Optional redemption dates: bounds on present value

Introduction: The Mystery of the Redemption Date

In our previous studies on bond pricing, we usually assumed the redemption date was fixed. But what happens if the borrower (the person who issued the bond) has the right to pay you back whenever they want within a certain window of time? This is called a bond with optional redemption dates.

Imagine you lend a friend money, and they say, "I'll pay you back sometime between 5 and 10 years from now." As an actuary, you need to know: what is that bond worth right now? Because the date is uncertain, we can't find a single exact value. Instead, we look for the upper and lower bounds—the maximum and minimum possible present values.

Don't worry if this seems tricky! We use a very simple "rule of thumb" to figure out which date the borrower will choose.

Note: This chapter builds on your knowledge of Bond pricing and yields. If you need a refresher on basic bond formulas, check out the previous chapter in this section.

The Borrower’s Strategy

The most important thing to remember is that the borrower is rational. They want to save money. Therefore, they will choose the redemption date that is most advantageous to them (and, conversely, least advantageous to you, the investor).

To find the bounds on the present value, we test the extreme ends of the redemption window (e.g., the earliest possible date and the latest possible date). We then determine which date results in the lowest price for a given yield.

The "Golden Rule" of Optional Redemption

Whether the borrower wants to redeem early or late depends on whether the bond is being priced at a premium or a discount relative to its redemption value.

• Case 1: Redemption Price \( > \) Net Income (The Bond is at a Discount)
  If the redemption payment is "expensive" for the borrower compared to the interest they are paying you, they will want to keep your money as long as possible. They will choose the latest possible redemption date.
• Case 2: Redemption Price \( < \) Net Income (The Bond is at a Premium)
  If the interest payments (coupons) are costing the borrower more than the final redemption payment would, they will want to stop paying those coupons as soon as they can. They will choose the earliest possible redemption date.

Quick Tip: Think of it like a subscription. If the monthly fee is high, you cancel as soon as possible. If the monthly fee is a bargain, you keep it going for as long as you can!

Calculating the Bounds

In the CM1 syllabus, we specifically look at the present value (\(P\)) for a given effective interest rate (\(i\)). To find the bounds, we usually calculate the price at the two extreme dates: \(n_{min}\) and \(n_{max}\).

The formula for the price of a bond with income tax and capital gains tax (CGT) is generally represented as:

\(P = C(1-t_1)a_{\overline{n}|}^{(p)} + Rv^n - t_2(R - P)v^n\)

Where:
\(C\) = Annual coupon
\(t_1\) = Income tax rate
\(R\) = Redemption value
\(t_2\) = Capital gains tax rate
\(n\) = Term to redemption

Did you know? Even though the formula looks scary, the logic for optional redemption usually boils down to comparing the net coupon rate with the investor's required yield.

How to determine the "Worst Case" Price

1. Calculate the price assuming redemption at the earliest date.
2. Calculate the price assuming redemption at the latest date.
3. The lower bound (minimum price) is the lower of these two values.
4. The upper bound (maximum price) is the higher of these two values.

Worked Example: Finding the Price Bounds

Scenario: An investor wants a yield of \(8\%\) per annum effective. They are looking at a bond with \(6\%\) annual coupons (paid annually in arrears). The bond is redeemable at \(100\%\) at any time between 10 and 15 years. For simplicity, we will ignore taxes in this example.

Step 1: Calculate Price for \(n = 10\) (Earliest)
\(P = 6 \cdot a_{\overline{10}|} + 100 \cdot v^{10}\) at \(8\%\)
\(a_{\overline{10}|} = \frac{1 - (1.08)^{-10}}{0.08} = 6.7101\)
\(P = 6(6.7101) + 100(0.4632) = 40.26 + 46.32 = 86.58\)

Step 2: Calculate Price for \(n = 15\) (Latest)
\(P = 6 \cdot a_{\overline{15}|} + 100 \cdot v^{15}\) at \(8\%\)
\(a_{\overline{15}|} = \frac{1 - (1.08)^{-15}}{0.08} = 8.5595\)
\(P = 6(8.5595) + 100(0.3152) = 51.36 + 31.52 = 82.88\)

Analysis:
The investor's required yield (\(8\%\)) is higher than the coupon rate (\(6\%\)). This means the bond is being sold at a discount (Price \( < 100\)).
As we predicted, the lower bound (minimum price) occurs at the latest date (\(n = 15\)), giving a price of \(82.88\).
The upper bound (maximum price) occurs at the earliest date (\(n = 10\)), giving a price of \(86.58\).

The Impact of Taxes

When Income Tax and Capital Gains Tax (CGT) are involved, the principle remains the same, but you must use the net cashflows.

If the bond is redeemable at a single date within a range at the borrower's option, the borrower will choose the date that results in the lowest present value for the investor. If you are asked for "the price" of such a bond, you should always provide the minimum value to be prudent.

Key Takeaway: The "Yield vs. Coupon" Shortcut

If there is no Capital Gains Tax:

• If Yield \( > \) Net Coupon Rate: Minimum Price is at the Latest date.
• If Yield \( < \) Net Coupon Rate: Minimum Price is at the Earliest date.

Common Mistakes to Avoid

• Assuming the middle date: Never just average the dates. The borrower will pick an extreme, not the middle.
• Forgetting Tax: Ensure you apply income tax to the coupons and CGT to the profit (Redemption Price minus Purchase Price) before discounting.
• Confusing Borrower vs. Investor: Always remember the borrower chooses the date that is cheapest for them, which is the "worst" result for the investor's valuation.