Think & Done

Yield to call YTC formula is explained and illustrated with example calculation using YTC formula. On this web page we look at the definition of yield to call on callable bonds, formula for YTC, and an example with YTC calculation using Linear interpolation, and a neat Yield to Call YTC Calculator.

What is YTC

When interest rates are expected to drop in the future, most bond issuing corporation will place a call option on bonds they issue. This will permit the corporation to buy back or call the bonds when interest rates drop, and the corporations will reissue the bonds with lower interest rates thus saving it money from interest payments that it would have to make otherwise at the older higher rate.

YTC Formula

There isn't a mathemtical formula that we can use to find YTC, actually there is an equation for bond prices that we can use instead to approximate the actual yield to call. In the YTC equation above the k is the YTC or the interest rate at which the interest payments and call price is at par with the market price of the bond. This rate is called yield to call.

YTC Example

Suppose McDonalds had issued a $1,000 callable bond at par value on Jan 2010 with a coupon rate of 10% and 20 years to maturity. McDonalds expects the interest rates to drop in 2020 and sets it as the year to call the bonds at a call price of $1,100. Now it's Jan 2012 and the interest rates on similar bonds have dropped to 9% thus market price of the bond has jumped to $1,088.You wish to buy such a bond and want to know the yield to call for the callable bond from McDonalds.
In the following section you will find the details for finding the yield to call on McDonalds callable bond as of Jan 2012

YTC Calculation

Bond Price at Lower YTC of 6.79%

Compounding = annual
Par Value = 1000
Market Price = 1088
Call Price = 1100
Coupon Rate = 10%
YTC Low = 6.79%
N = 8

 

Non Zero Bond Price Formula

Coupon Rate x Par Value x PVIFA(ytcl%, n) + Call Price x PVIF(ytcl%, n)

 

PVIFA Formula

PVIFA(ytcl%, n) = [1 - v] / ytcl%
v = 1 / (1 + ytcl%)^n
PVIFA(ytcl%, n) = [1 - { 1 / (1 + ytcl%)^n }] / ytcl%

 

PVIFA Calculation

v = 1 / (1+0.068)^8
v = 0.59078570518848
PVIFA(0.068, 8) = [1 - 0.59078570518848] / 0.068
PVIFA(0.068, 8) = 0.40921429481152 / 0.068
PVIFA(0.068, 8) = 6.01785727664

 

PVIF Formula

PVIF(ytvl%, n) = 1 / (1 + ytcl%)^n

 

PVIF Calculation

PVIF(0.068, 8) = 1 / (1+0.068)^8
PVIF(0.068, 8) = 1 / 1.6926611311304
PVIF(0.068, 8) = 0.59078570518848

 

Non Zero Bond Price Calculation

Price = 0.1 x 1000 x 6.01785727664 + 1100 x 0.59078570518848
Price = 601.785727664 + 649.86427570733
Price @ 0.068 = 1251.65

 

 

Bond Price at Upper YTC of 11.79%

Compounding = annual
Par Value = 1000
Market Price = 1088
Call Price = 1100
Coupon Rate = 10%
YTC High = 11.79%
N = 8

 

Non Zero Bond Price Formula

Coupon Rate x Par Value x PVIFA(ytch%, n) + Call Price x PVIF(ytch%, n)

 

PVIFA Formula

PVIFA(ytch%, n) = [1 - v] / ytch%
v = 1 / (1 + ytch%)^n
PVIFA(ytch%, n) = [1 - { 1 / (1 + ytch%)^n }] / ytch%

 

PVIFA Calculation

v = 1 / (1+0.118)^8
v = 0.40969962986328
PVIFA(0.118, 8) = [1 - 0.40969962986328] / 0.118
PVIFA(0.118, 8) = 0.59030037013672 / 0.118
PVIFA(0.118, 8) = 5.0025455096332

 

PVIF Formula

PVIF(ytch%, n) = 1 / (1 + ytch%)^n

 

PVIF Calculation

PVIF(0.118, 8) = 1 / (1+0.118)^8
PVIF(0.118, 8) = 1 / 2.4408125541478
PVIF(0.118, 8) = 0.40969962986328

 

Non Zero Bond Price Calculation

Price = 0.1 x 1000 x 5.0025455096332 + 1100 x 0.40969962986328
Price = 500.25455096332 + 450.66959284961
Price @ 0.118 = 950.92

 

 

YTC Approximation with Linear Interpolation

ytcL = 6.79%
ytcU = 11.79%
pvL = 1251.65
pvU = 950.92

ytc = ytcL + [(ytcH-ytcL)(pvL-pvYTC)] / [pvL-pvU]
ytc = 0.068 + [(0.118-0.068)(1251.65 - 1088)] / [1251.65-950.92]
ytc = 0.068 + [(0.05)(163.65)] / [300.73]
ytc = 0.068 + 8.1825 / 300.73
ytc = 0.068 + 0.0272
ytc = 0.0952
ytc = 9.52%

This is an approximation , the use of a computer provides a precise yield to call of 9.29%. It is important to keep in mind that interpolation gives only an approximation of the exact percentage.