You will be able to show with a worked example why a bond's price falls when market yields rise.
Pick a bond you own or are watching and look it up on your brokerage app or a bond price list. Note its coupon and its quoted price per 100 of face value. If the price is above or below 100, market interest rates have changed since the bond was first issued. Present value arithmetic lets you calculate how big that difference should be.
A bond's price can drop even when the issuer is in good shape and has never missed a coupon. Hui Min (an invented investor) paid 100 for a bond with a 3% coupon and six years until maturity (invented figures). Twelve months on, with five years remaining, her app quotes it at 95.55. Her coupons have all arrived on schedule and the issuer is sound. Even so, the screen shows almost S$45 less than she paid on every S$1,000 she holds. She assumes there has been an error somewhere, but there hasn't. Interest rates went up, and the higher rates lowered the price.
A bond's coupon is the yearly payment fixed in its terms. For Hui Min that is 3 per 100, which is S$30 each year on S$1,000 of face value. Face value is what the issuer pays back when the bond matures, and on that date it is also exactly what the bond is worth. Lesson 1.1, "A bond is a loan with a fixed timetable", described the fixed coupons and the repayment of face value together as the bond's "promise". A change in market rates leaves every part of that promise as it was.
Suppose similar borrowers are now issuing bonds at 4%, paying S$40 a year on S$1,000 (an example figure). Someone with S$1,000 to invest would get S$10 a year more from a new bond than from Hui Min's, so nobody will pay her face value. Her coupon is fixed, so the price is the only thing that can change. It has to fall below face value until a buyer would earn the same yield as the new 4% bonds. That yield counts both the coupons and the climb back to face value at maturity.
The same logic runs the other way when rates fall. If new bonds pay just 2% (an example figure), her 3% coupon is better than what's on offer, and buyers will pay more than face value to get it. Because the cash flows are fixed, price and yield always move in opposite directions. This is the price-yield relationship. When the market yield and the coupon are the same, the bond is priced at exactly face value.
To price a bond at a given market yield, find what each of its payments is worth today, then add them up. The method is discounting, which How money works, lesson 6.3, "Present value: what a future promise is worth today", introduced as present value. The total is the bond's price per 100 of face value.
Start by listing the cash flows: one coupon for each year, with the face value added to the last one. Next, divide each payment by (1 + market yield) raised to the number of years until it arrives. Finally, add up the discounted amounts to get the price.
Hui Min's bond now has five years to run, a face value of 100 and a 3% coupon paid once a year. It pays 3 in each of the five years, and 100 comes back at the end of year five, so the last payment is 103. At a 4% market yield, divide each payment by 1.04 raised to its year:
Year 1: 3 / 1.04 = about 2.885 Year 2: 3 / 1.04^2 = about 2.774 Year 3: 3 / 1.04^3 = about 2.667 Year 4: 3 / 1.04^4 = about 2.564 Year 5: 103 / 1.04^5 = about 84.658
The five values add up as 2.885 + 2.774 + 2.667 + 2.564 + 84.658 = about 95.55, the same figure Hui Min sees on her app.
Repeat the calculation at 2%, dividing each payment by 1.02 raised to its year. The discounted payments add up to about 104.71. If yields had fallen to 2% instead of rising, she could have sold at a gain of almost S$47 on every S$1,000 of face value.
You can also check a price from the buyer's side. First, express the coupon as a percentage of the purchase price. Then find the gain at maturity if the price is under 100, or the loss if it is over 100. Spread across the years left, the coupon income plus the gain, or minus the loss, should come to the market yield per year.
At 95.55, a buyer's coupon of 3 is about 3.14% of what they paid. They also gain about 4.45 when 100 is repaid at maturity. Across the five remaining years, the coupon income and that gain together come to 4% a year, which is the market yield.
At 104.71, the buyer still receives 3 each year but loses about 4.71 at maturity, because only face value is repaid. With that loss taken off the coupon income, the return is 2% a year, again the market yield.
Side by side, the prices are about 104.71 at a 2% yield, exactly 100 at 3% (where the yield equals the coupon) and about 95.55 at 4%. The two moves are not the same size. A one-point drop in yield added about 4.71 to the price, while a one-point rise took off about 4.45. Module 2 comes back to this uneven shape when it covers duration.
If Hui Min keeps the bond, she receives S$30 every year and S$1,000 at maturity, as long as the issuer can still pay. The market price is what she would get if she sold today. While she holds the bond, 95.55 is only a figure on her screen. The price also moves back towards 100 as maturity approaches, because a bond is worth exactly its face value on the day it matures.
Price swings matter most to people who may have to sell before the bond matures. If the money is for a goal with a fixed date, the risk is that rates rise shortly before you have to sell. Later lessons cover two ways to manage this risk. One is to match the bond's maturity to the date you'll need the money, which is how the ladder in module 8 works. The other is to check how sensitive a bond is to rate changes, which lesson 2.3 covers.
Why rates go up and down is taught in a separate course, "How the economy hits your wallet: rates, inflation and cycles". To price a bond, all you need is the direction of the change and the arithmetic.
Use the same five-year, 3% bond and work out its price at a few more yields. Plot each price against its yield, along with your 2%, 3% and 4% results, and look at the shape the prices make.
Use the worked method to reprice the course's made-up 5-year, 3% bond at yields of 2%, 3%, 4% and 5%. Then plot price against yield.
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