Question

Calculate the implied forward rates on one-year securties using the unbiased expectations theory. Keep the calculated value in one cell and make it as general as possible, i.e. use formulas such as COUNT, SUM, PRODUCT, etc. (Hint: Use COUNT() for the N value.)

??1=1+1???1+1??−1?−1−1 Nf_1=[(1+1R_N )^N/(1+1R_(N-1) )^(N-1) ]-1

Why is the calculated implied forward rate more than the zero-coupon?

	Zero-Coupon	Implied Forward
1-year	0.79%
2-year	1.08%
3-year	1.33%
4-year	1.59%
5-year	1.80%
6-year	1.96%
7-year	2.07%
8-year	2.17%
9-year	2.25%
10-year	2.33%
11-year	2.40%
12-year	2.45%
13-year	2.49%
14-year	2.53%
15-year	2.55%
16-year	2.59%
17-year	2.62%
18-year	2.67%
19-year	2.71%
20-year	2.75%
21-year	2.80%
22-year	2.84%
23-year	2.87%
24-year	2.90%
25-year	2.93%
26-year	2.96%
27-year	2.98%
28-year	3.00%
29-year	3.02%
30-year	3.03%

Answer 1

Information provided in the question:

Zero coupon rates for different tenors (ranging from 1 year to 30 years)

To calculate:

Implied Forward Rates for one-year securities using unbiased expectation theory

Solution:

Unbiased expectation theory states that long-term interest rates and future expectation of short-term interest rates are linked in such a manner that an investment in a long tenor security fetches same return as subsequent investments in a short tenor security. To clarify further, consider the zero coupon rates given in the question for 1 year and 2 year.

1 year Zero coupon rate (r₁) = 0.79%

2 year Zero coupon rate (r₂) = 1.08%

Unbiased expectation theory implies that return on an investment in a 2 year security is going to be same as return on an investment in a 1 year security for 2 years where the amount gets reinvested in the same security after 1 year.

(Return on 1 year security in 1st year) * (Return on 1 year security in 2nd year) = (Return on 2 year security in 2 years)

Let forward implied rate of one year security one year forward be denoted as f₁

(1 + r₁)^1 * (1 + f₁)^1 = (1 + r₂)^2

(1 + 0.79%)^1 * (1 + f₁)^1 = (1 + 1.08%)^2

(1.0079) * (1 + f₁) = (1.0108)^2

(1 + f₁) = (1.0217) / (1.0079)

1 + f₁ = 1.0137

f₁ = 1.37%

Implied forward rate for one year security one year forward is 1.37% which means investment in one year security will fetch 0.79% return in first year and 1.37% return in second year to make it equivalent to 1.08% return of a two year security.

To generalize the formula:

Let forward implied rate of one year security n year forward be denoted as f_n

Let zero coupon rate for n year security be denoted as r_n

f_n = ( (1 + r_n+1) ⁿ⁺¹ / (1 + r_n) ⁿ ) - 1

Using this formula, forward implied rates for different tenors can be calculated as:

Tenor	Zero-Coupon	Implied Forward	Remark
1-year	0.79%	0.79%
2-year	1.08%	1.37%	Implied Forward Rate 1 year forward
3-year	1.33%	1.83%	Implied Forward Rate 2 years forward
4-year	1.59%	2.37%	Implied Forward Rate 3 years forward
5-year	1.80%	2.64%
6-year	1.96%	2.76%
7-year	2.07%	2.73%
8-year	2.17%	2.87%
9-year	2.25%	2.89%
10-year	2.33%	3.05%
11-year	2.40%	3.10%
12-year	2.45%	3.00%
13-year	2.49%	2.97%
14-year	2.53%	3.05%
15-year	2.55%	2.83%
16-year	2.59%	3.19%
17-year	2.62%	3.10%
18-year	2.67%	3.52%
19-year	2.71%	3.43%
20-year	2.75%	3.51%
21-year	2.80%	3.81%
22-year	2.84%	3.68%
23-year	2.87%	3.53%
24-year	2.90%	3.59%
25-year	2.93%	3.65%
26-year	2.96%	3.71%
27-year	2.98%	3.50%
28-year	3.00%	3.54%
29-year	3.02%	3.58%	Implied Forward Rate 28 years forward
30-year	3.03%	3.32%	Implied Forward Rate 29 years forward

As noted above, implied forward rate is higher than zero coupon rate of equivalent maturity. If you will observe the given zero coupon rates, you will notice that zero coupon rate is increasing as the maturity/ tenor increases (i.e. an upward sloping curve where zero coupon rate for 2 year is greater than 1 year and so on).

For such an upward sloping curve, forward implied rate is higher than zero coupon rate as return on subsequent short term securities need to match return on long term securities (unbiased expectation theory). So, forward implied rate is higher to compensate for the low return received in initial years.