.

14. Six Month Coupon Bond
How much should you pay for a 20-year $1,000 Treasury bond with a coupon rate of 10% and semiannual coupons, when market rates are 8%?
The value of the bond is the sum of the present values of all the payments due.
Since coupons are semi-annual, the $100 coupon payment is split into two $50 payments.
The interest rate used for the calculation is the six month periodic rate and n = 40.

15. Useful Formula (1 + EAR)1/#periods per year = (1 + periodic rate)
To calculate the appropriate periodic rate, we use the effective annual rate and the number of payment periods per year, 2.
(1.08)1/2 =

16. Six Month Coupon Bond P = $1,221.00
(1.039) (1.039)2 (1.039) (1.039)39 (1.039)40
P = $1,221.00

17. The Annual Percentage Rate and the Periodic Rate
The periodic interest rate is the interest rate per compounding period.
Periodic rate = Annual Percentage Rate
Periods per Year
If the APR is 12%, the periodic rate per month is 1%.

18. The Effective Annual Rate
The effective annual rate is the interest accrued at the end of a year as a percentage of the principal amount.
To calculate the effective annual rate (EAR), we do the following:
Calculate the periodic rate.
Calculate the future value using the periodic rate.
Calculate the effective annual rate as the interest accrued as a percentage of the principal amount .

19. The Effective Annual Rate
Assume you receive an annual rate of 12%
on a one year $1,000 bond.
Calculate the periodic rate compounded monthly.
0.12/12 = 0.01
Calculate the future value.
$1,000 x (1.01)12 = $1,126.83
Calculate the effective annual rate.
($1, $1,000)/$1,000 = or 12.68%

20. The EAR
The EAR is useful when we want to compare the yield on instruments with different maturities, different interest rates, and/or different compounding periods.
For example, we can compare an instrument with a one year maturity paying 12% compounded monthly with another instrument that has a six month maturity paying 11.9% compounded weekly.

21. EAR: Example
The EAR on an instrument paying 12% compounded monthly is 12.68%.
$1,000 x (1.01)12 = $1,126.83
($1, $1,000)/$1,000 = or 12.68%

22. EAR: Example The EAR on the $1,000 asset earning 11.9% weekly is ????
Calculate the periodic rate.
0.119/52 =
Calculate the future value.
$1,000 x ( )52 = $1,126.31
Calculate the effective annual rate.
($1, $1,000)/$1,000 = = 12.63%

23. Summary: Formulas (1 + periodic rate )#periods per year = 1 + EAR
(1 + periodic rate) = (1 + EAR)1/#periods per year

24. Interest Rates and Returns
For any security, the rate of return is defined as the payments to the owner plus the change in its value, expressed as a ratio to its purchase price.
The return to a bond depends on its stream of coupon payments and the price the bond receives when it is sold.

25. Interest Rates and Returns
If the bond sells at a price in excess of its original purchase price, the owner receives a capital gain which increases his/her total return.
If the bond sells at a price below its original purchase price, the owner suffers a capital loss, which decreases his/her total return.

26. Return on a Bond The return on a bond may be expressed by the formula:
Ret = (C + (Pt+1 - Pt))/Pt
where
C = Coupon payment
Pt = Price of the bond in time t
Pt+1 = Price of the bond in time t + 1

27. Returns on Different Maturity 10% Coupon Rate Bonds
Term Initial i Initial P New i New P K Gain ROR
30 10% %
20 10% %
10 10% %
5 10% %
2 10% %
1 10% %

28. Things to Notice
The only bond whose return is certain to equal the initial yield is the one whose time to maturity is the same as the holding period.
A rise in interest rates is associated with a fall in bond prices, resulting in capital losses on bonds whose terms to maturity are longer than the holding period.

29. More Things to Notice
The longer the bond’s maturity, the greater is the size of the price change associated with an interest rate change.
The longer a bond’s maturity, the lower is the rate of return that occurs as a result of the increase in the interest rate.
Even though the bond had a good interest rate, its return became negative when interest rates rose.

30. Reinvestment Risk Reinvestment risk occurs
when an investor holds a series of short bonds over a long holding period and interest rates are uncertain.
If interest rates rise, the investor gains
If interest rates fall, the investor loses

31. Reinvestment Risk: Example
Assume a holding period of two years and an investor who has decided to buy two one year bonds sequentially.
Year 1 bond:
Face = $1000, initial interest rate = 10%
At the end of the year, the investor has $1100.
Year 2 bond:
Face = $1100, interest rate = 20%
At the end of year 2, the investor has $1320.

32. Reinvestment Risk: Example
The investor’s two year return will be:
($ $1000)/$1000 = 0.32 = 32% over two years.
In this case the investor has benefited by buying two one year bonds.
Conversely, if interest rates had fallen to 5%, the investor would done less well.

33. Reinvestment Risk: Example
Year 1:
($1000 x ( )) = $1100
Year 2:
($1100 x ( )) = $1155
Return = ($ $1000)/$1000 = 15.5% over two years.
The investor now loses from a change in interest rates.