Let's break this down into two parts: Yield to Maturity (YTM) and Yield to Call (YTC). Both are calculated differently because YTM assumes the bond will be held until maturity, while YTC assumes it will be called after a certain number of years.

1. Yield to Maturity (YTM)

The YTM is the internal rate of return (IRR) on the bond if held until maturity. The formula is complex, but we can approximate it using trial and error or a financial calculator. Here's the step-by-step calculation:

• Face Value (FV): $1,000
• Coupon Rate: 8.5% (paid semiannually, so each payment is $\frac{8.5\%}{2} = 4.25\%$ of face value, or $42.50 every six months)
• Price of Bond (PV): $950
• Number of periods (N): Since the bond matures in 18 years and pays semiannually, there are $18 \times 2 = 36$ periods.
• Coupon Payment: $42.50 (paid semiannually)
• Current Price: $950

The bond's price can be expressed as the present value of its future cash flows, which consist of the semiannual coupon payments and the face value repaid at maturity. The YTM is the interest rate that equates the bond's price to the present value of these cash flows.

Using the bond pricing formula:

$P = \sum_{t=1}^{N} \frac{C}{(1 + r/2)^t} + \frac{FV}{(1 + r/2)^N}$

Where:

• $P$ = Bond price ($950)
• $C$ = Semiannual coupon payment ($42.50)
• $FV$ = Face value ($1,000)
• $N$ = Number of periods (36)
• $r$ = YTM (annualized)

2. Yield to Call (YTC)

The Yield to Call assumes that the bond will be called at the first possible call date (5 years from now). The call price is $1,050.

• Call Price (CP): $1,050
• Years to Call: 5 years, with semiannual periods, so $5 \times 2 = 10$ periods.
• Coupon Payment: $42.50 (semiannual)
• Price of Bond (PV): $950
• Number of periods (N): 10

The formula to calculate YTC is similar to YTM but with the call price instead of face value and the number of periods until the call date instead of the full maturity period.

Let's now calculate both YTM and YTC using numerical methods.It seems that the internal rate of return (IRR) function from NumPy has been deprecated. I'll switch to a different approach using the bond price formula and a solver to calculate both the yield to maturity (YTM) and the yield to call (YTC). Let's proceed with that.The calculations yield the following results:

• Yield to Maturity (YTM): 17.93%
• Yield to Call (YTC): 19.61%

These are the annualized yields for both holding the bond until maturity and if it is called in 5 years.

Do you want more details on any of these calculations? Here are five related questions:

1. What factors affect the yield to maturity of a bond?
2. How does the price of a bond relate to its yield?
3. What is the significance of the coupon rate in bond pricing?
4. How do callable bonds impact an investor’s risk and return?
5. What is the difference between nominal yield and yield to maturity?

Tip: Callable bonds usually offer a higher coupon rate to compensate for the call risk faced by investors.