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Spot Rates

Spot prices are a basic building block in finance, but they are tricky when the commodity is money. When the commodity is money, spot prices are called spot rates (a.k.a., spot interest rate). A spot price is simply the market’s current price to buy or sell a commodity for immediate delivery. Spot prices are so common we tend to omit the “spot.” Today, the spot price of gasoline is about $2.29 (see http://www.eia.gov/todayinenergy/prices.cfm) and we are likely to just say “the price of gasoline is $2.29.” That’s if you want to buy a gallon of gas for immediate delivery.

What if you want to buy a gallon of gas in one year but you want to commit to the price today? That’s a forward price and the one-year forward price of gasoline is about $2.08 (see http://www.cmegroup.com/trading/energy/refined-products/rbob-gasoline.html). We don’t typically expect the forward price to be less than the spot price, although it happens in energy often, and it’s called backwardation (as opposed to contango which refers to an upward-sloping forward curve and seems more natural). Because these prices are for a certain standardized type of gasoline that trades on an exchange, the forward price is a futures price. So, the basic difference is between the spot price (i.e., for immediate delivery) and the forward price (i.e., for future delivery but at a price that is immediately determined).

### A Simple Example Starting With Swap Rates

So the basic difference is between the spot and forward price of a commodity; and if the commodity is money, the spot price is called the spot rate. Let’s look at an example (you can access my spreadsheet here). The only inputs, in yellow, are the face (par) value and a swap rate curve which is unrealistically but conveniently linear and steep. Notice the timeline is given in six month increments, so we have a six-month swap rate of 1.00%, a one-year swap rate of 2.00%, a 1.5-year swap rate of 3.00% and a two-year swap rate of 4.00%:

Please keep in mind that my example assumes semi-annual compound frequency. The swap rates are assumed. Swap rates are par rates, and there is a different par rate for each maturity. A par rate is the coupon rate, at the given maturity, that prices the bond to par. Say again?! Let’s illustrate. Notice the two-year par rate is a round 4.00%. This implies the following: if we have a two-year bond with a semi-annual coupon rate of 4.0%, and if we discount each of its four cash flows at their respective spot rates, the sum of these present values will equal the par value of $100.00. This bond has four cash flows and they each get discounted at different rates: they are discounted according to the spot rate which applies at each maturity. So, the swap rates are not the spot rates! Every coupon of this bond is the same (i.e., one half of 4.0% swap rate, or $2.00) but each spot rate is different.

### Starting at Six Months

The first time horizon of six months (0.5 years), above, is easy. The swap, spot and forward rates are each 1.00%. Let’s focus on the spot rate and its associated discount factor. The discount function is the series of discount factors (shown in green above). The discount factor and the spot rate are directly related. If the six-month swap rate is 1.0%, then the future cash flow is $100.50 which is the $100 par redeemed plus one-half of the 1.0% coupon. As 1.0% is a par rate, the bond must price to par. Now we can see how the discount factor is computed: the market will pay $100.00 today and expects $100.50 in six months (again, this is the definition of the 1.0% swap rate). As Tuckman explains, “The *discount factor* for a particular term gives the value today, or the *present value* of one unit of currency to be received at the end of that term. Denote the discount factor for t years by d(t). Then, for example, if d(0.5) equals 0.99925, the present value of $1.00 to be received in six months is 99.925 cents. Another security, which pays $1,050,000 in six months, would have a present value of 0.99925 × $1,050,000 or $1,049,213.”

In our example, therefore, the six-month discount factor is $100.00/$100.50 = 0.99502. The spot rate is a direct function of the discount factor; under semi-annual compounding, the discount factor, d(t) = 1/[1+r(t)/2]^(2t) where r(t) is the spot rate. Conversely, the spot rate r(t) = 2*[(1/d(t))^[1/(2t) – 1].

** Let’s just “test” these:**

- If we multiply the future cash flow by its discount factor, we should get the current price: $100.50*d(0.5) = $100.00. Indeed, $100.50 * 0.99502 = $100.00
- The spot rate should compound to the future cash flow: $100.00 * [1 + r(0.5)] = $100.50. Indeed, $100.00 * (1 + 1.0%/2) = $100.50

As Tuckman writes, “A spot rate is the rate on a spot loan, an agreement in which a lender gives money to the borrower at the time of the agreement to be repaid at some single, specified time in the future.” In this case, a six-month spot rate of 1.0% implies that $100.00 loaned will be repaid in six-months with an interest rate of 1.0% per annum with semi-annual compounding; i.e., in six months, the $100.00 will be returned plus $0.50 in interest. The spot rate is the fundamental building block because it is the appropriate rate for discounting the cash flow. The discount factor is a direct translation of the spot rate, but it has an additional feature: it incorporates the compound frequency. This is huge in practice. There is a saying in quantitative finance, “discount factors don’t lie.” A spot rate of 1.0% per annum varies depending on the compound frequency (do you see why?), but a discount factor is a simple multiplier that always applies.

### Now Let’s Look at the One-Year Rates

Now we can apply our understanding to see why the one-year swap rate is not exactly equal to the one-year spot rate. Recall we have assumed a one-year swap rate of 2.0%. What does this mean? It means that a one-year bond with a 2.0% semi-annual coupon will price exactly to par. Such a bond has two cash flows: a $1.0 coupon in six months and $101.00 in one year. We already know the six month discount factor, d(0.5).

** The definition of swap rate implies the following:**

$100.00 = $1.00*0.99502 + $101.00*d(1.0), such that d(1.0) = [$100 – ($1.00*0.99502)]/$101.00 = 0.98025

**And now we can retrieve the one-year spot rate from the one-year discount factor (or translate, really):**

r(t) = 2*[(1/d(t))^[1/(2t) – 1] = 2*[(1/0.98025)^[1/2 – 1] = 2.0050%; i.e., slightly higher than the swap rate!

What is the meaning of this one-year spot rate of 2.0050%? This is the market’s interest rate for a one-year loan. The 2.0% swap rate implies a one-year bond with a 2.0% coupon. Its first cash flow (the $1.0 coupon) is discounted with the six-month spot rate; but its final cash flow (of $101.00) will be received in one year and is therefore discounted at the one-year spot rate of 2.0050%.

### Summary

The spot rate is the interest rate charged to repay a loan in one single payment. The correct comparison is the interest rate on a zero-coupon bond.

If you can purchase a 10-year zero-coupon bond with a face value of $100.00 today for $70.00, then the implied semi-annual spot rate is about 3.60% = RATE(20, 0, -70, 100)* 2 or [(100/70)^(1/20) – 1]*2. And we can test to see that $70.00*(1+0.0360/2)^(10*2) = $100.00.

It’s not quite enough to say here that “the spot rate is 3.60%” because we need to specify the compound frequency, so we should say “the semi-annual spot rate is 3.60%.” We can further translate this spot rate into its discount factor: d(10) = 1/[1+r(t)/2]^(2t) = 1/[1+0.0360/2]^(2*10) = 0.70. And, indeed, $100.00*d(10) = $70.00.

This discount factor is great because it already accounts for compound frequency. So it’s enough here to say “the 10-year discount factor is 0.70” because it corresponds to the semi-annual spot rate of 3.60%.