2. You have just won $10 million in the state lottery which promises to pay you $1 million (tax free) every year for the next ten years. Have you really won $10 million?
No, because the present discounted value of these payments is necessarily less than $10 million as long as the interest rate is greater than zero.
3. If the interest rate is 10%, what is the present value of a security that pays you $1,100 next year, $1,210 the year after, and $1,331 the year after that?
PV = FV1/(1+i) + FV2/(1+i)2 + FV3/(1+i)3
PV = $1,100/1.1 + $1,210/1.12 + $1,331/1.13 = $3,000
5. Write down the formula that is used to calculate the yield to maturity on a twenty-year 10% coupon bond with $1,000 face value that sells for $2,000.
The present value is the purchase price of $2,000. The future value is the face value of $1,000. The annual coupon payment is 10% of the face-value, or $100. The bond term is 20 years. The present value formula for a coupon bond is:
PV = C/(1+i) + C/(1+i)2 + … + C/(1+i)n + F/(1+i)n
Plugging in the above information gives:
$2,000 = $100/(1+i) + $100/(1+i)2 + … + $100/(1+i)20 + $1000/(1+i)20
Using a financial calculator, you could find the yield to maturity as i = 3%. Please note that on an exam or quiz, I will only ask you for the formula, not the solution.
6. What it the yield to maturity on a $1,000 face-value discount bond maturing in one year that sells for $800?
PV = FV/(1+i) ( PV = $800 (current price), FV = $1,000 (future payment)
800 = 1000/(1+i) ( 1 + i = 1000/800 = 1.25 ( i = 25%
8. To pay for college, you have just taken out a $1,000 government loan that makes you pay $126 per year for 25 years. However, you don’t have to start making these payments until you graduate from college two years from now. Why is the yield to maturity necessarily less than 12%, the yield to maturity on a normal $1,000 fixed payment loan in which you pay $126 per year for 25 years?