# Understanding the Greeks

April 26, 2021

9:39 am

For our final discussion in this series, we hope to demystify the Greek letters that are key to understanding convertibles. Don’t worry, we will take it easy on the math, and there won’t be a test!

In our earlier discussions, we covered how the Black-Scholes option pricing formula revolutionised finance. The formula uses only a few inputs but is robust enough to generate consistent and predictable results. Based on this formula, investors can calculate sensitivity analysis to the price of their option.

We follow the practice of using Greek letters to stand for a variable, basically, how sensitive an option, or a convertible bond, would be to those key inputs. The Greeks can therefore be used to help assess both risk as well as potential reward.

Of course, these measures can also be calculated for a portfolio of convertible bonds, as well as for a single holding. There is some more math involved to get to the precise definition of each measure, but we will try to present a more intuitive view of what each of the Greeks means for investors.

Our review will cover:

• What are Delta and Gamma?

• What are Rho and Omega?

• Are there other Greeks?

• Why can convertibles have a negative yield to maturity or put?
##### What are Delta and Gamma

For most investors, understanding these two letters is about as much Greek as you need to know to understand convertibles. That is because they capture the sensitivity of a convertible to the price of the stock into which the bond converts, which usually drives most of the price change of a convertible itself.

You may also remember from our earlier discussions that we prefer mid-delta, or “balanced” convertible bonds with high gamma. That is because these convertibles usually offer the best ratio of upside capture versus downside risk, using a scenario where the underlying stock price moves both up and down an equal percentage change.

Let’s look at what each of these Greek letters means:

##### Delta

Expressed as a percentage from 0% to 100%, delta captures the change in the price of the convertible bond for a percentage move in the underlying stock price. A higher delta means a higher sensitivity, whereas a lower delta convertible bond will not move much in price as the underlying stock price changes.

With that said, delta is not static, and it will change as the underlying stock price changes. It is accurate for small changes in the underlying stock price, but you need the next Greek (gamma) to work out the price change for a large move.

##### Gamma

This measures the rate of change of delta itself. Basically, for a long-only convertible investor, as a stock price moves higher, a higher gamma convertible bond should see its own delta increase quickly. This means that the bond can earn more from potential equity upside than a lower gamma instrument might.

Also, gamma can serve as a proxy for convexity, which is the ratio of upside vs. downside for a convertible bond if the underlying stock price were to move up and down by a similar percentage. For a long-only investor, more convex convertibles are generally preferable to those that are not. Gamma is shown as a whole number. It is enough to know that for potential convexity, a gamma of more than 0.75 would be considered good, and more than 1.0 would be quite good.

##### What are Rho and Omega?

Rho and Omega help us to understand the price sensitivity of a convertible bond to interest rate risk (duration) and the credit risk of an issuer.

Convertibles have an embedded option, but they are also bonds. And while the price of an option changes (usually only slightly) when interest rates move, the calculation in the Black-Scholes formula only needs a risk-free rate and doesn’t factor in expected credit risk of an issuer.

Here is how these Greeks help us to understand bond-specific sensitivities:

##### Rho

Interest rate changes affect the price of both the bond component and the option component. Rho shows the total sensitivity of a convertible if all interest rates were to shift equally, that is, by the same amount across the yield curve.

Generally, higher interest rates have the greatest impact to a convertible in driving down the value of the bond component, but with lower interest rates pushing up that value. Higher duration bonds are affected to a greater degree by changes in interest rates vs. short duration bonds. The longer you must wait until maturity, the higher the duration of your bond.

Rho is shown from -1 to 1, which represents its sensitivity/participation rate to shifts in the yield curve. Rho itself shows the change in convertible price for a 1bpsparallel shift on the yield curve.

##### Omega

As a credit spread of an issuer changes, the value of the bond component will change. This measure shows the effect on a convertible’s price if credit spread alone were to change by a standard amount. Omega can also be referred to as Omicron.

Omega is shown from -1 to 1, which represents its sensitivity/participation rate to moves in credit spreads. Omega is the change in convertible price for a 1 basis point move in credit spread.

##### Are there other Greeks?

There are, and they are good to know, but less critical to understand. The majority of price moves for a long-only holding in a convertible bond will be explained via delta and gamma, as well as rho and omega.

##### Vega

Remember that the option to convert into equity will rise in value as volatility increases, but it will fall if volatility were to decrease. Vega shows the forecast change in the convertible price as volatility changes by a standard amount. Higher vega convertibles are more sensitive to changes in volatility than a lower vega convertibles.

Vega is the change in convertible price for a 1% move in volatility.

##### Theta

Imagine if the value of the option to convert was divided among the number of days remaining in its life. If there was a trading day where the underlying stock closed below the strike price, you wouldn’t choose to convert, and your option therefore lost that day’s value.

Theta is a standardised measure that if negative shows the daily loss in value to the convertible bond and is also referred to as time decay.

Generally, all convertibles with a call option have a negative theta. Theta changes as the other Greeks change, and as the time to maturity draws nearer.

Theta is shown from -1 to 1, which represents sensitivity/participation rate to time decay. Theta is shown as the change in convertible price per day based on time decay.

There are two additional Greeks that show sensitivity to changes in the underlying stock dividend (phi) and recovery rate on the bond component (upsilon). They matter, but are less crucial to know than the other Greeks that we have reviewed.

##### Why do some convertibles have a negative yield?

It used to be that negative yields were uncommon for bonds. But while having a negative yield is less of a novelty for a convertible bond, let’s look at why this happens, and what the implications may (or may not be) for investors.

In general, bond investors receive coupons with a final repayment of principal. The final return from a bond depends on when they get their principal back from an issuer, and bonds have different features that can impact the timing and value of repayment. For example, not all bond investments are held to maturity, especially if an issuer or an investor can force an early repayment of principal through a call or put option.

But for convertible bond investors, there is another difference to consider, and that is because holders have the right to convert a bond into shares of stock. Because of that conversion option, the value of the bond if converted into stock can be higher or lower than the par amount of the bond that the issuer must repay at maturity. It just means that some of the traditional yield measurements for fixed income might not fully reflect the value of that option.

As an example, we may hold a convertible where the underlying stock price has risen in value by a moderate amount, and we still have a positive view on that underlying stock. If the yield to maturity or put of the convertible has become negative, does that mean that our investment is now going to earn a negative return? Do we need to convert the bond to lock in our gains, which is what the negative yield seems to suggest?

Looking at the Greeks, if we see that the delta of the convertible remains moderate (say, no more than 0.75) and that gamma of the convertible is still good (above 0.5), we still have a good ratio of reward to risk. So long as we believe that it is likely that the underlying stock stays above a level where it makes more sense to convert than to wait for repayment at maturity, that would still be an attractive convertible, even if the level of yield is negative.

The negative yield simply tells us the received return if we waited for maturity and chose not to convert the bond into stock. There is also no need to choose to convert before maturity – options always have time value! We can always choose to sell our holding in this hypothetical bond before it matures.

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In case you missed it…

The term “RWC” may include any one or more RWC branded entities including RWC Partners Limited and RWC Asset Management LLP, each of which is authorised and regulated by the UK Financial Conduct Authority and, in the case of RWC Asset Management LLP, the US Securities and Exchange Commission; RWC Asset Advisors (US) LLC, which is registered with the US Securities and Exchange Commission; and RWC Singapore (Pte) Limited, which is licensed as a Licensed Fund Management Company by the Monetary Authority of Singapore.

RWC may act as investment manager or adviser, or otherwise provide services, to more than one product pursuing a similar investment strategy or focus to the product detailed in this document. RWC seeks to minimise any conflicts of interest, and endeavours to act at all times in accordance with its legal and regulatory obligations as well as its own policies and codes of conduct.

This document is directed only at professional, institutional, wholesale or qualified investors. The services provided by RWC are available only to such persons. It is not intended for distribution to and should not be relied on by any person who would qualify as a retail or individual investor in any jurisdiction or for distribution to, or use by, any person or entity in any jurisdiction where such distribution or use would be contrary to local law or regulation.

This document has been prepared for general information purposes only and has not been delivered for registration in any jurisdiction nor has its content been reviewed or approved by any regulatory authority in any jurisdiction. The information contained herein does not constitute: (i) a binding legal agreement; (ii) legal, regulatory, tax, accounting or other advice; (iii) an offer, recommendation or solicitation to buy or sell shares in any fund, security, commodity, financial instrument or derivative linked to, or otherwise included in a portfolio managed or advised by RWC; or (iv) an offer to enter into any other transaction whatsoever (each a “Transaction”). No representations and/or warranties are made that the information contained herein is either up to date and/or accurate and is not intended to be used or relied upon by any counterparty, investor or any other third party.

RWC uses information from third party vendors, such as statistical and other data, that it believes to be reliable. However, the accuracy of this data, which may be used to calculate results or otherwise compile data that finds its way over time into RWC research data stored on its systems, is not guaranteed. If such information is not accurate, some of the conclusions reached or statements made may be adversely affected. RWC bears no responsibility for your investment research and/or investment decisions and you should consult your own lawyer, accountant, tax adviser or other professional adviser before entering into any Transaction. Any opinion expressed herein, which may be subjective in nature, may not be shared by all directors, officers, employees, or representatives of RWC and may be subject to change without notice. RWC is not liable for any decisions made or actions or inactions taken by you or others based on the contents of this document and neither RWC nor any of its directors, officers, employees, or representatives (including affiliates) accepts any liability whatsoever for any errors and/or omissions or for any direct, indirect, special, incidental, or consequential loss, damages, or expenses of any kind howsoever arising from the use of, or reliance on, any information contained herein.

Information contained in this document should not be viewed as indicative of future results. Past performance of any Transaction is not indicative of future results. The value of investments can go down as well as up. Certain assumptions and forward looking statements may have been made either for modelling purposes, to simplify the presentation and/or calculation of any projections or estimates contained herein and RWC does not represent that that any such assumptions or statements will reflect actual future events or that all assumptions have been considered or stated. Forward-looking statements are inherently uncertain, and changing factors such as those affecting the markets generally, or those affecting particular industries or issuers, may cause results to differ from those discussed. Accordingly, there can be no assurance that estimated returns or projections will be realised or that actual returns or performance results will not materially differ from those estimated herein. Some of the information contained in this document may be aggregated data of Transactions executed by RWC that has been compiled so as not to identify the underlying Transactions of any particular customer.

The information transmitted is intended only for the person or entity to which it has been given and may contain confidential and/or privileged material. In accepting receipt of the information transmitted you agree that you and/or your affiliates, partners, directors, officers and employees, as applicable, will keep all information strictly confidential. Any review, retransmission, dissemination or other use of, or taking of any action in reliance upon, this information is prohibited. The information contained herein is confidential and is intended for the exclusive use of the intended recipient(s) to which this document has been provided. Any distribution or reproduction of this document is not authorised and is prohibited without the express written consent of RWC or any of its affiliates.

Changes in rates of exchange may cause the value of such investments to fluctuate. An investor may not be able to get back the amount invested and the loss on realisation may be very high and could result in a substantial or complete loss of the investment. In addition, an investor who realises their investment in a RWC-managed fund after a short period may not realise the amount originally invested as a result of charges made on the issue and/or redemption of such investment. The value of such interests for the purposes of purchases may differ from their value for the purpose of redemptions. No representations or warranties of any kind are intended or should be inferred with respect to the economic return from, or the tax consequences of, an investment in a RWC-managed fund. Current tax levels and reliefs may change. Depending on individual circumstances, this may affect investment returns. Nothing in this document constitutes advice on the merits of buying or selling a particular investment. This document expresses no views as to the suitability or appropriateness of the fund or any other investments described herein to the individual circumstances of any recipient.

AIFMD and Distribution in the European Economic Area (“EEA”)

The Alternative Fund Managers Directive (Directive 2011/61/EU) (“AIFMD”) is a regulatory regime which came into full effect in the EEA on 22 July 2014. RWC Asset Management LLP is an Alternative Investment Fund Manager (an “AIFM”) to certain funds managed by it (each an “AIF”). The AIFM is required to make available to investors certain prescribed information prior to their investment in an AIF. The majority of the prescribed information is contained in the latest Offering Document of the AIF. The remainder of the prescribed information is contained in the relevant AIF’s annual report and accounts. All of the information is provided in accordance with the AIFMD.

In relation to each member state of the EEA (each a “Member State”), this document may only be distributed and shares in a RWC fund (“Shares”) may only be offered and placed to the extent that (a) the relevant RWC fund is permitted to be marketed to professional investors in accordance with the AIFMD (as implemented into the local law/regulation of the relevant Member State); or (b) this document may otherwise be lawfully distributed and the Shares may lawfully offered or placed in that Member State (including at the initiative of the investor).

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The representative and paying agent of the RWC-managed funds in Switzerland (the “Representative in Switzerland”) FIRST INDEPENDENT FUND SERVICES LTD, Klausstrasse 33, CH-8008 Zurich. Swiss Paying Agent: Helvetische Bank AG, Seefeldstrasse 215, CH-8008 Zurich. In respect of the units of the RWC-managed funds distributed in Switzerland, the place of performance and jurisdiction is at the registered office of the Representative in Switzerland.