As a hypothetical example, if you deposit 100 yen at 2% compound interest, compounded semiannually, how much do you have after one year?
Quiz: Interest and the Time Value of Money
Last reviewed: September 24, 2026
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5 questions to check your understanding of the chapter “Interest and the Time Value of Money”. Choose an answer and select “Check answer” to see the answer and explanation. You can also open “Answer and explanation” to see the answer first.
Question 1
Answer and explanation
Answer: C
With semiannual compounding, you divide the annual rate of 2% by the two periods in a year and add 1% interest every six months. The 100 yen becomes 101 yen after six months, and after one year 1% of 101 yen is added, giving 102.01 yen. That is, 100 × 1.01 × 1.01 = 102.01.
The 102.00 yen option is the amount with annual compounding. Semiannual compounding gives 0.01 yen more, because the 1 yen of interest added after six months itself earns interest over the second half of the year. The 101.00 yen option is the amount after six months, and the 102.02 yen option is the amount after one year with continuous compounding, where the number of periods grows without limit.
Learn more: Compounding and Discounting
Question 2
As a hypothetical example, at 2%, what happens to the present value of 100 yen received in 10 years as you compound more often (annually, semiannually, monthly, continuously)?
Answer and explanation
Answer: A
The more often interest is compounded, the larger the amount that 100 yen today grows to in 10 years. Discounting reverses this calculation and divides the future amount by the same growth factor, so the discount factor and the present value become smaller. The present value of 100 yen received in 10 years is 82.03 yen with annual compounding, 81.95 yen with semiannual, 81.89 yen with monthly, and 81.87 yen with continuous compounding.
"It stays the same" is wrong because the same 2% gives different future amounts and discount factors under different compounding conventions. "It rises" confuses the larger future amount with the direction of the present value.
Learn more: Compounding and Discounting
Question 3
What does a discount factor represent?
Answer and explanation
Answer: D
A discount factor is the value today of 1 yen to be received in the future. With annual compounding, you get it by dividing 1 by "1 + the rate" once for each year, so the one-year discount factor at 2% is about 0.9804. Multiplying the amount of a future cash flow by the discount factor for the date it is received gives its present value.
"The amount that 1 yen deposited today grows to" is the calculation in the opposite direction: at 2%, 100 yen today becomes 102 yen in one year. A discount factor is not an amount of interest but a number you multiply a future amount by to find its value today. The change in present value when rates for all periods rise by 1bp (0.01%) is called BPV (basis point value), and it is used to measure interest rate risk.
Learn more: Present Value and Discount Factors
Question 4
As a hypothetical example, a three-year bond has a principal of 10,000 million yen and pays 2% interest once a year, and the discount rate is the same for all periods. By how much does the bond's present value fall when the discount rate rises from 2% to 5%?
Answer and explanation
Answer: C
The higher the discount rate, the smaller the discount factor. When the rate goes from 2% to 5%, the total present value goes from 10,000 million yen to 9,183 million yen. This is because the one-year discount factor falls from 0.9804 to 0.9524 and the three-year factor from 0.9423 to 0.8638.
Subtracting 9,183 million yen from 10,000 million yen, the present value falls by about 817 million yen. The bond's interest is fixed at 200 million yen a year, so a higher discount rate does not change the cash flows received. Each is multiplied by a smaller discount factor, so the present value falls.
The 9,183 million yen option is the present value after the rise, not the amount of the fall. The 300 million yen option multiplies the 3-point rise in the rate by the principal, and the 600 million yen option is the total of three years of interest. Neither is the change in present value.
Learn more: Present Value and Discount Factors
Question 5
As a hypothetical example, if the 1-year spot rate is 2.00% and the 2-year spot rate is 2.25% (both with annual compounding), which value is closest to the forward rate from year 1 to year 2?
Answer and explanation
Answer: B
The forward rate is the rate at which two ways of investing give the same amount after two years. One is to invest for two years at the 2-year spot rate. The other is to invest for one year at the 1-year spot rate and then reinvest at the forward rate. Investing 100 gives about 104.55 after two years in the first case and 102.00 after one year in the second. So the forward rate is 104.55 ÷ 102.00 − 1, or about 2.50%.
If is the forward rate for one year starting one year from now, you can find it from the two spot rates as follows.
Where spot rates rise with maturity, the forward rate is higher than the spot rate, so the answer is above the 2-year spot rate of 2.25%. The 2.00% option is the 1-year spot rate, and the forward rate equals it only when spot rates are the same for every maturity. The 4.25% option is the sum of the two spot rates, and adding rates does not give the rate from year 1 to year 2.
Learn more: Zero Rates and Forward Rates