Q: bond price change when yield goes to zero

Can bond yield become negative? Yes 2015-2017 many bonds traded at negative yield. https://www.ft.com/content/312f0a8c-0094-11e6-ac98-3c15a1aa2e62 shows a realistic example of a vanilla bond trading at $104. Yield is negative –You pay $104 now and will get $100 repayment so you are guaranteed to lose money.

Mathematically, when yield approaches negative 100 bps, price goes to infinity.

When yield approaches zero, bond price would go to the arithmetic sum of all coupons + repayment.

Mark Hendrick’s YC rules of thumb, clean n simplified

Mark Hendricks’ lecture on Fixed Income introduced a nice, simplified methods of looking at many important mathematical rules of thumb on the yield curve.
We used only zero coupon government bonds without any call feature.
We used log yield, log fwd rate, log spot rate, log return, log price (usually negative) etc. This reduces compounding to addition! There’s no “1+r” factor either.
As much as possible we use one-period (1Y) loans. All the interest rates quoted are based on some hypothetical (but realistic) loan, and the loan period is for one-period, though it can forward-start 3 periods from time of observation. One of the common exceptions — 5Y point on the yield curve is the yield on a bond with 5Y time to live, so this loan period is 5Y, not one-period.
If the shortest unit of measurement is a month, then that’s the one-period, otherwise, 1Y is the one-period. All the rates and yields are annualized.
Perhaps the best illustration is the rule on fwd curve vs YC, on P4.12.

Modified duration^Macaulay duration, briefly again

The nice pivot diagram on http://en.wikipedia.org/wiki/Bond_duration is for Macaulay duration — dollar-weighted average maturity. Zero bond has duration equal to its maturity. (I think many textbooks use this diagram because it’s a good approximation to MD.)

The all-important sensitivity to yield is …. MD i.e. modified duration. Dv01 is related to MD (not Macaulay) — http://bigblog.tanbin.com/2012/05/bond-duration-absolute-1-relative-x.html

MD is the useful measure. It turned out that MD is different from Macaulay duration by a small factor.

quantitative feel of bond duration – mapping absolute 1% -> relative x%

In the simplest illustration of modified duration, if a bond has modified duration == 5 years, then a 100bps yield change translates to 5% dollar price (valuation) change.

Note that 100 bps is an Absolute 1% change in yield, whereas the 5% is a Relative 5% change in valuation. If original valuation == $90 [1], then 100 bps =>> $4.5 change.

After we clear this little confusion, we can look at dv01. Simply set the absolute yield change to 1 bp. The valuation change would be a Relative 0.05% i.e. $0.045. The pattern is

Duration == 5 years => dv01 == 0.05% Relative change
Duration == 6 years => dv01 == 0.06% Relative change
Duration == 7 years => dv01 == 0.07% Relative change

Note 0.05% Relative change means 0.05% times Original price, not Par price.  Original price can be very different from par price, esp. for zero bonds.

[1] 90 in bond price quote means 90% of par value. For simplicity we would assume par is $100, though smallest unit is $1000 in practice.

(See P10 of YieldBook publication on Duration.)

question@Reo pricing engine: effective duration

Hi Jerry,

I recently worked on eq derivative pricing. I realized traders need to know their sensitivities to a lot of variables. That made me start thinking about “your” pricing engine — If a bond trader has 100 open positions, she also need to know her aggregate sensitivity to interest rate (more precisely the yield curve).

To address this sensitivity, I know Reo displays dv01 at position level (and rolls up to account/sector levels), but how about effective duration?

If we do display duration on a real time basis, then is it calculated using dv01 or is there option-adjusted-spread factored in for those callable bond positions?

dv01 ^ duration – software algorithm

Q: Do dv01 and duration present the same level of software complexity? Note most bonds I deal with have embedded options.

I feel answer is no. dv01 is “simulated” with a small (25 bps?) bump to yield… Eff Duration involves complex OAS. See the Yield Book publication on Durations.

In AutoReo, eff duration is computed in a separate risk system — a batch system… No real time update.

By contrast, eq option (FX option probably similar) positions need to have their delta and other sensitivities updated more frequently.

bond duration vs option sensitivity greeks

Just as an option’s market value has sensitivities to underlier price, vol, time to expiration… collectively known as the greeks, a fixed-income portfolio valuation has a much-watched sensitivity to yield, known as Modified-Duration.

Interestingly, The most complex part of bond duration is … option adjusted spread for bonds with embedded options!

A 2nd common type of bond-with-options are caps/floors. Is any Greek widely used for these? Yes since these instruments are mathematically equivalent to calls/puts on an underlying bond.

I feel caps/floors are less popular than swaption, whose market is quite large.

bond duration(n KeyRateDuration) #learning notes 2

Jargon warning: yield is best written in bps/year, like 545bps/year. If you say 5.45% it gets ambiguous in some contexts such as modified duration. “1% rise in yield” could mean 2 things

– 5.45% —-> 6.45%,
– 5.45% –x-> 5.50% is a misunderstanding

This is not academic; this is real. Portfolio sensitivity to yield fluctuations is a key concern of banks on Wall St or Main St. It’s all about x bps change in yield. (From now on, always use bps to describe yield; avoid percentage.)

DV01 is dollar value of a “basis point”, free of any ambiguity.

DV01 and modified duration are 2 of the most widely used bond math numbers. Both are derived from bond cash flow.

Mac duration — definition — weighted average of wait time for the cash flows.
Mac duration — usage — not much in real world trading

Modified duration — definition — Mac duration modified “slightly”, by a tiny factor. REDUCED by (1+r)
Modified duration — usage — more useful than Mac duration. It measures price sensitivity to a yield shift, on a given bond.

For a simple example of a bond with modified duration of 5 years. 100 bps yield change results in a 5% dollar price change.

Key Rate Duration is an natural (and intuitive) extension of the duration concept, useful in MBS etc.

discount curve – cheatsheet

Based on P245 of [[complete guide to capital markets]]

Discount curve is designed to tells us how to discount to NPV $1 received x days from today, where x can be 1 to 365 * 30. If the curve value for Day 365 is 0.80, then SPOT rate is 25% or 2500bps — 80 cents invested today becomes $1. Note the value on the curve is not bps or spot rate, but a discount factor value from 0 to 1.

Q: how do I get forward rate between any 2 dates?
A: P246. Simple formula.

Discount curve is “built” from and must be consistent with
+ ED CD rates (spot rates) of 1,7,14..,90 days. As loans, these loan term always starts today.
+ ED futures rates (forward rates). Loan term always last 3 months, but start on the 4 IMM dates of this year, next year, next next year ….
(Note ED futures rates are determined on the market; ED CD rates are announced by BBA.)

(Note I have ignored IR swaps, which are more liquid than ED futures beyond 10Y tenor.)

Discount curve needs to cover every single date. First couple of months are covered by latest announced ED CD rates, interpolating when necessary. After we pass 90, all subsequent dates (up to 30 years) are covered by ED futures rates observed on CME. Being forward rates, these aren’t directly usable as those CD rates, but still very simple — If the 3-month forward rate 3/19/2008 – 6/19/2008 is 200bps, and discount factor for 3/19 is 0.9, then 6/19 discount factor is (0.9 / 1.02)


For a given bond with a pre-determined series of payouts, you discount every payout with the same YTM to get the PV. Sum up the present values and you get the price — From YTM, get price. Therefore, given the pre-determined payouts, from the price, you can derive YTM numerically.

For a given bond, the higher you set its YTM, the deeper the discount, the lower the PV and price.

Q: how do people compare bonds with different coupons and maturities?
A: YTM. prices aren’t comparable. Therefore, YTM is a way to *characterize* a bond’s price, coupon rate and maturity.

Q: for a given bond, how is price/YTM determined by the market.
A: a bond trader set price (or YTM) on his bond. A buyer probably bid at another price. Offer is lifted when they match.

Among AA bonds for example, the higher the YTM, the more worthwhile(?) is this investment? I don’t think so. If it’s such a bargain, then the offer would be grabbed right away. Trader is forced to set the YTM so high (and price so low) perhaps because maturity is in the *distant* future.

YTM is not closely related to ROI. For a beginner, I would say it’s nothing to do with return. YTM is a *discount-rate*. Across all bonds, the higher this rate, the deeper the discount. I feel YTM is mostly influenced by credit rating and also maturity. I don’t think it’s influenced by coupon rate — Everything else being equal[Q1], a low coupon bond is priced below a high coupon bond. But I guess identical YTM.

Everything else being equal[Q2], a CCC bond is priced below a AAA bond. Therefore, that CCC’s YTM is much higher than that AAA bond. Sellers have to set the YTM high to attract buyers. More precisely, sellers must discount the CCC’s payouts more than the AAA’s payouts.

If a trader increases his bond’s YTM, he is applying deeper discount to lower his asking price.

In reality, premium bonds are discounted deeply compared to par bonds. Remember discount rate (ie yield) is chosen by sellers and buyers. Compared to a comparable par bond (comparable rating), a premium bond has higher price, slightly higher yield and higher coupon.

I don’t think FI developers need this level of familiarity, but If you need to get thoroughly comfortable with basic yield concepts, then master the reasoning behind scenarios below.

Q: For a single bond (ie same coupon rate), what does it mean when price drops?
A: Trader is discounting the payouts more deeply, so yield rises.

Q1: 2 bonds of same issurer and maturity but different coupon rates. Yield should match. what about price?
A: price probably follows coupon rate

Q: 2 bonds of same issurer and same price. What about yield and coupon rate?
A: yeild reflects credit rating so should match. price probably follows coupon rate, so coupon should match too. All 3 attributes should show no difference.

Q2: 2 bonds of different issuer but same maturity selling at the same price. What can you say about yields and coupon rates.
A: the higher coupon is discounted deeply to give the same NPV ie price.

foundation — all rates are quoted as semi-annual compound

This is a basic concept worth repeating. The common basis/assumptions allow fair comparison of rates.

A 2 year spot rate of 10% means

“take my $1m today. in 2 years pay me (1+10%/2)(1+10%/2)(1+10%/2)(1+10%/2)*$1m”

A 6-month rate of 10% 2 years forward means

“in 2 years take my $1m. 6 months later pay me (1+10%/2)*$1m “

a 10% yield means

“This bond makes $100 coupon payments every 6 months from now. If you discount the payout 1.5 years away by (1+10%/2)(1+10%/2)(1+10%/2), and apply the same formula, and add up the present values, then you will get the price i’m proposing to you.”

fundamental choices — roll every6month^long term coupon bond

To understand forward rates (and to characterize market sentiments on a given day), it’s always useful to compare the 2 basic choices:

Investor Sam (short term investor) invests his $1m in a 6-month zero, and reinvest 6 months later, rolling every 6 months. If I expect rates to rise, i follow Sam.

Investor Leo (long term investor) invests his $1m in a 30-year bond at 4% (like Singapore’s CPF special account). If I expect interest to stay below 4%, I follow Leo.

Now let’s look at the market sentiment over the next x years. From today’s bond prices you can derive discount factors and forward rates. Leo is locking in all of those forward rates for 30 years.

yield^spot rate

(this is less precise than the other posts….)

both rate apply the same 500bps/6month over years

> spot rate covers an entire market (market for cash loan?)
< yield is quoted for a particular bond. More generally, it can describe many bonds of the same credit quality and same tenor, but with different coupon rates.

> spot rate is derived from prices and is a faithful observation/characterization of market sentiment on a given day.
< yield is wishful thinking — hoping interest rate will remain unchanged

yield of a fairly priced 2-year strip should equal SR2 ie today’s sport rate for a 2-year loan.

yield — simplified discount factor

* the concept of yield is a simplification of the precise model of discount factor.

* discount factors for 1 year vs 2 years (DF1, DF2) may look unreasonable. To be precise, you can’t mathematically derive one from the other. For example, Investors should not assume interest rate and inflation will stay constant.

* The simplistic concept of bond yield ignores this strict observation of market reality, by assuming interest rate to remain constant. Under that idealized assumption, DF1*DF1 == DF2.

* I call it wishful thinking. Interest rate is determined by supply/demand and will change.

foundation — discount factor

* discount factor (say 0.8) for 5-year is used to discount any income received on the day 5 years from now. $100 received then is valued at $80 as “present value”. Each periodic receipt from a bond must be discounted by the discount factor for the term. Since this is crucial to the yield concept, let’s be specific.
** payout[1.5] (ie payout in 1.5 years) is discounted by discount factor [1.5] or DF1.5
** payout[2] (ie payout in 2 years) is discounted by discount factor [2] or DF2

* 1-year discount factor is close to 1. The longer the time span, the lower the discount factor, the deeper the discount.

* discount factor is derived from real prices, so reflects supply and demand, and reflects how much investors are/were willing to pay.

* discount factor is primarily a function of “distance” from today (ie wait-period), and not a function of asset class. If you like, the asset in question is liquid cash.

* Discount factor measures “how willing investors are to save”. In high interest/inflation years, discount is deep and heavy. discount factor is lower. investors aren’t willing to put money under their pillow since it devalues too fast.