Question

If you have a four year bond with a coupon of 7% and a YTM of 5% verify that investing in this bond for the period of the bond’s Macauly duration results in no change to the overall future value of your cash flows if interest rates fall by 1%. Show all calculations.

Answer 1

Based on 7% coupon rate and 5% YTM we will calculate the Present Bond value based on 5% YTM and 100 par value

We have below cash flows:-

Years	0	1	2	3	4
Cash Flow		7	7	7	107

Yet we do not know cash flow at 0 year.

We can calculate Present value by below formula:-

i.e. present value of bond is 107.09. For above calculation of PV i strongly recommed to use excel or Financial Calculator.

Now we will calculate Macauly Duration by using weighed average value of all the cash flows

Years	0	1	2	3	4
Cash Flow	-107.09	7	7	7	107
cash flow / (1+0.05)^n		6.666667	6.349206	6.046863	88.02916
above row * year		6.666667	12.69841	18.14059	352.1167

In above table, first column reperesents the formula used for calculation of another row.

i.e. First two rows we know from our cash flow.

Third row is that each cash flow is calculated at present value i.e.

for year 3 - 7 / (1+0.05)^3 = 6.05

For year 4 - 107 / (1+0.05)^4 = 88.03

Fourth row has been caculated by multiplying each flow flows from row three with respective year

i.e. for year 3 - 6.05*3 = 18.14

for year 4 - 88.03 * 4 = 352.12

etc.

So now we will calculate the sum of all fourth row elements and divide that value with the initial cash outflow i.e. bond's value for calculating Macauly Duration =

= (6.67+12.7+18.14+352.12) / 107.09

= 3.64 years

Now if we invest the amount upto Macauly duration than our cash flows will be as follows:-

Years	0	1	2	3	3.64
Cash Flow		7	7	7	102.56

Here coupon payment will be the same 7%

the last column has been calculated as below:-

At 3.64 year we will get par value which is 100 + the interest payment for 0.64 years with 4% interest rate

so

= 100 + 0.64 * 0.04 * 100

= 100 + 2.56

= 102.56

Now knowing these we can calculate our present value for these cash flows with 4% interest rate.

Same present value as we have calculated with 5% YTM.

This can be done in another way as below:-

When investment horizon is equal to Macauly duration, we will have approximate same YTM as the bon's YTM

As bond was for 4 years and duration was 3.64 so bond was taken (4-3.64) = 0.36 years before.

so the bond value was

However i will be calculating Present value in Excel using below formula in excel=

= PV (0.04,0.36,7,100,0)

From above we can calculate bond value at 3.64 year

= 101.05

and we are getting coupon payment of 7 which is reinvested at 4% and gives the value after 3.64 years as below:-

So the total value after 3.64 years is:-

= 26.85 + 101.05

= 127.91

and total invested amount was = 107.09

So return is given by

= approximate 5%

which is equal to 5% YTM even though interest rate dropped by 1%.

Thank You!!