Tutorials › VEE Accounting and Finance › Face-Value Scaling in Coupon-Bond Prices

Home  /  Tutorials  /  VEE Accounting and Finance / Bond Cash Flows and Pricing

VEE Accounting and Finance · Bond Cash Flows and Pricing · Tutorial 820 of 996

Face-Value Scaling in Coupon-Bond Prices

Use the linearity of present value to scale coupon-bond prices reliably when a bond’s face amount changes but its terms and required yield do not.

Beginner 10 min read

What You'll Learn

  • State the conditions under which a coupon-bond price scales in direct proportion to face value.
  • Derive the scaling rule from the present-value formula for a bond’s cash flows.
  • Scale face value, coupon payments, redemption payment, and price by a common factor.
  • Apply the rule to annual and quarterly coupon schedules without changing the yield convention.
  • Recognize cases where a changed yield, payment schedule, or fixed fee prevents direct price scaling.

Why Face-Value Scaling Matters

Coupon bonds are often quoted or analyzed at one face amount even when an investor is considering a different holding size. The key question is whether the bond’s price can be adjusted proportionally without repricing every cash flow. Under specific conditions, it can: if the bond’s face value and all its contractual payments are multiplied by the same positive factor, while the required yield and payment timing remain unchanged, its value is multiplied by that factor too.

This is a property of present value, not a special feature of a particular bond denomination. It is useful when moving between a price per bond, a price per larger face amount, and a portfolio position. The conditions matter: scaling face value alone while leaving coupon dollars unchanged would change the bond’s terms and would not preserve the same proportional relationship.

Definition: Face-value scaling means multiplying a bond’s face value and each contractual coupon and redemption payment by the same positive factor \(k\), with \(k>0\). If the payment dates and required yield per payment period are also unchanged, the bond’s price is multiplied by \(k\).

The Scaling Rule

Consider a coupon bond with face value \(F\), coupon payment \(C\) each period, \(n\) remaining periods, and required yield \(j\) per period. Assume the coupons are paid at the end of each period and the face value is repaid with the final coupon. The price, excluding any separate transaction charge, is the present value of those promised cash flows:

$$ P=\sum_{t=1}^{n-1}\frac{C}{(1+j)^t}+\frac{C+F}{(1+j)^n} $$

If \(F\) and every coupon are multiplied by \(k\), the scaled coupon is \(kC\), the scaled face value is \(kF\), and the final payment is \(kC+kF=k(C+F)\). The required yield \(j\) and the discounting periods have not changed. Therefore:

$$ \begin{aligned} P_{\text{scaled}} &=\sum_{t=1}^{n-1}\frac{kC}{(1+j)^t} +\frac{kC+kF}{(1+j)^n}\\ &=k\left(\sum_{t=1}^{n-1}\frac{C}{(1+j)^t} +\frac{C+F}{(1+j)^n}\right)\\ &=kP. \end{aligned} $$

The justification is the linearity of addition and multiplication by a constant: each present-value term is multiplied by \(k\), so their sum is also multiplied by \(k\). Equivalently, the price per dollar of face value is unchanged. This conclusion does not require the bond to sell at par; it applies whether its price is above, below, or equal to face value.

Theorem: If every contractual cash flow of a bond is multiplied by \(k>0\), while payment dates and the required yield per period remain fixed, then its present value is multiplied by \(k\). In coupon-bond terms, scaling face value and the coupon rate-preserving coupon dollars together gives \(P_{\text{scaled}}=kP\). The rule does not apply as stated if only some cash flows change, the required yield changes, or payment timing changes.

Worked Applications

Worked Example: Scaling an Annual-Coupon Bond

A hypothetical bond has face value $1,000, pays a 6% annual coupon, and has three years remaining. Its required annual yield is 5%, with annual payments. Find its price, then find the price of a bond with identical terms but face value $4,000.

Method. First value the original cash flows at the annual yield. Then apply the scaling rule with \(k=4\). The coupon is \(0.06(1{,}000)=\$60\) each year, and the final payment is \(\$60+\$1{,}000=\$1{,}060\).

Discounting each payment at 5%:

$$ \begin{aligned} P &=\frac{60}{1.05}+\frac{60}{1.05^2}+\frac{1{,}060}{1.05^3}\\ &=57.142857143+54.421768707+915.667854443\\ &=1{,}027.232480293\ldots \end{aligned} $$

The price of the original bond is \(\$1{,}027.23\) to the nearest cent. The face value rises from $1,000 to $4,000, so the factor is \(k=4\). Each coupon becomes \(\$240\), and the final payment becomes \(\$4{,}240\). The scaled price is:

$$ P_{\text{scaled}}=4(1{,}027.232480293\ldots) =4{,}108.929921172\ldots $$

Thus the scaled bond costs \(\$4{,}108.93\) to the nearest cent. As a check, discounting its payments directly gives \(240/1.05+240/1.05^2+4{,}240/1.05^3\), which is four times the original price before rounding. Common mistake avoided: multiplying the face value but leaving the annual coupon at $60 would not represent a bond with the same 6% coupon rate.

Worked Example: Scaling a Bond Priced Below Face Value

A hypothetical annual-coupon bond has face value $1,000, a 5% coupon rate, and two years remaining. The required annual yield is 10%. Find the price of a bond with face value $1,500 that has the same coupon rate, maturity, and required yield.

Method. Price the original bond by discounting its payments, then use \(k=1.5\). Its annual coupon is \(0.05(1{,}000)=\$50\), and the second-year payment is \(\$1{,}050\).

$$ \begin{aligned} P &=\frac{50}{1.10}+\frac{1{,}050}{1.10^2}\\ &=45.454545455+867.768595041\\ &=913.223140496\ldots \end{aligned} $$

The original price is \(\$913.22\) to the nearest cent. The larger bond has coupon payments of \(0.05(1{,}500)=\$75\) and a final payment of \(\$1{,}575\). Its price is:

$$ \begin{aligned} P_{\text{scaled}} &=1.5(913.223140496\ldots)\\ &=1{,}369.834710744\ldots\\ &=\$1{,}369.83\text{ to the nearest cent.} \end{aligned} $$

The price remains below face value: the original price is below $1,000, and the scaled price is below $1,500. Direct discounting confirms the scaled result: \(75/1.10+1{,}575/1.10^2=68.181818182+1{,}301.652892562=1{,}369.834710744\). Common mistake avoided: the face-value factor is \(1.5\), not the difference between the new and old face values. The new face value is 150% of the old one, so all payments and price are multiplied by 1.5.

Worked Example: Scaling Quarterly Payments

A hypothetical bond has face value $1,000 and two quarterly periods remaining. Its stated annual coupon rate is 8%, with coupons paid quarterly. The required yield is a nominal annual rate of 8% convertible quarterly. Find the original price and the price when every contractual payment is scaled by 3.5.

Method. Convert both annual rates to the quarterly amounts used by the cash flows. The coupon each quarter is \(1{,}000(0.08)/4=\$20\). The yield per quarter is \(0.08/4=0.02\), or 2%. Cash flows are $20 at the end of the first quarter and $1,020 at the end of the second quarter.

$$ \begin{aligned} P &=\frac{20}{1.02}+\frac{1{,}020}{1.02^2}\\ &=19.607843137+980.392156863\\ &=1{,}000 \end{aligned} $$

The original bond is priced at $1,000. Under the factor \(k=3.5\), the face value becomes $3,500, the quarterly coupon becomes $70, and the final payment becomes $3,570. Discounting those payments:

$$ \begin{aligned} P_{\text{scaled}} &=\frac{70}{1.02}+\frac{3{,}570}{1.02^2}\\ &=68.627450980+3{,}431.372549020\\ &=3{,}500. \end{aligned} $$

The scaled price is $3,500, also obtained as \(3.5(1{,}000)\). Common mistake avoided: the quarterly coupon is one-fourth of the annual coupon amount, not the full annual amount. The yield must also be expressed per quarter before discounting quarterly payments.

Conditions and Limits of the Rule

The scaling result is simple, but it is only valid when the financial comparison is genuinely proportional. Check the following before applying it:

  • Every contractual payment scales. The coupon dollars and redemption amount must all be multiplied by the same factor as face value. If the coupon rate is held fixed, coupon dollars scale automatically with face value.
  • The required yield per payment period is unchanged. A different yield changes the discount factors, so price is not obtained just by multiplying the old price by the face-value factor.
  • Payment timing and remaining maturity are unchanged. A different coupon frequency or a different number of remaining payment periods changes the present-value calculation.
  • Use the same price basis. The rule concerns the present value of the bond’s cash flows. A separately added fixed transaction charge does not scale with face value.

For example, if a bond’s face value doubles but its coupon dollars remain fixed, the coupon portion of the cash flows has not doubled. The new bond is not simply a twice-sized version of the original, so the scaling rule cannot be used. Similarly, even if all contractual payments double, changing the required yield changes the present value of each payment by a different discount factor.

Scaling is also useful for checking a calculation. If a bond’s face value and every payment have been multiplied by 2.4 but the stated yield and payment dates are unchanged, a calculated price that is not 2.4 times the original price signals an arithmetic or input inconsistency. This is a proportionality check, not a replacement for using the correct cash flows and yield convention.

Key takeaway: Under unchanged payment timing and required yield, multiplying face value and every coupon and redemption payment by \(k\) multiplies the bond’s price by exactly \(k\). Verify that the cash flows—not just the face amount—are scaled consistently.

Check Your Understanding

For each problem, identify the scaling factor and keep the yield and payment periods consistent.

  1. A two-year annual-coupon bond has face value $800, coupon rate 6%, and required annual yield 6%. What is its price? What is the price of an otherwise identical bond with face value $1,800? (Answer: $800 and $1,800.)
  2. A hypothetical one-year annual-coupon bond has face value $2,000, coupon rate 4%, and required annual yield 5%. Its face value and all payments are scaled by 0.5. Find the scaled price, to the nearest cent. (Answer: $981.41.)
  3. A hypothetical one-year annual-coupon bond has face value $1,000, coupon rate 8%, and required annual yield 4%. If face value and all payments are scaled by 0.8, find the new price to the nearest cent. (Answer: $860.36.)
  4. A hypothetical bond has face value $1,000, coupon rate 12%, four quarterly periods remaining, and a nominal annual required yield of 12% convertible quarterly. What is the price if face value and all payments are scaled by 1.2? (Answer: $1,200.)
  5. A bond’s cash flows have a present value of $2,500 at a stated required yield. Its face value and all contractual payments are tripled, and the required yield is unchanged. A fixed transaction fee of $15 is then added to the purchase outlay. What are the scaled bond value and total outlay? (Answer: Bond value $7,500; total outlay $7,515.)

Previous

Next

Back to Tutorials
Clinical Biostats