Introduction to Quant Finance · Lesson 15
Time value, interest rates and bonds
Why €100 next year is worth less than €100 today, how that sets a bond's price, and why bond prices fall when interest rates rise.
Know the basics? Read the lesson notes€100 today, or €100 in a year?
Someone offers you a choice: €100 today, or €100 exactly one year from now. One word first. Interest is what a borrower pays for the use of someone else's money. The interest rate is that payment for one year, as a percent of the amount.
Picture a made-up bank that pays 5% a year on savings: leave €100 with it for a year and it pays you €5 on top. Which choice is worth more to you today?
Interest on interest
Today's €100 wins because it can earn interest while you wait. A euro now is worth more than a euro later: this is the time value of money.
Now leave the €100 in the made-up bank for ten years at 5%. Each year's interest is 5% of what the account holds at the start of that year, so last year's interest earns interest too. Press Next year, then play the rest.
€100 at a made-up 5% a year
The account at the end of each year. Grey: the €100 you put in. Colour: the interest added so far.
Year one adds €5 (to €105.00), year two €5.25 (to €110.25), year three €5.51 (to €115.76). Each year multiplies the account by 1.05, so ten years multiply it by 1.05 ten times over, which is 1.6289: the €100 ends at €162.89.
This is compounding: each year's return multiplies the account, so returns multiply rather than add up. What €100 grows to is its future value.
Run it backwards
Now turn the question round. How much would you need in the bank today to have exactly €100 in three years, at 5%? Drag the amount until it grows to €100.
Today's amount, grown for three years at 5% (made up)
Dashed: the €100 you want in three years.
€86.50 grows to €100.13, close. The exact amount comes from undoing the growth. Three years multiply by 1.05 × 1.05 × 1.05 = 1.157625, so divide instead: €100 ÷ 1.157625 = €86.38.
€86.38 is the present value of €100 paid in three years at 5%: what that later €100 is worth today. Working back like this is called discounting.
Further away, worth less
The curve shows what €100 is worth today if it arrives after 1 year, 2 years, and so on up to 30, at the rate you set. It starts at 5%. Slide the rate up to 10%.
Today's value of €100 paid later (made-up rates)
Across: years until the €100 arrives. Up: what it is worth today. Dots: 10 and 30 years.
At 5%, €100 paid in 10 years is worth €61.39 today, and €100 in 30 years only €23.14. At 10%, the 30-year €100 is worth €5.73.
Two things shrink a payment's present value: a longer wait, because it is divided by (1 + rate) once for every year; and a higher rate, because each of those divisions is bigger.
Present value: bring every amount to today
Amounts paid on different dates can only be compared once they are moved to the same date. Growing an amount forward at the interest rate gives its future value; dividing a later amount back gives its present value. Both are compounding, one year at a time.
- PV
- the present value: the amount today
- FV
- the future value: the amount at the later date
- r
- the interest rate per year, as a decimal: 5% is 0.05
- t
- the number of years between today and the later date
- (1 + r)t
- (1 + r) multiplied by itself t times: 1.053 = 1.05 × 1.05 × 1.05 = 1.157625 (on a calculator, the x^y key)
Growing: €100 × 1.0510 = €100 × 1.6289 = €162.89.
Discounting: €100 ÷ 1.053 = €100 ÷ 1.157625 = €86.38. Check: €86.38 × 1.157625 = €100.00. The 30-year payment: €100 ÷ 1.0530 = €100 ÷ 4.3219 = €23.14.
- the rate stays the same in every year;
- interest is added once a year and left in to earn interest;
- the later amount is certain to be paid, in full.
The payment might not come, or rates differ by how long you lend for. The end of this lesson shows the second; the first is left for a later course.
Try it on new numbers. You are promised €500 in 4 years. At a made-up 6% a year, what is that worth today? Type euros.
A bond is a list of payments
A bond is a loan you can buy and sell. The borrower, a government or a company, promises fixed payments: a yearly interest payment called the coupon, and at the end the amount borrowed, called the face value.
Take a made-up 3-year bond with a face value of €100 and a €5 coupon. It pays €5 after one year, €5 after two, and €5 + €100 = €105 after three. Say loans like this one pay a made-up 6% a year today. We call this the going rate. Bring each payment to today: press each year.
One made-up bond, discounted at 6%
Outlines: what each year pays. Colour: what it is worth today. The bar on the left adds up what you have brought to today.
€5 ÷ 1.06 = €4.72, €5 ÷ 1.062 = €4.45 and €105 ÷ 1.063 = €88.16, which add up to €97.33. That is the bond's price at 6%: what its payments are worth today.
It pays €115 in all, but most of that comes late and is discounted. The price is below €100 because the coupon, 5% of the face value, is less than the 6% a lender could get on loans like it.
Find the yield
In a market it usually goes the other way round: you see the price and want the rate. The same bond now sells for €94.75. Which single rate makes its three present values add up to exactly €94.75? There is no simple formula for the rate, so try rates: drag until the sum matches the price.
The made-up bond at the rate you try
Colour: each payment's present value, and their sum on the left. Dashed: the €94.75 price.
At 7.0% the present values are €4.67, €4.37 and €85.71, which add up to €94.75. That rate is the bond's yield (in full, its yield to maturity): the one rate at which its payments are worth exactly its price. Roughly, it is the yearly return from buying at €94.75 and holding to the end, if every payment arrives.
As you dragged, a higher rate gave a lower sum, because the payments stay fixed while each division grows. So a bond's price and its yield move in opposite directions.
A bond's price is the sum of its present values
Each payment is discounted for its own number of years, and the price is the sum. Written for any bond that pays once a year, this is the bond pricing formula. Given the going rate, it gives the price. Given the price, the rate that makes it true is the yield, found by trying rates as you just did.
- P
- the bond's price today
- C
- the coupon, paid at the end of each year
- N
- the face value: the amount repaid at the end, with the last coupon
- T
- the number of years left, so the number of payments
- y
- the yearly rate: the going rate when you work out a price; the yield when the price is given
At 6%: 5 ÷ 1.06 + 5 ÷ 1.1236 + 105 ÷ 1.1910 = 4.72 + 4.45 + 88.16 = €97.33.
At 7%: 5 ÷ 1.07 + 5 ÷ 1.1449 + 105 ÷ 1.2250 = 4.67 + 4.37 + 85.71 = €94.75, so a price of €94.75 means a yield of 7.0%.
At 5%: 4.76 + 4.54 + 90.70 = €100.00. When the rate equals the coupon as a percent of the face value (the coupon rate), the price equals the face value.
- the payments are fixed in advance and paid in full and on time;
- one rate fits every year;
- the next payment is exactly one year away (a bond bought between payment dates needs an extra step, left out here).
The yield is your yearly return only if you hold to the end and each coupon, when it arrives, earns that same yield until the end.
Breaks whenThe borrower may not pay: then the yield overstates what you get (a later course). Or rates differ by how far away a payment is: the yield curve, at the end of this lesson.
New numbers. A made-up 2-year bond has a €4 coupon and a €100 face value. Loans like it pay 5% a year. What is its price? Type euros.
Guess first: rates rise
The time left until a bond's last payment is its maturity. Take two made-up bonds, each with a €5 coupon and a €100 face value: one has a maturity of 1 year, the other 10 years. Loans like them pay 5%, so both are priced at €100. Then the going rate rises to 7%. Their payments don't change. Which price falls more?
Move the rate
Three made-up bonds, each with a €5 coupon and a €100 face value, with maturities of 1, 3 and 10 years. The 3-year one is the bond from before. Each curve is one bond's price at every rate from 1% to 10%. Slide the rate from 5% down to 3%, then up to 7%.
Price against the going rate, three made-up bonds
Across: the rate. Up: the price. Dashed line: the 1-year bond. Dashed grid line: €100, where all three sit at 5%.
Each payment is divided by 1.07 instead of 1.05 once for every year it is away. A payment 1 year away keeps 1.05 ÷ 1.07 = 0.981 of its present value, so it loses 1.9%; one 10 years away keeps (1.05 ÷ 1.07)10 = 0.828 of it and loses 17.2%. The 10-year bond falls less, 14.0%, because its €5 coupons come sooner and lose less. So a bond whose money comes later moves more. At 3% it works the other way: all three rise, the 10-year one most, to €117.06.
This sensitivity to rates is measured by a bond's duration, which is longer when its payments come later.
For the precise
Durations at 5% (Frederick Macaulay's definition, 1938): 1-year bond 1.00 year, 3-year bond 2.86 years, 10-year bond 8.11 years. The rise to 3% is a little larger than the fall to 7% for every bond (the 10-year: +17.1% against −14.0%), because the curves bend; that bend, called convexity, is not covered here.Rates by maturity: the yield curve
So far one rate fitted every year. In a market, the yield on a 1-year loan and on a 30-year loan from the same borrower need not be the same. Plot the yields of one borrower's bonds against their maturities and you get its yield curve. Press each made-up shape.
Three made-up yield curves
Across: maturity in years. Up: the yield. Colour: the shape you pressed. Invented numbers; no market, no date.
A curve that rises with maturity is called normal, one that stays level is flat, and one that falls with maturity is inverted. All three here are invented: this lesson says nothing about where rates are at any date.
Only a flat curve matches the one-rate assumption. With any other shape, a careful price uses a rate that fits how long each payment takes to arrive, not one rate for all.
Now or later?
What is the €1,100 worth today? Type euros.
Hint: divide by 1.04 once for each of the two years: €1,100 ÷ 1.042.
Two bonds, one rise
What happens to their prices?
Coupon or yield?
What is its yield?
Lesson done
A euro today is worth more than a euro later, because it can earn interest meanwhile: dividing a later amount by (1 + rate) once for every year gives its present value. A bond is a list of fixed payments, and its price is the sum of their present values; its yield is the one rate that makes that sum equal the price. Because the payments are fixed, a bond's price falls when rates rise, and falls more when its payments come later. All of this assumes payments made in full and on time and one rate for every year; default risk (a borrower who doesn't pay), inflation (prices rising, so a later euro buys less) and taxes are left for a later course.
Key ideas
- Interest rate
- what a borrower pays for one year's use of money, as a percent of the amount
- Time value of money
- a euro now is worth more than a euro later, because it can earn interest meanwhile (with a positive rate)
- Compounding
- each year's interest is earned on what the account holds, earlier interest included, so the growth multiplies
- Future value
- what an amount today grows to by a later date at a given rate
- Present value
- what an amount paid later is worth today at a given rate
- Discounting
- working from a later amount back to its present value, by dividing by (1 + rate) once for every year
- Bond
- a loan that can be bought and sold, with fixed payments promised by the borrower
- Coupon
- a bond's regular interest payment
- Face value
- the amount borrowed, paid back with a bond's last coupon
- Maturity
- the time left until a bond's last payment
- Yield
- the one rate at which a bond's payments, discounted, add up to its price (in full: yield to maturity)
- Duration
- a measure of how strongly a bond's price moves when rates change; longer when its payments come later
- Yield curve
- the yields of one borrower's bonds plotted against their maturities: normal (rising), flat or inverted (falling)
Formulas
FV = PV × (1 + r)t · PV = FV ÷ (1 + r)t
P = C ÷ (1 + y) + C ÷ (1 + y)2 + … + (C + N) ÷ (1 + y)T
At a made-up 5% a year, €100 paid in 3 years is worth €86.38 today, and in 30 years €23.14.