You have been asked to estimate the value of a 10-year bond with a coupon that will be low initially but it is expected to grow later in the bond’s life. The coupon is expected to be 5% of the face value of the bond (which is $ 1000) for the first 5 years, and will increase by 1% every year for the next 5 years; the coupon rate will be 6% in year 6, 7% in year 7, 8% in year 8, 9% in year 9 and 10% in year 10. Estimate the value of this bond.

To estimate the value of this bond, we need to calculate the present value of each cash flow (coupon and face value) and then sum them up.

Step 1: Calculate the present value of each coupon payment.
- For the first five years, the coupon rate is expected to be 5% of the face value ($1000). Since this coupon is paid annually, we can calculate its present value using the formula for the present value of an ordinary annuity:

PV = C x (1 - (1 + r)^(-n)) / r,

where PV is the present value, C is the coupon payment, r is the discount rate, and n is the number of periods.

The coupon payment for each year will be:
Year 1: $1000 x 5% = $50
Year 2: $1000 x 6% = $60
Year 3: $1000 x 7% = $70
Year 4: $1000 x 8% = $80
Year 5: $1000 x 9% = $90

Using a discount rate of your choice (let's assume 4% for this example), we can calculate the present value of each coupon payment:

PV1 = $50 / (1 + 4%)^1
PV2 = $60 / (1 + 4%)^2
PV3 = $70 / (1 + 4%)^3
PV4 = $80 / (1 + 4%)^4
PV5 = $90 / (1 + 4%)^5

Step 2: Calculate the present value of the face value payment.
Since the face value of the bond is $1000 and it will be paid at the end of the 10th year, we can calculate its present value using the formula for a single future cash flow:

PV = FV / (1 + r)^n,

where PV is the present value, FV is the future value (the face value of the bond), r is the discount rate, and n is the number of periods.

Using a discount rate of 4% again, we can calculate the present value of the face value payment:

PV6 = $1000 / (1 + 4%)^6

Step 3: Sum up the present values of the coupon payments and face value payment.

Bond Value = PV1 + PV2 + PV3 + PV4 + PV5 + PV6

Please note that the discount rate used in this estimation is just an assumption. In practice, you would typically use the prevailing market interest rate to calculate the present value of a bond.