Final answer:
Dan's effective annual yield rate is calculated by determining the future value of reinvested coupons at a nominal rate of 5% convertible monthly, adding the $2000 par value received at maturity, and comparing the total amount received to the initial investment of $1825 over a ten-year period.
Step-by-step explanation:
Calculating the Effective Annual Yield Rate
To calculate Dan's effective annual yield rate over the ten-year period for a $2000 par value bond with 8% annual coupons purchased for $1825, we need to account for the interest from coupon payments and the capital gains from the bond. Dan will reinvest his coupon payments at a nominal rate of 5% convertible monthly.
First, we calculate the annual coupon payments, which are 8% of the $2000 par value, resulting in $160 per year. Then, to find the reinvestment rate per month, we divide the nominal rate of 5% by 12, giving us approximately 0.4167% per month.
To determine the future value of reinvested coupons, we assume that each coupon payment is made at the end of each year and reinvested monthly until the end of the tenth year. We use a financial calculator (BA-II) for this step:
- N = 120 (total reinvestment period in months for the first coupon)
- I/Y = 0.4167 (monthly reinvestment rate)
- PMT = -160 (negative because it's an outflow for reinvestment)
- FV = 0 (initially, since we are calculating the future value)
- Calculate Future Value (FV) for the first coupon.
Repeat this step for each subsequent coupon, decreasing the total reinvestment period (N) by 12 months for each year closer to maturity.
Add up all the future values of the coupon payments and add the $2000 par value received at maturity to get the total amount received.
The effective annual yield is then:
Effective annual yield = [(Total amount received - Initial investment)/Initial investment]¹/¹⁰ ⁻ ¹
This yield represents total returns, which include interest payments and any capital gains or losses.