Originally Posted by

**Aidie** I was always told that Modified Duration = Effective Duration when there are no embedded options, although have not been able to prove this numerically as I get quite different answers using the two formulas. I guess it is as this formula is really just an approximation of the modified duration?

**Effective Duration = (P**_{-} - P_{+} ) / [(2)*(P_{0})*(Y_{+} - Y_{-})]

How would you explain the difference? Partial Convexity? or just that this is only meant to be an approximation?

Thanks for any thoughts!

**Duration** is a very useful tool that helps a bond analyst to help immunize the bond from sudden changes in interest rates.

The yield from bond may change from its initial value thus the bond holder would want to find her actual rate of return which happens to be the YTH - yield to horizon.

YTH - yield to horizon is a better indicator of a return as compared to YTM - yield to maturity. But finding YTH is no easy task as this will entail forecasting interest rates at future dates.

Having said that let me show you how using duration, the bond owner can immunize her investment in fixed income security by making use of duration as the horizon.

But first let us calculate the YTH - yield to horizon for the same bond that was issued at 4% semi-annual coupon at par of 1000 dollars with 10 years to maturity. The initial yield on the bond was 6% but after the purchase of the bond the very next day, the interest rate jumped to 7%.

Now if she wanted to hold on to the investment for 5 years, she would like to know her YTH - yield to horizon

Her YTH turns out to be 5.38% if she decided to hold on to the investment for 5 years

Code:

Semi-annual compounding of interest
Par value = 1000
YTM0 = 6%
**YTM1 = 7%**
Coupon = 4%
Years to maturity = 10
**Horizon = 5**
Bond price before change in YTM
Interest compounded semi annually
Par value of bond is 1000
Coupon rate on bond is 0.02
Initial YTM on bond is 0.03
Years till maturity are 20
Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
PVIFA(0.03, 20) = 14.877474860456
PVIF(0.03, 20) = 0.55367575418633
Price = 0.02 x 1000 x 14.877474860456 + 1000 x 0.55367575418633
Price = 297.54949720911 + 553.67575418633
Price = 851.23
Bond price after change in YTM
Interest compounded semi annually
Par value of bond is 1000
Coupon rate on bond is 0.02
Initial YTM on bond is 0.035
Years till maturity are 20
Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
PVIFA(0.035, 20) = 14.212403301952
PVIF(0.035, 20) = 0.50256588443167
Price = 0.02 x 1000 x 14.212403301952 + 1000 x 0.50256588443167
Price = 284.24806603905 + 502.56588443167
Price = 786.81
Horizon rate of return
Bond price using initial YTM = 851.23
Bond price after change in YTM = 786.81
Change in YTM = 0.035
Bond horizon = 10
horizon rate = (786.81/851.23)^(1/10) * (1+0.035) - 1
horizon rate = (0.9243310735682)^(0.1) * 1.035 - 1
horizon rate = 0.99216237890656 * 1.035 - 1
horizon rate = 1.0268880621683 - 1
horizon rate = 2.69%
**Annual horizon rate = 5.38%**

Now she remembers from her college financial management course, that her YTH - yield to horizon will turn out to be the same as initial YTM if she were to use **Duration** as her Horizon.

Thus she would be able to immunize the bond from any change in interest rates. It turns out that indeed her YTH - yield to horizon is 6%, the same as initial YTM when her horizon is the same as **Duration** of the bond

The rates have jumped to 7% from 6% but YTH has remained at 6%

Code:

Semi-annual compounding of interest
Par value = 1000
YTM0 = 6%
**YTM1 = 7%**
Coupon = 4%
Years to maturity = 10
**Horizon = Duration = 8.1688353010109609266845229091446**
Bond price before change in YTM
Interest compounded semi annually
Par value of bond is 1000
Coupon rate on bond is 0.02
Initial YTM on bond is 0.03
Years till maturity are 20
Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
PVIFA(0.03, 20) = 14.877474860456
PVIF(0.03, 20) = 0.55367575418633
Price = 0.02 x 1000 x 14.877474860456 + 1000 x 0.55367575418633
Price = 297.54949720911 + 553.67575418633
Price = 851.23
Bond price after change in YTM
Interest compounded semi annually
Par value of bond is 1000
Coupon rate on bond is 0.02
Initial YTM on bond is 0.035
Years till maturity are 20
Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
PVIFA(0.035, 20) = 14.212403301952
PVIF(0.035, 20) = 0.50256588443167
Price = 0.02 x 1000 x 14.212403301952 + 1000 x 0.50256588443167
Price = 284.24806603905 + 502.56588443167
Price = 786.81
Horizon rate of return
Bond price using initial YTM = 851.23
Bond price after change in YTM = 786.81
Change in YTM = 0.035
Bond horizon = 16.337670602022
horizon rate = (786.81/851.23)^(1/16.337670602022) * (1+0.035) - 1
horizon rate = (0.9243310735682)^(0.06120823612861) * 1.035 - 1
horizon rate = 0.99519541111377 * 1.035 - 1
horizon rate = 1.0300272505027 - 1
horizon rate = 3%
**Annual horizon rate = 6.01%**

Now the rates have dropped to 5% from 6% but here YTH is the still 6% or the same as initial YTM

Code:

Semi-annual compounding of interest
Par value = 1000
YTM0 = 6%
**YTM1 = 5%**
Coupon = 4%
Years to maturity = 10
**Horizon = Duration = 8.1688353010109609266845229091446**
Bond price before change in YTM
Interest compounded semi annually
Par value of bond is 1000
Coupon rate on bond is 0.02
Initial YTM on bond is 0.03
Years till maturity are 20
Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
PVIFA(0.03, 20) = 14.877474860456
PVIF(0.03, 20) = 0.55367575418633
Price = 0.02 x 1000 x 14.877474860456 + 1000 x 0.55367575418633
Price = 297.54949720911 + 553.67575418633
Price = 851.23
Bond price after change in YTM
Interest compounded semi annually
Par value of bond is 1000
Coupon rate on bond is 0.02
Initial YTM on bond is 0.025
Years till maturity are 20
Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
PVIFA(0.025, 20) = 15.589162285647
PVIF(0.025, 20) = 0.61027094285883
Price = 0.02 x 1000 x 15.589162285647 + 1000 x 0.61027094285883
Price = 311.78324571294 + 610.27094285883
Price = 922.05
Horizon rate of return
Bond price using initial YTM = 851.23
Bond price after change in YTM = 922.05
Change in YTM = 0.025
Bond horizon = 16.337670602022
horizon rate = (922.05/851.23)^(1/16.337670602022) * (1+0.025) - 1
horizon rate = (1.0832082190468)^(0.06120823612861) * 1.025 - 1
horizon rate = 1.0049041899641 * 1.025 - 1
horizon rate = 1.0300267947132 - 1
horizon rate = 3%
**Annual horizon rate = 6.01%**

The toolkit

Yield to Horizon - YTH

Bond horizon rate of return calculation

YTH calculator

http://finance.thinkanddone.com/tadBHRR.exe