To solve this, we need to determine how much Betty and Bob should invest in the zero-coupon bond (ZCB) to guarantee at least $250,000 at the end of 8 years, even if their stock market investment turns out to be worthless.

Step-by-Step Solution:

1. Zero-Coupon Bond Calculation: The zero-coupon bond (ZCB) grows at an annual yield of 12%, compounded semiannually. The future value $F$ of a ZCB is given by the formula: $F = P \left( 1 + \frac{r}{n} \right)^{nt}$ Where:

   • $F =$ Future value (desired amount at maturity)
   • $P =$ Present value (amount to invest in the bond)
   • $r =$ Annual interest rate (12% = 0.12)
   • $n =$ Number of compounding periods per year (semiannually, so $n = 2$)
   • $t =$ Number of years (8 years)

   We know $F = 250,000$, $r = 0.12$, $n = 2$, and $t = 8$. We need to solve for $P$, the amount they should invest in the ZCB.

   $250,000 = P \left( 1 + \frac{0.12}{2} \right)^{2 \times 8}$ $250,000 = P \left( 1 + 0.06 \right)^{16}$ $250,000 = P (1.06)^{16}$ $250,000 = P (2.518170)$ $P = \frac{250,000}{2.518170}$ $P \approx 99,283.06$

   So, Betty and Bob should invest approximately $99,283.06 in the zero-coupon bond.

2. Remaining Amount for Stock Market Investment: The total amount Betty and Bob have to invest is $250,000. Therefore, the remaining amount to be invested in the stock market is: $250,000 - 99,283.06 = 150,716.94$

   So, they should invest approximately $150,716.94 in the stock market.

Conclusion:

• Invest $99,283.06 in the 8-year ZCB yielding 12% compounded semiannually.
• Invest the remaining $150,716.94 in the stock market.

This guarantees that, even if the stock market investment becomes worthless, the zero-coupon bond will still grow to $250,000 at the end of 8 years.

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Would you like any further details or clarifications? Here are some related questions for deeper exploration:

1. How would the investment strategy change if the ZCB had annual compounding instead?
2. What would happen if the stock market investment yielded positive returns instead of being worthless?
3. How much money would Betty and Bob have if they invested all their funds in the ZCB?
4. What if the interest rate on the ZCB was different, say 10% or 15%?
5. How would inflation affect the $250,000 goal in real terms over 8 years?

Tip: When working with compounded interest, always carefully check the frequency of compounding periods, as this can significantly affect the calculations.