Master Chapter One

Time Value of Money

Understood literally this means the value of a dollar erodes because of inflation and with passage of time. Value of a dollar in the future is less than the value of a dollar today if we are in a positive interest rate (motivated by inflation) environment. To clarify further – nominal interest rates above inflation i.e. positive real yields would add to the purchasing power of money (assuming it’s invested) but negative real yields would erode it which is when consuming the dollar today is better than investing it.

Future value (FV) of a dollar is a function of the time period and the expected return over that period. It’s this expected return that we refer to as the discount rate to calculate the equivalent present value (PV) of the future cash flows.

Expected return can take the form of simple or a compounded interest rate. I’ll skip any discussion on simple interest and focus instead on the power of compounding. Intuitively interest on interest is a reflection of a compounded return, which can be mathematically expressed as:

F V = P V \times {(1 + R_{t})}^{t}

Where R_t = Annual Rate of Interest, t = time period in years

The above formula when used to calculate PV of a future cash flow,

P V = \frac{F V}{{(1 + R_{t})}^{t}}

And the term

\frac{1}{{(1 + R_{t})}^{t}}

is referred to as the discount factor which when multiplied by the respective future cash flow will give the present value of the same.

For higher frequency of compounding the formula above will be amended to:

F V = P V \times {1 + \frac{R_{t}}{f}}^{(t \times f)}

Where f is the frequency of compounding i.e. f = 2 for semi-annual compounding, f= 4 for quarterly compounding, f = 12 for monthly compounding, f = 365 for daily compounding.

Continuous compounding

In today’s markets we have seen increasing use of daily compounded rates (OIS). which would be depicted by:

F V = P V \times {1 + \frac{R_{o i s}}{365}}^{365} - 1

Where, R_ois is the annual interest rate compounded daily.

Overnight Indexed swap (OIS) markets mimic daily funding rates hence daily compounding of the floating rate is done only for business days and not calendar days. In other words, simple interest accrual is followed for Friday over Monday and or over public holidays.

In the extreme case where we want to compound every instant, we need to solve the above equation as:

F V = \lim_{t \to \infty} {P V \times {[1 ​ + \frac{R_{t}}{t}]}^{t}}

F V = {P V \times e^{R_{t} \times t}}

Concept of YTM or IRR

The popular “internal rate of return” (IRR) term is used to refer to the singular yield (same as the reinvestment yield) for calculating returns on multiple cash flows. Note that IRR and Yield to Maturity are interchangeable for defining the expected return on investments.

Present Value of cash flows (assuming annual compounding) is denoted as:

P V = \frac{C}{{(1 + I R R)}^{t_{1}}} + \frac{C}{{(1 + I R R)}^{t_{2}}} + .... + \frac{(T V + C)}{{(1 + I R R)}^{t_{n}}}

Where,
PV : Present Value, TV : Terminal Value, C : Coupon
t₁,t₂… : intermediate cash flow periods, tₙ : Tenor (Final Cashflow time period),
IRR : Internal Rate of Return

A key assumption for the concept of IRR is that all intermediate cash flows are reinvested at an average rate equal to the IRR itself to generate the same compounded return. As an example, a 5% annual coupon bearing 5y bond with an IRR or YTM of 4% (annually compounded) has a present value of:

P V = \frac{5}{{1 + 4 %}^{}} + \frac{5}{{(1 + 4 %)}^{2}} + \frac{5}{{(1 + 4 %)}^{3}} + \frac{5}{{(1 + 4 %)}^{4}} + \frac{(100 + 5)}{{(1 + 4 %)}^{5}} = 104.452

Par curve

Par curve is the spot interest rate curve for coupon bearing instruments as traded in the market, interest rates on which discount the cash flows to a present value (PV) of 0. You can think of it as the YTM or IRR yields across tenors to signify a term structure of rates.

Zero curve

Zero curve is a theoretical expected yield curve as derived from the par curve trading in the market. It can be understood as a single period compounded return for a specific tenor. As an example consider the following par bond yield curve paying annual coupons for the sake of simplicity:

1y: 6.12% 2y: 6.52% 3y: 6.72% 4y: 6.87%

Now to find zero coupon yields:

- 1y par coupon would itself equal the 1y zero coupon given the annual (hence one time) payment. This would be the expected return for the 1y horizon that can discount the future value to equal the current par value.

- As for the 2y zero rate:
  - - - Calculate the PV of the 1y coupon using 1y zero rate
      - Now calculate the 2y zero rate as the discount rate for the second year such that the sum of PV of coupon received at the end of first year and the final principal + coupon received at the end of second year equals par

- Similarly for 3y zero rate:
  - - Calculate the PVs of 1y and 2y coupon cash flows using the respective zero rates
    - Now calculate 3y Zero rate as the discount rate for the third year such that the sum of PV of coupons received at the end of first and second year and the final principal + coupon received at the end of third year equals par

Mathematically 2y zero rate can be calculated as:

100 = \frac{6.52}{(1 + 6.12 %)} + \frac{(100 + 6.52)}{{(1 + 2 y_{z c})}^{2}}

Solving above for the 2y zero coupon rate (denoted as 2y_zc ) = 6.53%… and so on

This zero rates calculation is popularly referred to as bootstrapping. In fact arriving at breakeven forward rates from zero rates or even par rates is a form of bootstrapping.

Forward Rate Calculation - Zero Arbitrage

For zero arbitrage world assuming annual compounding:

{(1 + R_{2 y})}^{2} = {(1 + R}_{1 y}) \times {(1 + R}_{1 y 1 y})

Where,
R_2y : 2 year rate
R_1y: 1 year Rate
R_1y1y: 1 year fwd 1y rate

Extending the above for a general notation below:

R_{x y} = {\frac{{[1 + R_{x + y}]}^{(x + y)}}{{[1 + R_{x}]}^{x}}}^{(\frac{1}{y})} - 1

Where,

R_xy is defined as the annual compounded rate for y years at the end of starting in x years time
R_x+y is the annual compounded rate for tenor (x+y) years

Day count conventions

It’s the number of days for which interest on a fixed income security accrues and is paid. Importantly this impacts the discount factor for calculating the present value of cash flows as different accrual periods would have different discount rates. There are different day count conventions at play in different markets (Actual/365, Semi Bond 30/360, Actual/Actual …) and one has to be cognizant of what applies to which market when we implement pricing or actual cash flow calculations for the same.

For instance while calculating bond swap spreads in India one has to consider that bonds trade on a semi 30/360 convention while swaps trade on an Actual/365 convention. Therefore, for the month of February accrual cash flow on 1st March will have a 30 day coupon for a bond but only 28 or 29 days (leap year) for the swap leg.

Duration

Duration quite literally is the average time taken to receive the promised cash flows on a financial instrument. We say financial instrument and not merely a fixed income instrument because in theory anything with future cash flows is sensitive to interest rates (time value of money) and therefore would have duration risk in it. Three main drivers of duration are: a) tenor of investment b) coupon rate c) nominal yields (YTM).

For instance assuming the same tenor of 5 years:

a. A zero coupon bond that has no intermediate cash flows would take exactly the tenor of the bond i.e. 5 years to realise the instrument’s cash flows

b. A 5% coupon-bearing bond with intermediate coupon earnings would have a shorter time period/duration to realise all cash flows

c. And a 7% coupon-bearing bond would have a still shorter time period and hence the lowest duration among all

Duration as described above can be mathematically denoted as below:

\frac{Σ_{i = 1 t o n} {T_{i} \times \frac{C_{i}}{{(1 + y)}^{T_{i}}}}}{B}

, where:

- T is the time period in years and Tᵢ = i^th time period

- C denotes the cash flow and Cᵢ is the cash flow at time period i

- y is the annualised yield to maturity (YTM) of the bond payable annually

- B is the current bond price which if you recall is the present value of all bond cash flows discounted at y, i.e.

B = \sum_{i = 1 t o n} {\frac{C_{i}}{{(1 + y)}^{T_{i}}}}

For the sake of simplicity, we have assumed annual compounding above, but in case of coupon payment frequency of more than once a year do remember to adjust the YTM for that frequency. For instance an 8% annual coupon payable quarterly should have

{(\frac{1 + 8%}{4})}^{0.25}

as the discount factor for the first quarter and so on.

You can think of the formula above as the weighted average time period for receiving all cash flows where the time period T_i is weighted by the proportion of the bond’s total present value received at time Tᵢ. Sum of these weights would naturally be equal to 1. Only in case of a non-zero coupon-bearing bond or an instrument with intermediate cash flows will the duration (measured in years above) be lesser than the tenor/maturity of the bond. This is also called Macaulay’s Duration, named after the Canadian Economist Frederick Macaulay.

As an extension of the above what’s used in practice is the Modified Duration which is a practical representation of the price sensitivity of a bond given a marginal change in its yield i.e.

\frac{Δ B}{Δ y}

.

Mathematically this is explained applying the power and chain rule of differentiation on the bond price notation B to arrive at:

\frac{Δ B}{Δ y} = \sum_{i = 1 t o n} {- T_{i} \times \frac{C_{i}}{{(1 + y)}^{(T_{i} + 1)}}}

= \frac{- 1}{(1 + y)} \times \sum_{i = 1 t o n} {T_{i} \times \frac{C_{i}}{{(1 + y)}^{T_{i}}}}

= \frac{- 1}{(1 + y)} \times {\sum_{i = 1 t o n} \frac{{T_{i} \times \frac{C_{i}}{{(1 + y)}^{T_{i}}}}}{B}} x B

= - \frac{1}{(1 + y)} \times M a c a u l a y' s D u r a t i o n \times B

First term, {\frac{1}{(1 + y)} \times M a c a u l a y ‘ s D u r a t i o n}

on the right-hand side of the equation above is the modified duration carrying the negative sign to reflect the inverse yield vs price relationship.

First term, {\frac{1}{(1 + y)} \times M a c a u l a y ‘ s D u r a t i o n}

on the right-hand side of the equation above is the modified duration carrying the negative sign to reflect the inverse yield vs price relationship.

The expression can be approximated to read as:

D V 01 = - M o d i f i e d d u r a t i o n \times B

Δ B = - \frac{1}{(1 + y)} \times M a c a u l a y' s D u r a t i o n \times B \times Δ y

Where ΔB is the dollar change in the bond’s present value for a marginal change in its yield. To standardize this concept across bond tenors/types we assess present value sensitivity for a 1 basis point change in yield also termed as price value per basis point (PVBP) or dollar value per basis point (DV01). Elaborating further,

Δ B = D V 01 = - M o d i f i e d d u r a t i o n \times M a r k e t v a l u e o f t h e B o n d N o t i o n a l \times 0.01%

For example a long bond position for a face value of USD 5 mio, with a modified duration of 3.55 and market price at 101.75 would have a DV01 (assuming 1bp move up in yield) of:

-3.55 x 101.75% x 5,000,000 x 0.01% = ~ – USD 1806

Or in case of a 1bp move down in yield the dollar value change in the bond price would be ~ +USD 1806.

It follows then higher the duration higher the price sensitivity of the bond.

For dollar based investors trading local currency EM bonds, dollar value per basis point can be calculated from the PVBP of the local currency bond as below:

\frac{P V B P \times L o c a l C u r r e n c y N o t i o n a l}{S p o t F X}

DV01 is sometimes incorrectly confused with duration though latter is a key component in computing the former. The notation and example above clarify that DV01 is the dollar value change in the Bond price in response to a 1bp change in yield while Duration is a percentage metric i.e. change in cents versus a marginal change in yield.

Now let’s get to the second order impact of the change in bond yields on its prices i.e. the change in duration with a marginal change in yield. We already know from the duration formulae above that a higher/lower yield corresponds with a lower/higher duration. From the approximate notation above we can see that:

\frac{Δ B}{B} = \frac{- M a c a u l a y' s D u r a t i o n}{(1 + y)} \times Δ y

Interestingly as yields fall not only does the price of the bond rise but even the percentage change in the bond price is higher given the same movement in yields. This is the same as saying that Duration (both Macaulay and Modified) also rises as yields fall and vice versa (notation above proves it). Conversely when yields rise the bond prices fall but the rate of fall also decreases (as duration decreases) and hence the bond price fall is not as much as the bond price rise for the same rise vs fall in yields. This leads to a convex (or normally referred to as positively convex) behaviour of prices for a long bond position.

Graph 1 - Bond price vs yield relationship to show convexity

Source: Pandemonium.

Table 1 – Duration across different coupon tenor bonds vs IRRs/YTM

IRR	5% 2y	5% 5y	5% 10y	0% 2y	0% 5y	0% 10y	10% 2y	10% 5y	10% 10y
1.0%	1.9	4.5	8.3	2.0	4.9	9.8	1.9	4.3	7.5
2.0%	1.9	4.5	8.1	2.0	4.9	9.7	1.9	4.2	7.3
3.0%	1.9	4.4	8.0	1.9	4.8	9.6	1.9	4.1	7.2
4.0%	1.9	4.4	7.8	1.9	4.8	9.5	1.8	4.1	7.0
5.0%	1.9	4.3	7.7	1.9	4.7	9.4	1.8	4.0	6.9
6.0%	1.8	4.3	7.5	1.9	4.7	9.3	1.8	4.0	6.7
7.0%	1.8	4.2	7.4	1.9	4.7	9.3	1.8	3.9	6.6
8.0%	1.8	4.2	7.2	1.9	4.6	9.2	1.8	3.9	6.4
9.0%	1.8	4.1	7.1	1.8	4.6	9.1	1.8	3.8	6.2

Source: Pandemonium.

I’ve tried to assess several combinations of bond tenors and coupons to exhibit the change in  duration (convexity) at different yields. Segregating them into 3 groups you’ll notice:

1. Group 1 – Same coupon for different tenors – longer duration bond i.e. 10y is more
convex then 5y is more convex than 2y. Hence longer the tenor for the same coupon
more convex it is.

2. Group 2 – Different coupons for same tenors – there’s more duration in lower
coupons but lower convexity in them and vice versa.

3. Group 3 – Higher coupons (consider 10% above) that tend to annuities would be
lowest on duration for any specific tenor but highest on convexity.

Thus intuitively portfolio managers who are uncertain about direction of rates tend to buy higher coupon (more convex) bonds as those would give them more protection in adverse moves. However the trade-off is that when yields fall they would make less money because of its lower duration. If one is convinced that yields would fall then they would tend towards the longest duration bond – aka a zero coupon bond – to maximise gains.

For any given level of rates you would notice that duration of coupon bearing bonds tend to cap out irrespective of how long the tenors as the longer maturity cash flows being discounted by higher exponentials tend to have smaller and smaller impact on the duration value.

For instance in a 5% interest rate regime, a 30y or 50 year coupon bearing bonds would tend to have duration in a fairly tight range peaking around 16-19 years whereas in a 10% regime the similar peak is observed around the 10 year duration mark. Low single digit coupons do create more dispersion of duration even for the longer tenors as – firstly the max duration level is higher than that seen on higher coupons, secondly the incremental exponential impact of lower coupons is larger than the higher coupons which causes more dispersion.

Positive vs Negative Convexity

The regular usage of convexity is to denote positive convexity which comes into play as duration goes lower with higher interest rates and vice versa. All vanilla fixed income instruments whether bonds or swaps are positively convex in that regard. Negative convexity is just the reverse i.e. an increase in duration as interest rates increase. The magnitude of convexity for a fixed income portfolio is the sum of convexity across all its holdings – long/short positions would have positive/negative convexity and portfolio managers would tend to construct portfolios with positive convexity for obvious reasons.

Being exposed to Bonds with an embedded call option in them are a classic example of negative convexity. Holders of mortgage loans (banks) are effectively short an option whereby the borrower could prepay the loan and refinance it at a much lower rate. (i.e. the borrower is long a put option on the higher rate loan) thereby denying the banks to earn a higher interest rate for a longer duration. The loan pre-payment not only reduces duration in the books of banks as rates go lower, it also delivers them a new long-duration exposure at a much lower rate. The opposite would happen when borrowers choose to run the loan for longer as rates go higher, delivering the banks a longer duration exposure in a rising rates environment implying higher mark-to-market losses.

It would be easier to identify a negatively convex situation for a portfolio as being short gamma (that comes with being short an option) and positively convex as being long gamma. We would discuss that in detail in later sections on option greeks.

Duration and PVBP for a Floating Rate Bond

It’s important to cover this base too, as for floating rate bonds if we apply the ‘time taken to receive the promised cash flow’ or Macaulay’s reference to duration it wouldn’t intuitively tie up with the ‘price sensitivity to interest rates’ or reference to Modified Duration. As an example for a 10 year floating rate bond with coupon linked to a 6 month floating benchmark one wouldn’t think that the average time taken for receiving all (coupon + principal) cash flows is 6 months, but the price of the bond would intuitively exhibit sensitivity to interest rates only up to a 6 month period at any given time. This is because impact of the coupon rate on the bond’s present value would be offset by the discount rate (same as the coupon rate). Here’s a mathematical notation for a 3 year (annually compounded) floating rate bond benchmarked on a 1y index:

P V = \frac{P \times R_{01}}{((1 + R_{01}))} + \frac{P \times R_{11}}{((1 + R_{01})) \times ((1 + R_{11}))} + \frac{P \times (R_{21} + P)}{((1 + R_{01}) \times (1 + R_{11}) \times (1 + R_{21}))}

Where,
PV: Present value of floating rate bond

P: Principal Amount or Face Value

Rates in the notation are what’s currently ‘implied’ or ‘projected’ by market:

R₀₁: current projected zero rate for 1y, R₁₁: current projected zero rate for 1y forward 1y, R₂₁: current projected zero rate for 2y forward 1y

Algebraic expansion of the right-hand expression reduces it to principal value P i.e. the present value of a floating rate bond equals its initial principal value or par. The notation above can be generalized for any tenor floating rate bond and the result would be the same.

The result PV = P proves that a floating rate bond would always have a value of Par at the start with no resets having occurred. As soon as the first reset occurs we can see that the numerator of the first term gets fixed but the denominator can continue to change as the underlying interest rate (discount rate) changes with passage of time and the bond starts behaving like a fixed rate bond out to the first reset tenor. By the same corollary, we can also say that a floating rate bond will pull to par in the run-up to the next reset period.

At any point in time therefore a floating rate bond is a package of a short dated fixed rate bonds – starting with the residual tenor of the current reset period in case the underlying index has fixed – with coupons based on the index at the start of the current period and/or the implied forward starting floating rates.

At the start of every reset period then, our example of the 10y floating rate bond would behave like a 6m fixed rate bond exhibiting the same duration and convexity properties. PVBP of this 10-year floating rate bond will be equal to the PVBP of the 6m fixed rate bond with a coupon equal to the current 6m floating rate fix.

Vanilla Cash Products

Let’s begin with the most basic building block of the fixed income cash market that would become the foundation of derivative instruments ahead.

- Money Market Instruments – Are single cash flow simple interest-bearing products, typically up to a 12-month tenor. They could be issued at par or discount, can be negotiable or non-negotiable though economically there isn’t any difference between the two.

- - - Negotiable instruments are those that do not need the issuer’s/originator’s permission to trade onward. The most basic form it can take is a bearer cheque i.e. if one issues a bearer cheque to person A they are free to pass it onward for the receiver to encash it. Other real market examples are a treasury bill, a certificate of deposit, a commercial paper et al.

- - - Examples of non-negotiable instruments i.e. those that cannot be transacted onward without both parties’ consent are bank deposits, inter-bank call/notice money, repo borrowing/lending, trade receivables, bill discounting et al.

- A simple formula for understanding the return on a money market instrument:

F V = P V {1 + r \times (\frac{t}{365})}

Where:

FV: Future Value, PV: Present Value, r: simple rate of interest, t: Tenor

- Coupon Bearing Instruments – these are non-zero interest/coupon bearing financial products with multiple cash flows during the tenor of the product. Examples are sovereign and corporate bonds, medium term notes, or any other interest bearing security. One could also think of Bank Loans, External Commercial Borrowings in the same vein just that these are non-negotiable in nature. Yield on these instruments is understood as the internal rate of return (discussed earlier) or the yield to maturity.

- We have discussed fixed cash flows above, but coupon bearing instruments could also have floating coupons tied to a benchmark index decided at the time of issuance reset at regular intervals for eg. mortgage rates linked to prime lending rate plus a spread, inflation linked notes/bonds, floating rate bonds linked to SOFR.

- Another popular variation are the zero coupon bonds i.e. for the present value formula above the coupon is set at zero with IRR calculated as the compounded rate for the tenor of the bond. While this is a single cash flow instrument the return on it is referenced as its IRR (as it’s typically longer than 1y and a compounded rate) that’s different from a money market instrument return (also zero coupon in nature but simple interest up to 12 month tenor).

- As an example a 5y Zero coupon bond with a maturity value of 100 @ an IRR of 4% has a PV as denoted below:

P V = \frac{100}{{(1.04)}^{5}} = 82.1927

- A coupon bearing bond can be considered as a package of cash flows occurring at different tenors. Each of the component cash flow is effectively a zero coupon bond out to their respective tenor and can be traded / valued independently. In certain markets one could essentially separate the tenor-specific cash flows (coupon or principal) in a coupon bearing bond and trade each of them as independent securities (aka zero coupon bonds) called coupon and principal STRIPS (Separately Traded Registered Interest and Principal Securities). STRIPS typically trade as deep discount bonds, where the independent STRIPS are sold to a diverse investor base who would likely pay a higher price for it than where it trades as part of a package of cash flows.

- One could also in theory reconstitute individual components of the STRIPS and trade them back as a single bond package. In theory, the ability to strip and reconstitute allows for a better price discovery of the different tenor cash flows in a bond.

Long term liability holders (Insurance/ pension funds) prefer zero coupon long dated bonds (still better long dated STRIPS) to lock in the IRR and mitigate the reinvestment risk effectively maximising the duration of their portfolio.

Foreign Exchange - Spot and Forward

Spot denotes the price of currency A in terms of currency B settled typically T+2. Any contract with non-standard settlement (that’s generally not Spot) is termed as an outright forward. Difference between outright forwards and spot denotes the premium/discount on currency A in terms of currency B, termed as FX points. Or the other way as in the real world – FX forwards are generated by adding FX points to Spot. In a zero arbitrage world that adheres to interest rate parity, forward points should be an exact reflection of the ratio of the FX-implied local currency zero rate and the benchmark/foreign currency zero rate. Effectively if one borrowed in one currency (paying that currency’s interest rate for borrowing) and invested the proceeds in the other currency (earning the other currency’s interest rate) then the forward points should neutralise any net gains.

$ K R W_{f o r w o r d (t)} = $ K R W_{s p o t} \times \frac{(1 + R_{K R W} \times \frac{t}{365})}{(​ 1 + R_{U S D} \times \frac{t}{365})}

Where t = tenor of the forward expressed in days,
R_KRW and R_USD are the respective annual interest rates for tenor t

The above trade can be deconstructed into individual cash flows as below (assuming we buy $KRW spot and sell $KRW forward):

- Buy $KRW at spot rate at time t₀

- Lend USD to earn the below in time t

U S D \times (1 + R_{U S D} \times \frac{t}{365})

- Borrow KRW and pay the cost in time t as below

K R W \times (1 + R_{K R W} \times \frac{t}{365})

- Sell USD

\times (1 + R_{U S D} \times \frac{t}{365})

against buy KRW

\times (1 + R_{K R W} \times \frac{t}{365})

which implies selling in the forward leg.

If the $KRW Forward rate as implied by the notional values in the last bullet is different from the $KRW formula denoted above one could arbitrage the above cash flows against the forward rate for a net gain.

The local currency interest rate as derived from the above formula is termed as Implied FX interest rate (different from the actual local currency deposit or risk free rate).

Spot + FX forward transaction is effectively borrowing in one currency against lending in the other currency at their respective interest rates and FX forward points are thus a reflection of the respective interest rate differentials.

In reality however interest rate parity isn’t met due to the following reasons:

1. Limited capacity of counterparties to borrow and/or lend in each of the individual currencies at their respective benchmark rates given balance sheet/credit constraints
2. Bid-offers on borrowing vs lending of currencies
3. Limits on products such as FX forwards imposed by central banks
4. Demand and supply for foreign exchange hedging

Synthetic Borrowing using FX Forwards – An example of how local counterparties use FX Forward market to borrow hard currency is discussed below:

Korean Counterparty does a Buy/Sell $ KRW FX Swap with a bank in Korea. Effectively it has bought USD against KRW for the near date and sold USD against KRW for the far date i.e. USD has been borrowed and KRW has been lent. Large flows of this nature will create a disparity leading to forward points being different (lower in this case) from what should be denoted by the actual yield differential / Interest Rate parity. Countries with large dollar-denominated investments of local asset managers (Korea, Japan, Taiwan typical cases) exhibit this behaviour.

In other words – the return on KRW funds to generate USD is lower than the local KRW interest rate or the cost of borrowing USD using KRW cash is higher than the benchmark USD interest rate.

Deliverable and Non-deliverable forwards

Currencies subject to restrictions on capital account convertibility have forward contracts that are non-deliverable i.e. they do not involve notional exchange and are net-cash settled at the end of the contract. The net cash is typically exchanged in the base/hard currency (eg USD or EUR) depending on the contract.

N e t c a s h s e t t l e m e n t = U S D N o t i o n a l \times \frac{(K - F X_{f i x})}{F X_{f i x}}

K = forward contract strike
FX_fix = FX as of settlement date at the relevant published fixing rate (proxy for spot on maturity of the contract)

Deliverable FX forwards (onshore) and non-deliverable forwards (offshore) are economically (i.e. in terms of P&L) the same though settlement cash flows are different. For instance a deliverable 3 month USDINR outright forward at 85 would make the same P&L (hold the same value) as a non-deliverable forward contract at 85 on maturity as long as the FX_fix is a good proxy for the spot rate at maturity.

Currencies like SGD, HKD and THB trade on a deliverable basis even in the offshore markets.

In Asia NDF markets exist for currencies like INR, IDR, KRW, CNY, TWD, PHP as an offshore expression of the currency view. In addition for China there is a deliverable unit namely CNH meant for offshore usage that closely tracks onshore RMB oftentimes with a steady basis owing to the varying demand for CNH relative to RMB.

Activity in forward markets comprises hedging, receivables, financing, and speculative flows.

Deliverable NDF (Onshore NDF)

In recent times there’s been an emergence of an NDF market onshore (eg. Indonesia) as BI felt the need for exerting more control over its FX markets while catering to the hedging needs of investors. The D-NDF market is typically used to hedge actual onshore investments as compared to offshore NDFs which do not require underlying exposure. Also Indonesia being an indexed market has boasted of sizeable offshore bond ownership, hence the supply of FX forwards (potential hedges) in the D-NDF market has helped contain panic in times of broader EM risk-off.

Again as stated above – from an economic perspective the D-NDF and NDF contracts are economically same and should price at the same level – but there exists supply demand dynamics which can cause differences to emerge between the two.

The D-NDF market typically settles the contract using the local currency while an NDF contract offshore settles in hard currency (eg USD).

Related Resources

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Master Chapter One

Time Value of Money

Continuous compounding

Concept of YTM or IRR

Par curve

Zero curve

Forward Rate Calculation - Zero Arbitrage

Day count conventions

Duration

Graph 1 - Bond price vs yield relationship to show convexity

Positive vs Negative Convexity

Duration and PVBP for a Floating Rate Bond

Vanilla Cash Products

Foreign Exchange - Spot and Forward

Deliverable and Non-deliverable forwards

Deliverable NDF (Onshore NDF)

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Related Resources

1: Time Value of Money & Power of Compounding

2: Bootstrapping of Yield Curves – Par, Zero, Forward

3: Duration & Convexity

4: Duration and PVBP for a Floating Rate Bond

5: Money Market and Cash Bonds

6: Foreign Exchange & Interest Rate Parity

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Varda Pandey