Bond pricing and yields | bond pricing actuarial | Lemma

Lemma/CM1 Foundation/Equation of Value and Applications

CM1 Foundation · Equation of Value and Applications

Bond pricing and yields

Price a fixed-interest bond as the present value of coupons plus redemption, and interpret running and redemption yields.

Skills in this lesson

1. Pricing a fixed-interest bond0/2 verified
2. Running and redemption yieldsLocked

Prerequisites

Recommended before this lesson — you can still read and practice below:

Net present value and internal rate of return (locked)

Teach

A fixed-interest bond pays a regular coupon and returns its redemption value at maturity. Its fair price is the present value of those cashflows at the investor's required yield $i$ .

Pricing equation

For a bond of nominal $N$ , coupon rate $D$ per annum (paid annually for simplicity), redemption $R$ at time $n$ :

n

P = D\,N\, a_{\overline{n}|} + R\, v^n,

where $a_{\overline{n}|} = (1 - v^n)/i$ and $v = (1+i)^{-1}$ . The coupons form a level annuity; the redemption is a single discounted payment.

Worked example: Price of a 3-year bond

Problem. A bond of nominal Rs. 100 pays 6% annual coupons and is redeemed at par after 3 years. Price it to yield 8%.

Solution. Coupon $= 6$ , $R = 100$ , $i = 8\%$ .

a_{\overline{3}|} = \frac{1 - 1.08^{-3}}{0.08} \approx 2.5771,\qquad v^3 \approx 0.7938.

P = 6(2.5771) + 100(0.7938) \approx 15.46 + 79.38 = \text{Rs. } 94.85.

It trades below par because the yield exceeds the coupon rate.

Running and redemption yields

The running (income) yield ignores capital gain:

\text{running yield} = \frac{\text{annual coupon}}{P} = \frac{DN}{P}.

The redemption (gross) yield is the IRR that equates price and cashflows — the $i$ solving the pricing equation above. It captures both income and the pull-to-redemption capital gain or loss.

Practice

Practice 1

$DN$

$DN\, a_{\overline{n}|} + R\, v^n$$DN\, s_{\overline{n}|} + R$$R\, v^n$ only$DN \cdot n + R$

Hint

Coupons form a level annuity; redemption is a single payment.

Practice 2

$A Rs. 100 nominal bond pays 6% annual coupons, redeemed at par after 3 years, priced to yield 8%. Find the price (nearest 0.01).$

Hint

$6\,a_{\overline{3}|} + 100v^3$ with $a_{\overline{3}|}\approx 2.5771$.

Practice 3

$A 5% annual coupon bond redeemed at par is priced to yield exactly 5%. Give the price per Rs. 100 nominal.$

Hint

When the yield equals the coupon rate the bond is at par.

Practice 4

$The running (income) yield on a bond is:$

annual coupon ÷ nominalannual coupon ÷ priceredemption ÷ pricethe IRR of all cashflows

Practice 5

$A bond pays a Rs. 6 annual coupon and trades at Rs. 120. Give the running yield as a percentage.$

Hint

$6/120$.

Practice 6

$If the required yield exceeds the coupon rate, the bond trades:$

above parat parbelow parat the redemption value

Mastery checks

Pass each skill by answering correctly on 2 of 3 checks (fewer if fewer are available). Practice questions above are rehearsal — only mastery checks count toward progress. Retry wrong answers until you succeed.

Pricing a fixed-interest bond

0/2 verified

Check 1

$A 2-year bond with 10% annual coupons, redeemed at par Rs. 100, priced to yield 10%. Give the price.$

Hint

Yield equals coupon rate, so price = par.

Check 2

$A Rs. 100 nominal bond pays 5% annual coupons, redeemed at par after 2 years, priced to yield 6%. Find the price (nearest 0.01).$

Hint

$5\,a_{\overline{2}|} + 100v^2$ with $a_{\overline{2}|}\approx 1.8334$.

Check 3

$If the required yield is below the coupon rate, the price is:$

below parabove parexactly parzero

Running and redemption yields

Locked — complete prior skills first

Check 1(Complete prior skills first)

$A bond pays a Rs. 8 annual coupon and trades at Rs. 80. Give the running yield as a percentage.$

Hint

$8/80$.

Check 2(Complete prior skills first)

$The redemption (gross) yield differs from the running yield because it:$

ignores the couponscaptures both income and the capital gain or loss to redemptionuses the nominal not the priceis always lower

Check 3(Complete prior skills first)

$Which yield measure ignores any capital gain or loss at redemption?$

redemption yieldinternal rate of returnrunning (income) yieldgross yield

≈

0.7938.